treatiseontheory00watsuoft

=—w =o ==. =Sh ==3,—— ——s1 ra J =o —— — ATREATISE ONTHE THEORY OF BESSEL FUNCTIONS BY Gf"N.'WATSON, Sc.D., F.R.S. PROFESSOR OB'MATHEMATICS INTHEUNIVERSITY OFBIRMINGHAM LATELY FELLOW OFTRINITY COLLEGE, CAMBRIDGE 'Cp/o 1>^ CAMBRIDGE ATTHEUNIVERSITY PRESS 1922 PREFACE THISbook hasbeen designedwithtwoobjectsinview. The first isthe developmentofapplicationsofthefundamentalprocessesofthetheoryof functions ofcomplexvariables. ForthispurposeBessel functions areadmirably adapted;whiletheyoffer atthesame time arather widerscopefortheappli- cation ofpartsofthetheoryoffunctions ofarealvariable than isprovided by trigonometricalfunctions inthetheoryofFourier series. Thesecondobjectisthecompilationofacollection ofresults which would beofvalue totheincreasing number ofMathematicians andPhysicists who encounter Bessel functions inthecourse oftheir researches. Theexistence of suchacollection seems tobedemanded bythegreaterabstruseness ofproperties ofBessel functions(especiallyoffunctions oflarge order) which havebeen requiredinrecentyearsinvariousproblemsofMathematical Physics. Whilemyendeavour hasbeen togiveanaccount ofthetheoryofBessel functions which aPureMathematician wouldregardasfairly complete,Ihave consequentlyalsoendeavoured toinclude allformulae, whethergeneralor special, which, althoughwithout theoretical interest, arelikelytoberequired inpractical applications;andsuch results aregiven,sofaraspossible,ina formappropriateforthesepurposes. Thebreadth ofthese aims, combined with thenecessityforkeepingthesizeofthebook within bounds, hasmade itnecessarytobeasconcise asiscompatiblewithintelligibility. Since thebookis,forthemostpart,adevelopmentofthetheoryoffunc- tions asexpoundedintheCourseofModern Analysis byProfessor Whittaker andmyself,ithasbeen convenient toregardthat treatise asastandard work ofreference forgeneral theorems, rather than torefer thereader tooriginal sources. Itisdesirable todraw attention here tothefunction which Ihaveregarded asthecanonical function ofthesecond kind,namelythefunction which was definedbyWeber andusedsubsequently bySchlafii, by(irafand(Jubk'r and byNielsen. Forhistorical andsentimental reasons itwould havebeenpleasing tohave feltjustifiedinusingHankel's function ofthesecond kind; butthree considerationspreventedthis.The first isthenecessityforstandardizingthe function ofthesecond kind; and, inmyopinion,theauthorityoftliegroup ofmathematicians who useWeber's function hasgreater weightthan the authorityofthemathematicians whouseanyother onefunction ofthesecond kind. Thesecond istheparallelismwhich theuseofWi'ber's function exhibits between thetwokinds ofBessel functions andthetwokinds (cosineandsine) PREFACE igonometricalfunctions. Thethird istheexistence ofthedevice bywhich jrpolationismadepossibleinTables IandIIIattheendofChapter XX, ichseems tomake theuseofWeber's function inevitable innumerical work. Ithasbeenmypolicytogive,inconnexion witheach section, references COanymemoirs ortreatises inwhich theresults ofthesection havebeen previously enunciated; but itisnottobeinferred thatproofs giveninthis book arenecessarilythosegiveninanyofthesources cited. Thebibliography attheendofthebook hasbeenmade ascompleteaspossible, thoughdoubtless omissions willbefound init.While Idonotprofesstohave insertedevery memoir inwhich Bessel functions arementioned, Ihavenotconsciouslyomitted anymemoircontaininganoriginal contribution, howeverslight,tothetheory ofthefunctions; withregardtotherelatedtopicofRiccati'sequation,Ihave been eclectic totheextent ofinserting onlythose memoirs which seemed to berelevant tothegeneralscheme. Inthecase ofananalyticaltreatise such asthis, itisprobablyuseless to hopethatnomistakes, clerical orother, haveremained undetected; butthe munber ofsuchmistakes hasbeenconsiderablydiminishedbythecriticisms findthevigilanceofmycolleagues MrC.T,Preece andMrT.A.Lumsden, /whoselabours toremove errors and obscurities have been ofthegreatest /value. Tothesegentlemenand tothestaff oftheUniversity Press,whohave *given every assistance, withunfailing patience,inawork ofgreat typographical complexity,Ioffermygratefulthanks. G.N.W. August 21,1922. CONTENTS CHAP. I.BESSEL FUNCTIONS BEFORE 1820 II.THEBESSEL COEFFICIENTS III.BESSEL FUNCTIONS IV.DIFFERENTIAL EQUATIONS V.MISCELLANEOUS PROPERTIES OFBESSEL FUNCTIONS VI.INTEGRAL REPRESENTATIONS OFBESSEL FUNCTIONS VII.ASYMPTOTIC EXPANSIONS OFBESSEL FUNCTIONS VIII.BESSEL FUNCTIONS OFLARGE ORDER IX.POLYNOMIALS ASSOCIATED WITH BESSEL FUNCTIONS . X.FUNCTIONS ASSOCIATED WITH BESSEL FUNCTIONS . XI.ADDITION THEOREMS XII.DEFINITE INTEGRALS XIII. INFINITE INTEGRALS XIV.MULTIPLE INTEGRALS XV.THEZEROS OFBESSEL FUNCTIONS XVI.NEUMANN SERIES ANDLOMMEL'S FUNCTIONS OFTWO VARIABLES XVILKAPTEYN SERIES XVIIL SERIES OFFOURIER-BESSEL ANDDIN I . XIX.SCHLOMILCH SERIES XX.THETABULATION OFBESSEL FUNCTIONS TABLES OFBESSEL FUNCTIONS BIBLIOGRAPHY INDEX OFSYMBOLS LISTOFAUTHORS QUOTED GENERAL INDEXPAGE 1 U 38 85 132 160 194 225 271 308 358 373 383 450 4/ / 522 551 576 618 654 665 753 789 791 706 CORRIGENDA Page 62,line 11,for"2^- ,^" (i^)^'""yectc^"2^- ,-^^{hzf-K'm=0 '^ •5/1=0 "^ • Page 91,line5from thefootofthepage, for"Brassiue" read"Brassinne." Page 228, informula(2),/or"98720 sec^^" read"78720 .sec^^." Page 327,line 7from thefootofthepage,/or"Bruhns"read"Bruns." CHAPTER I BESSEL FUNCTIONS BEFOUE 1826 I'l. Riccati'sdifferential equation. ThetheoryofBessel functions isintimatelyconnected with thetheoryof acertaintypeofdifferentialequationofthe first order, known asRiccati's equation.InfactaBessel function isusuallydefined asaparticularsolution ofalinear differentialequationofthesecond order(known asBessel'sequation) which isderived from Riccati'sequation byanelementarytransformation. The earliestappearanceinAnalysisofanequationofRiccati'stypeoccurs inapaper*oncurves which waspublished byJohn Bernoulli in1694. In thispaperBernoulligives,asanexample,anequationofthistypeandstates thathehasnotsolved itf. Invariousletters:J:toLeibniz, written between 1697 and1704, James Bernoulli refers totheequation,which hegivesintheform dy—yydx-'r xxdx, and states, more than once, hisinabilitytosolve it.Thus hewrites(Jan. 27, 1697):"VellemporroexTescirenum etbanc tentaveris dy=yydx+xxdx. Egoinmille formas transmutavi, sedoperam mefvm improbum Problemaper- petuolusit." Fiveyearslaterhesucceeded inreducingtheequationtoalinear equationofthesecond order andwrote§toLeibniz(Nov. 15,1702): "Qua occasione recordoraequationesaliasmemoratae dy=yydx+x'-dx inquanun- quam separare potuiindeterminatas aseinvicem, sicutaequatio maneret simpliciterdifferentialis :sedseparaviillasreducendoaequationemadbanc differentio-differentialem|| ddy.y=—x-dx"." When thisdiscoveryhadbeenmade, itwasasimple steptosolve thelast equationinseries, andsotoobtain thesolution oftheequationofthe first order asthequotientoftwopower-series. *ActaEruditorum puhlicata Lipsiae, 1694, pp.435—437. t"Esto proposita aequatiodifferentialis haecxHx+y'^dx=(i-dyquaeanperseparationem indeterminatarum construi possitnoudum tentavi"(p.436). +SeeLeihnizens gesamellte Werke, Dritte Folge (Matliematik),iii.(Halle, 1855), pp.50—87. §Ibid.p.65.Bernoulli's procedure was, effectively,totakeanewvariable udefined bythe formula 1du_ udx~-^ iutheequation d!/ldx=x^+y",andthen toreplace ubyy. IITheconnexion between thisequation andaspecialform ofBessel's equationwillbeseen in§4-3. W.B.F.^ 2 THEORY OFBESSEL FUNCTIONS[CHAP.I And, infact, thisform ofthesolution wascommunicated toLeibnizby James Bernoulli within ayear (Oct. 3,1703)inthefollowingterms*: "Reduce autemaequationem dy=yydx-\-xxdx adfractionemcujus uterque terminusperseriemexprimitur,ita 1 L pff. _33.4.7 3.4.7.8.11 3.4 .7 .8 .1] .12 .15 3 .4 .7.8.11 .12 .15 .]6.19^~ ,^ X^ X^- .*•!« 1 4- — 4- ._pf,o3.4^3.4.7.8 3.4.7.8.11.12^3.4.7.8.11.12.15.16 quaeseries quidemactuali divisione inunam conflaripossunt,sedinqua ratioprogressionisnontam facilepatescat,scil. ^~ 3"^37377"^3.3.3.7.11"^37373737 57777Tn^ Ofcourse, atthat time, mathematicians concentrated theirenergy,sofar asdifferentialequationswere concerned, onobtainingsolutions infinite terms, andconsequentlyJames Bernoulli seems tohave receivedhardlythefullcredit towhich hisdiscoveryentitled him. Thus, twenty-two years later, thepaper f, inwhich Count Riccati first referred toanequationofthetypewhich now bears hisname, wasfollowed byanote;]: byDaniel Bernoulli inwhich itwas stated thatthesolution oftheequation^ aa-"dx+uudx=hdu wasahitherto unsolvedproblem. Thenoteended withanannouncement in ananagramofthesolution :"Solutioproblematisab 111.Riccatoproposito characteribus occultis involuta 24a, 66,6c, 8rf,33e, bf,2g, 4>h,SSi, 61,21in, 2Qn, 16o,Sp,hq,l7r, 16s,2U,32w, ^x,3y,+,-, ,±,=,4,2,1." Theanagram appearsnever tohavebeen solved;butBernoullipublished hissolution 11oftheproblemabout ayearafter thepublicationoftheanagram. The solution consists ofthedetermination ofasetofvalues ofn,namely—4!m/(2m ±1),wheremisany integei',foranyoneofwhich theequationis soluble infinite terms; thedetails ofthissolution willbegivenin§§4-1,4*11. Theprominence giventothework ofRiccati byDaniel Bernoulli, combined with thefactthat Riccati'sequation wasofaslightlymoregeneral typethan *SeeLcihnizens gesamellte Werke, Dritte Folge (Mathematik),iii.(Halle, 1855), p.75. fActa Eniditorum, Suppl.viii.(1724), pp.66—73.Theform inwhich Kiccati took tlie equation was x'"''dq=du+xiudx:q, where q=.r". XIbid. pp.73—75.Daniel Bernoulli mentioned that solutions hadbeen obtained bythree othermembers ofhisfamily—John, Nicholas andtheyounger Nicholas. §Thereader should observe thatthesubstitution bdzU=--r-zdx givesrisetoanequation which iseasilysoluble inseries. IIExercitationes quaedammathematicae (Venice, 1724), pp.77—80;Acta Eruditonim, 1725, pp.465—473. 1*2] BESSEL FUNCTIONS BEFORE 1826 3 John Bernoulli'sequation*hasresulted inthename ofRiccatibeing associated notonlywith theequation which hediscussed withoutsolving,butalsowith astillmoregeneral typeofequation. Itisnowcustomarytogivethenamef Riccati'sgeneralised equationto anyequationoftheform where P,Q,Maregivenfunctions ofx. Itissupposed thatneitherPnorRisidenticallyzero. IfB=0, theequationislinear; ifP=0, theequationisreducible tothehnear formbytaking 1/?/asanew varialjle. The lastequation wasstudiedbyEuler;]:;itisreducible tothegeneral linearequationofthesecond order, andthisequationissometimes reducible toBessel'sequation byanelementarytransformation(cf.§§3"1, 4"3,4'31). Mention should bemade here oftwomemoirsbyEuler. Inthefirst§it isproved that,when aparticular integral t/iofRiccati'sgeneralised equation isknown, theequationisreducible toalinearequationofthe firstorderby replacing yhy i/i+l/u,and sothegeneralsolution canbeeffectedbytwo quadratures.Itisalsoshewn(ibid. p.59)that, iftwoparticularsolutions are known, theequationcanbeintegrated completely byasingle quadrature; and thisresult isalso tobefound inthesecond ||ofthetwopapers. Abrief dis- cussion ofthese theorems \villbegiveninChapteriv. 1'2.Daniel Bernoulli's rnechanicallirohleni. In1738 Daniel Bernoullipublishedamemoir ITcontainingenunciations of anumber oftheorems ontheoscillations ofheavychains. Theeighth**of these isasfollows:"DeJiguracatenaeuniforniiteroscillantis. SitcatenaAG uniformitergravisetperfecteflexilissuspensadepuncto J.,eaqueoscillationes facere uniformesintelligatur: perv'eneritcatena insitumAMF\ fueritque longitudocatenae =^:longitudo cujuscunque partisFM=x,sumatur nejus valoris+f utfit 1 1 -I —Vetc.=0. n 4/i/i 4.9n3 4.9.16n^ 4.9.1G.25/r' *-SeeJames Bernoulli, Opera Omnia,ii.(Geneva, 1744), pp.1054—1057 ;itisstated that the pointofEiccati's problemisthedetermination ofasolution intinite terms, andasolution which resembles thesolution byDaniel Bernoulli isgiven. tTheterm'Kiccati's equation'wasusedbyD'Alembert, Hist, deVAcad. Ii.desSci.deBerlin, XIX.(1763), [published 1770], p.242. tInstitutiones CalculiIiiteiiraliii,ii.(Petersburg, 1769), §831, pp.88—89. luconnexion with thereduction, seeJames Bernoulli's letter toLeibniz already quoted. §Novi Comvi. Acad. Petrop.viii.(1760—1761), [published 1703], p.82. IIIbid. IX.(1762—1763), [published 1764], pp.163—164. H"Theoremata deoscillationibus coriwrumfile flexili connexorura etcatenae verticaliter suspeusae," Comm. Acad. Sci.Imp. Petrop.vi.(1732—3), [published 1738], pp.108—122. **Luc. cit.p.116. t+Thelength ofthesimple equivalent pendulumisii. 1—2 4 THEORY OFBESSEL FUNCTIONS [CHAP.I Ponatiirporrodistantia extremipunctiFahlinea vertical! =1,dico fore distantiampuncti ubicunque assumptiMabeadem linea verticali aequalem OCXX x^ x'^ x^ „ ~w 4)in 4.9?i3 4.9.16n' 4.9.16.25/1^ Hegoesontosay:"Invenitur brevissimo calculo ?i=proxime0'691 I.... Habet autem littera ninfinitos valores alios." The last series isnowdescribed asaBessel function*oforder zeroand argument2'\/(x}n); andthe lastquotationstates that thisfunction hasan infinite number ofzeros. Bernoullipublished fproofsofhistheorems soon afterwards; intheorem VIII,heobtained theequationofmotion byconsideringtheforcesactingon theportionFM oflengthx.Theequationofmotion was alsoobtained by Eulerj many yearslaterfromaconsideration oftheforcesactingonanelement ofthechain. Thefollowingisthesubstance ofEuler'sinvestigation: Letpbethelinedensityofthechain (supposed uniform) and letTbethetension at height Xabove thelowest pointofthechain initsundisturbedposition. Themotion being transversal, weobtain theequation 8T=gp8x byresolving verticallyforanelement of chain oflength8x.TheintegraloftheequationisT=gp,v. Thehorizontal componentofthetensionis,e&ectiYQly,' T{dyldx) where yisthe(hori- zontal) displacementoftheelement;andsotheequationofmotion is p»^S=<^^) Ifwesubstitute forTandproceedtothelimit, wefindthat df^^dx\ dx)' If/isthelengthofthesimple equivalent pendulumforanyonenormalvibration, we write y=-.n(?)si„(f«^?), whereAandfareconstants;andthenn{x\f)isasolution oftheequation dfdv\ V dx\dx)f Ifxlf=u, weobtain thesolution intheform ofBernoulli'sseries, namely ^= 1- ,+ ^i— .- 1—.—r.^ 11.4 1.4.9 1.4.9.16 *OntheContinent, thefunctions areusually calledcijlinder functions, or,occasionally, /«nc- tions ofFourier -Besisel, after Heine, Journal furMath. lxix.(1868), p.128; seealsoMath. Ann. 111.(1871), pp.609—610. tCovim. Acad. Petrop.vii.(1734—5), [published 1740], pp.162—179. +Acta Acad.Petrop.v.pars1(Mathematiea), (1781), [published 1784], pp.157—177. Euler took theweightoflengtheofthechain tobeE,andhedefined gtobethemeasure ofthe distance(nottwice thedistance)fallen byaparticle from restundergravity inasecond. Euler's notation liasbeen followed inthetextapart from thesignificance ofgandtheintroduction of pand 5(for d). 1*3] BESSEL FUNCTIONS BEFORE 1826 5 —t,whereCand Uareconstants. Sinceyisfinitewhen j;=0,Cmust bezero. Ifaisthewhole lengthofthechain, y=whenx—a,andsotheequation todetermine/'is a a? <fi~ \7f^T:\p~ 1.4. 9r'+ ••=0- Byanextremely ingenious analysis, which willbegiven fullyinChapter xv,Euler proceededtoshew thatthethree smallest roots oftheequation ina//are1-445795, 7-6658 and 18-63. [Moreaccurate values are1-4457965, 7-6178156 and18-7217517.] Inthememoir* immediately followingthisinvestigation Euler obtained thegeneral solution(intheform ofseries)oftheequation^\\i—\-\-v=-0^ buthisstatement ofthe lawofformation ofsuccessive coefhcients isratherincomplete. Thelawofformation had, however, been stated inhisInstitutiones CalculiIntegralis\,ir.(Petersburg, 1769), i^977, pp.233-235. I'S.Euler smechanicaljjvohlem. The vibrations ofastretched membrane wereinvestigated byEuler ;J:in 1764.Hearrived attheequation ldP^_drzIdz 1d'z ?~df-~d?'^r(h''^r-d(l>-' where zisthetransversedisplacementattime tatthepointwhosepolar coordinates are(r, </>);and eisaconstantdependingonthedensityand tension ofthemembrane. Toobtain anormal solution hewrote z=usin(at+A)sin(/5(/)+B), where a,A,^,Bareconstants anduisafunction ofr;andtheresult of substitution ofthisvalue ofzisthedifferentialequation d-aIdu(d- /3-\ dr- rdr \e- r-j Thesolution ofthisequationwhich isfinite attheoriginisgivenonp.256 ofEuler's memoir;itis u-r 11- 2-(,7:nye-^2 .4(«.+1)(n+3)e^'"' f' where nhasbeenwritten^inplaceof2/3+1. This differential equationisnowknown asBesscl'sequationforfunctions oforder /3 ;and^mayhave||anyofthevalues 0,1,2,.... Save foranomitted constant factor theseries isnow called aBessel coefficient oforder^andargument ar\e.Theperiodsofvibration, 2-7r/a,ofa *ActaAcad. Petroj). v.pars1(Mathematica), (1781), [published 1784], pp.178—190. tSeealso §§935,93G(p.187 etseq.)forthesolution ofanassociated eiiuationwhich willbe discussed in§3-52. +Novi Coinm. Acad. Petrojj.x.(17G4), [published 17t)6J, pp.-243— -200. §Thereason whyEulermade thischangeofnotation isnotobvious. IIIf/3were notaninteger,thedisplacemeut would notbeaone-valued function ofposition, inview ofthefactor sin(/30+B). 6 THEORY OFBESSEL FUNCTIONS [CHAP.I circular membrane ofradius awith afixedboundary*aretobedetermined from theconsideration thatuvanishes when r=a. Thisinvestigation byEuler contains theearliestappearanceinAnalysisof aBessel coefficient ofgeneral integralorder. 1'4.TheresearchesofLagrange,Carlini andLaplace. Onlyafewyearsafter Euler hadarrived atthegeneralBessel coefficient inhisresearches onvibrating membranes, thefunctionsreappeared,inan astronomicalproblem.Itwasshewn byLagrange fin1770 that, intheelliptic motion ofaplanetabout thesunatthefocusattracting accordingtothelaw oftheinversesquare,therelations between theradius vectorr,themean anomalyMandtheeccentric anomaly E,which assume theforms M^E -€iimE, r=a{l-€cosE), giverisetotheexpansions (6 y.E^M^ tAn&inuM,-=l+i6^+SjB„cos»ilf, n=\ ^ w=l inwhich aand earethesemi-majoraxisandtheeccentricityoftheorbit, and '^'*" ^m=o 2''+''^m\{n +my.'"'""'' ^^^^^ 2"+^""m\(n+m)l^ Lagrange gavetheseexpressionsforw=1,2,3.Theobjectoftheexpansions istoobtainexpressionsfortheeccentricanomalyandtheradius vector in terms ofthetime. Inmodern notation these formulae arewritten ^„=2/„(ne)/n, 5,=-2(e/n)J^(ne). Itwasnoted byPoisson, Connaissance desTerns, 1836[published 1833], p.6that nae amemoir byLefort, Journal deMath. xi.(1846), pp.142—152,inwhich anerrormade by Poisson iscorrected, should alsobeconsulted. Aremarkableinvestigationoftheapproximatevalue ofA.^when nislarge and <e<1isdue toCarlini:|:;thoughtheanalysisisnotrigorous (andit would bedifficult tomake itrigorous)itisofsufficient interest forabrief account ofittobegivenhere. *Cf.Bourget, Ann. Sci. deVEcoIe norm.siij}.in.(1866), pp.55—95,andChree, Quarterly/ Journal, xxi. (1886), p.298. tHist, deVAcad. R.desSci.deBerlin, xxv.(1769), [published 1771], pp.204—233. [Oeuvres,. III.(1869), pp.113—138.] XRiceixhe siiUa courergenza delta serie cheserva alia soliizione delproblemadiKeplero (Milan, 1817).Thiswork was translated intoGerman byJacobi, Astr. Nach. xxx.(1850), col.197—254 {Werke,vii.(1891), pp.189—245]. Seealsotwopapers byScheibner dated 1856, reprinted inMath. Ann, xvii.(1880), pp.531—544, 545—560. 14] BESSEL FUNCTIONS BEFORE 1826 7 Itiseasytoshew thatA^isasolution ofthedifferentialequation Define ubytheformulaJ„=2?i"-ie-^"''7«!''^u*ithen Hence when nislarge either worm^ordu/de must belarge. Ift<=0(?«<')weshouldexpect w^anddujde tobe0(?j2o) and(«»)respectively; and onconsidering thehighest powersofninthevarious terms ofthelastdifferential equation, wefindthat a=1.Itisconsequently assumed thatuadmits ofanexpansionindescending powersofnintheform u=du^,+Ui+v.yjn+..., whereWq,«i, u^.,...areindependentof«. Onsubstitutingthis series inthedifferentialequationofthe firstorder andequatingto zerothecoefficients ofthevarious powersof??,wefindthat where u^^=duJdf;sothat ??o=±-'^ ,?«i=t^— :,,andtherefore e 1—f- [«c7.= M|log-p^-^^,^±^/(l-.^')+l}-ilog(l-e2)+..., and, since thevalue ofAnshews that^iide~nlog^e when eissmall, theupper signmust betaken andnoconstant ofintegrationistobeadded. FromStirling's formula itnowfollows atonce that 6«exp {nj{\-e^)}An~ v/(U).«t(l-e2)i{l+V(l-e2)}"' andthis istheresult obtained byCarlini. Thismethod ofapproximationhasbeen carried much further byAleissel(see §8-11), while Cauchy*hasalsodiscussed approximate formulae forA^inthecase ofcomets movinginnearly parabolicorbits(see §8'42),for which Carlini's approximationisobviously inadequate. Theinvestigationofwhich anaccount hasjustbeengivenismuch more plausiblethan thearguments employed byLaplaceftoestablish thecorre- sponding approximationfori?„. Theinvestigation given byLaplaceisquite rigorousandthemethod which heuses isofconsiderable importancewhen thevalue ofB^ismodified by takingallthecoefficients intheseries tobepositive—or,alternatively, by supposingthat eisapure imaginary.ButLaplace goesontoarguethatan approximationestablished inthecase ofpurely imaginaryvariables maybe used'sans crainte'inthecaseofreal variables. Toanyonewho isacquainted with themodem theoryofasymptotic series, thefallacious character ofsuch reasoningwillbeevident. *Comptes Rendus, xxxvni. (1854), pp.990— 9;i3. tMecanique Celeste, supplement,t.v.[first published 1827].Oeuvres, v.(raris. 1882), pp.486—489. 8 THEORY OFBESSEL FUNCTIONS [CHAP.I The earlierportionofLaplace's investigationisbased ontheprinciple that, intheeaseofaseries ofpositiveterms inwhich theterms steadilyin- creaseuptoacertainpointandthensteadily decrease, theorder ofmagnitude ofthesum oftheseriesmayfrequentlybeobtained from aconsideration of theorder ofmagnitudeofthegreatestterm oftheseries. Forotherandmore recent applicationsofthisprinciple,seeStokes,Proc. Camb. Phil. Soc. VI.(1889), pp.362—366[Math, andPhys. Papers,v.(1905), pp.221—225], andHardy, Proc.London Math. Soc.(2)ii.(1905), pp.332—339; Messenger,xxxiv.(1905), pp.97—101. Astatement oftheprinciplewasgiven byBorel, Acta Mathematica, xx.(1897), pp.393— 394. Thefollowing expositionoftheprinciple appliedtotheexampleconsidered b}^Laplace maynotbewithout interest : The .series considered is J,I(?;,+2m)?i»-^-'"-^f»+-'» " ~^,„=o 2»+ =i»»»i!(» +«i)!' inwhich nislargeand ehasafixedpositivevalue. Thegreatest term isthat forwhich m= /x,wherefiisthegreatest integer such that 4^(n+,x){n+2^-2)^(n+2n)nh% andso fj.isapproximately equalto Now,if?<„,denotes thegeneral term inII,S^\itiseasytoverify byStirling's theorem that, toafirstapproximation, -^^'r^qi-, where' log2=-2V(l4-.2)/(ne2). HenceB„^'^)r^Uf,{l +2q+2q^+2q^+...} .,. , , ~2«^^/W(l-g)},since*qisnearly equalto1. Now, byStirling's theorem, e"-iexp{ns/(H-62)} andso5„(i)<^|MI±li)l^e"exp{nv-(l+.^)} Theinference whichLaplace drew from thisresult isthat V Trn' J{l+V(l-e-)}«" Thisapproximate formulahappenstobevalidwhen e<1(though thereason forthis restriction isnotapparent, apartfrom the factthat itisobviously necessary), but itisdifficult toproveitwithoutusingthemethods ofcontour Theformula l+^^^f~s'N{^-q)} may beinferred from general theorems onseries; of.Bromwich, Theory ofInfinite Series, §.51. Itisalsoaconsequence ofJacobi's transformation formula inthetheory ofelliptic functions, ^3(0|r)=(-;r)-H3(0|-T-i); seeModernAnalysis, %21-5l. 1-5] BESSEL FUNCTIONS BEFORE 1826 9 integration (cf.§8*31).Laplaceseems tohavebeendubious astothevalidity ofhisinference because, immediatelyafter hisstatement about realand imaginary variables, hementioned, bywayofconfirmation, that hehad anotherproof;butthelatterproofdoesnotappeartobeextant. 1'5.TheresearchesofFourier. In1822appearedtheclassical treatise byFourier*, LaTheorieanalytique delaChaleur; inthiswork Bessel functions oforder zero occur inthedis- cussion ofthesymmetricalmotion ofheat inasolid circularcylinder.Itis shewn byFo-urier(§§118—120)thatthetemperature v,attimet,atdistance Xfrom theaxisofthecylinder,satisfies theequation dvKfd'-v 1dv dtCD Kda--xdx where K,C,DdenoterespectivelytheThermalConductivity, SpecificHeat andDensityofthematerial ofthecylinder;andheobtained thesolution I2-^2^4- 2^4^6- , where g=mCDjKandmhastobesochosen that hv+K{dvldx)= attheboundaryofthecylinder,where histheExternal Conductivity. Fourierproceededtogiveaproof (§§307—309)byRolle's theorem that theequationtodetermine thevalues ofmhasfaninfinityofreal roots and nocomplexroots. Hisproofisslightly incompletebecause heassumes that certain theorems which havebeenprovedforpolynomialsaretrueofintegral functions; thedefect isnot difficult toremedy,andamemoir byHurwitz^ hastheobjectofmakingFourier's demonstration quite rigorous. Itshould alsobementioned "thatFourier discovered thecontinued fraction formula(§313)forthequotientofaBessel function oforder zeroand its derivate;generalisationsofthisformula willbediscussed in§§5-6, 9'6o. Another formulagiven byFourier, namely 1—TT-\-7^—T—7i— -.—^+...—— \COS(asmx)dx, 2- 2".4- 2-.4^6- ttJo hadbeenproved someyearsearlier byParseval§;itisaspecialcase ofwhat arenowknown asBessel's andPoisson'sintegrals (§§2'2, 2"3). *Thegreater partofFourier's researches wascontained inamemoir depositedinthearchives oftheFrench Institute onSept. 28,1811, andcrowned onJan. 6,1812. Thismemoir istobe found intheMem. deVAcad. desSci.,iv.(1819), [pubHshed 1824], pp.185— 555;v.(1820), [published 1826], pp.153—246. tThis isageneralisationofBernoulli's statement quotedin§1-2. +Math. Ann. xxxiii.(1889), pp.246—266. §Mem. dessavans etranijers,i.(1805), pp.639—648. This paperalsocontains theformal statement ofthetheorem onFourier constants which issometimes called Parseval's theorem; another paper bythis littleknown writer, Mem. dessavans etrangers,i.(1805), pp.379—398, con- tains ageneral solution ofLaplace's equationinaform involving arbitraryfunctions. 10 THEORY OFBESSEL FUNCTIONS [CHAP.I Theexpansionofanarbitraryfunction intoaseries ofBessel functions of order zerowasalsoexamined byFourier(§§314—320); hegavetheformula forthegeneralcoefficient intheexpansionasadefiniteintegral. ThevalidityofFourier's expansionwasexamined much more recently byHankel, Math. Ann. viii.(1875), pp.471—494; Schlafli, Math. Ann. x.(1876), pp.137—142; Diui, Serie diFourier,i.(Pisa, 1880), pp.246—269;Hobson, Proc.London Math. Soc. (2)vii. (1909), pp.359—388; andYoung,Proc.London Math. Soc.(2)xviii.(1920), pp.163—200. Thisexpansionwillbedealt with inChapterxviir. 1'6.TheresearchesofPoisson. Theunsymmetricalmotions ofheat inasolidsphereand also inasolid cylinderwereinvestigated byPoisson* inalengthymemoirpublishedin1823. Intheproblemofthesphere f,heobtained theequation where rdenotes thedistance from thecentre, /?isaconstant, nisapositive integer (zero included), andRisthat factor ofthetemperature,inanormal mode, which isafunction oftheradius vector. Itwasshewn byPoisson that asolution oftheequationis j.n+iIQQg(^y,pQQgjy^sin^"^+^ ft)dw Jo andhediscussed thecases n=0,1,2indetail. Itwillappear subsequently (§3'3)that thedefiniteintegralis(saveforafactor)aBessel function of order n+^. Intheproblemofthecylinder {ibid. p.340 etseq.)theanalogous integralis X" cos{h\cos (o)sin-^codto, .'o wheren=0, 1,2,...andXisthedistance from theaxisofthecylinder. The integralisnowknown asPoisson'sintegral (§2-3). Inthecasen=0,animportant approximateformula forthe lastintegral and itsderivate wasobtained byPoisson(ibid., pp.350—352)when thevariable islarge;thefollowingisthesubstance ofhisinvestigation: LetXJq(k)=- Icos{kcosw)da, J^Ik)=Icoswsin(kcosw)da. ThenJ^{k)isasolution oftheequation *Journal deI'Ecole R.Pohjtechnique, xii.(cahier 19),(1823), pp.249—403. tIbid.p.300etseq. Theequation wasalsostudied byPlana, Mem. della R.Accad. delle Sci. diTorino, xxv. (1821), pp.532—534,andhassince been studied bynumerous writers, some of whom arementioned in§4-3. SeealsoPoisson, LaTheorie Mathematiqite delaChaleur (Paris, 1835), pp.366, 369. +SeealsoEohrs, Proc.London Math. Soc. v.(1874), pp.136—137. TienotationJg{k)was notusedbyPoisson. 1-6] BESSEL FUNCTIONS BEFORE 1826 11 When kislarge, 1/(4^^^ j^^^yj^gneglectedincomparison withunity andsowemayexpect thatjQ{k)^lkisapproximatelyoftheformAcosk+Bam kwhere xiandBareconstants. Todetermine AandBobserve that 1/"tcosk.J(j{k)-sink.J,,'(/)=~I(cos'^ ^o)cos(2^-sin-ico)+sin-iwcos(2^-cos-iw)}da. Write TT-COfor u>inthelatter halfoftheintegral andthen 2(^cosk ..Tji(k)-sin X- .J,/ (X-)=- /cos- Acocos(2/:sin^^w)c/w 2v/2 /*V(•-'/.•)/,r--i\4=,, /1-^; Icos;t'-a.r, TTv^'y \ 2^•/ andsimilarlysink .,7|, (/{;)+cosk .J^(k)=^^-^ / (1—t7 )'^i'lx'^dx. TTxjkJo \2a7 V(2A-) cosBut lim I•-'(i-^)^"\t'2,c^A-=I^"^^•2.dr =iV(i'r), A-^Qo j(I V 2/t7 suij(,sm byawellknown formula*. [Note.Itisnoteasytoprove I'igorouslythatthepassagetothelimit ispermissible; thesimplest procedureistoappealtoBromwich'sintegral form ofTannery's theorem, Bromwich, Theory ofInfinite Series, §174.] Itfollows that cosk .Jy(k)-sink.J^;(k)= /n(1+«a), 1 sink .Jqyk)+cosk.J^{k)= where fj-^-O and/;fc-^0ask-^cc;andthereforef(nk)(i+'/O, J,(k)=Jink) 1[(1+6j(.)cosk+{l+j/j.)sin/], I^0(^0= ^(^-^[- (1-i-6,)sill ^'+(1+'m)cos/•]. Itwasthenassumed byPoisson thatJj){k)isexpressibleintheform 1 ^{^k)<.1'A" \,/„5'B" sink whereA=B=\. The series are,however, notconvei-gent butasymptotic,andthevalidity ofthisexpansion wasnotestablished, untilnearly forty years later,when itwas investi- gated byLipschitz, Journal furMath. LVi.(1859), pp.189—196. Theresult offormally operatingontheexpansion assumed byPoisson forthefunction d^ 1 ./q{k)sJiirk) with theoperator -Tr.i+1+jTi'^ dk-^ '2.1.B'-jA2.2B"-{\.2 +^)A'2 .3i?'"-(2.3+^).4'' _F"* k^"*k^ .,2.1.^' +J^.2.2A"+i\.2+'j)B',2.3.i"'+(2.3+i:)yr\+SinA-| Y^— 1 V,1 r-. r... [-P k^ /{:* *Cf.Watson, Complex Integration andCauchifs Theorem (Carab.Math. Tracts, no.15,1914), p.71,foraproofofthese results byusing contour integrals. 12 THEORY OFBESSEL FUNCTIONS [CHAP.I andso,byequatingtozerothevarious coefficients, wefindthat A'^-Ib, A'-=-,-^,A.A":i^B,... andhence theexpansionofPoisson's integralis /•tt /ttX^r/1 9 9.25 \, j^cos(/&cos0,)cfo,~ ^^jLVl-8|-278^^+2.^:W3+-r^'^' \_ 9^9.25 , jsink . But, since theseries ontherightarenotconvergent,theresearches ofLijischitz and subsequentwriters areanecessary preliminarytotheinvestigationofthesignificanceof thelatterportionofPoisson'sinvestigation. Itshould bementioned thatanexplicitformula forthegeneral term intheexpansion was firstgiven byW.E.Hamilton, Trans. R.IrishAcad. xrx.(1843), p.313; hisresult wasexpressedthus : -rcos(2/3sina)da=-^1[0]— ([-\ff(4^)-»cos(2/3-\nn-\n\ andhedescribed theexpansionassemi-convergent;the-expressions [0]~" and[-i]"are tobeinterpretedasIjn!and{-\) {—%)••.{-n+\). Aresult ofsomeimportance,which wasobtained byPoisson inasubsequent memoir*,isthatthegeneralsolution oftheequation Jo Jo whereAandBareconstants. Itfollows atonce thatthegeneralsolution oftheequation d-y1dy , .^_ dx-Xdx" is2/=^ Ie-''^cos<- ^(y^.^ g-7wcos<oiQg(^sij^2^)^^ .'o 7o This result wasquoted byStokesfasaknown theorem in1850, and itis likelythathederived hisknowledgeofitfrom theintegral giveninPoisson's memoir; butthefactthat theintegralissubstantiallydue toPoisson has beensometimesoverlooked|. *Journal deI'EcoIe R.Polytechnique,xii.(cahier 19), (1823), p.476.Thecorrespondiug general integral ofanassociatedpartial differential equation wasgiven inanearlier memoir, ibid. p.227. tCamb. Phil. Trans, ix.(1856), p.[38],[Math, andPhys. Papers, iii.(1901), p.42]. JSeeEncyclopedic desSci.Math. u.28(§53), p.213. r7] BESSEL FUNCTIONS BEFORE 1826 13 1'7.TheresearchesofBessel. Thememoir* inwhich Bessel examined indetail thefunctions which now bear hisname waswritten in1824, butinanearlier memoiri* hehadshewn thattheexpansionoftheradius vector inplanetarymotion is -=1+ie-+SBncosllM, where Ba= sinusin(nu—nesinu)du;nTTJo thisexpressionforB^should becomparedwith theseriesgivenin§1'4. Inthememoir of1824 Besselinvestigated systematicallythefunctionIj/^ defined bytheintegral :|: 1r-'"//=^r—cos(hu—ksinu)du. Hetookhtobeanintegerandobtained manyoftheresults which willbe givenindetail inChapterii.Bessel'sintegralisnotadaptedfordefiningthe function which ismostworthstudywhen hisnotaninteger (see §lO'l) ;the function which isofmost interest fornon-integralvalues ofhisnotIj/^but thefunction definedbyLommel which willbestudied inChapterIII. After thetime ofBesselinvestigationsonthefunctions became sonumerous that itseems convenient atthisstagetoabandon thechronologicalaccount andtodevelopthetheoryinasystematicandlogicalorder. Anhistorical account ofresearches from thetime ofFourier to1858ha«beencom|)iled byWagner, BeniMittheilungen, 1894, pp.204—266; abriefer account oftheearly history wasgiven byMaggi,Attidella R.Accad. deiLincei, {Transunti), (3)iv.(1880), pp.259—263. *Berliner Ahh.1824 [published 1826], pp.1—52.Thedate ofthismemoir,"Untersuchung desTheils derplanetarischen Storungen, welcher ausderBewegung derSonne entsteht,"' is Jan. 29,1824. tBerliner Ahh.1816—17 [published 1819], pp.49—55. JThis integral occurs intheexpansionoftheeccentric anomaly ;with thenotation of§1-4, aformula given byPoisson, Connaissancc desTeins, 1825 [published 1822], p.383. CHAPTER II THEBESSEL COEFFICIENTS 2"1.Thedefinition oftheBesselcoefficients. Theobjectofthischapteristhediscussion ofthefundamentalproperties ofasetoffunctions known asBesselcoefficients. There areseveralwaysof definingthese functions;themethod which willbeadoptedinthiswork isto define them asthecoefficients inacertainexpansion.Thisprocedureisdue toSchlomilch*, whoderived many propertiesofthefunctions from hisdefi- nition, andproved incidentallythatthefunctions thusdefined areequaltothe definiteintegrals bywhichtheyhadpreviously been definedbyBesself.It should, however, bementioned that theconverse theorem that Bessel's inte- gralsareequaltothecoefficients intheexpansion, wasdiscoveredbyHansen :J: fourteen yearsbefore thepublicationofSchlomilch's memoir. Some similar results hadbeenpublishedin1836byJacobi(§2-22). Thegeneratingfunction oftheBessel coefficients is Itwillbeshewn that thisfunction canbedevelopedintoaLaurent series, quafunction oft;thecoefficient ofP^intheexpansioniscalled theBessel coefficient ofargumentzandorder n,and itisdenotedbythesymbol /„(z), sothat (1) eH'-d= it-J,,{z).n=-cc Toestablish thisdevelopment,observe that e^'^canbeexpandedintoan absolutely convergentseries ofascending powersoft;and forallvalues oft, with theexceptionofzero, e~^^'^ canbeexpandedintoanabsolutelyconver- gentseries ofdescending powersof t.When these series aremultiplied together,theirproductisanabsolutely convergent series, andsoitmaybe arranged accordingtopowersoft;that istosay,wehaveanexpansionofthe form(1),which isvalid forallvalues of^:andt,t=excepted. *ZeitschriftfiirMath, undPhys.ii.(1857), pp.137—165.Forasomewhat similar expansion, namely that ofe^cosO^ gggPrullani, 3Iem. Soc. Ital.(Modena),xviii.(1820), p.503. Itmust be pointedoutthatSchlomilch, following Hansen, denotfed byJj,,^whatwenow write asJ,^{2z); butthedefinition giveninthetext isnowuniversally adopted. Traces ofHansen's notation aretobefound elsewhere, e.g. Schlatli, Math. Ann. in.(1871), p.148. tBerliner Ahh.1824 [published 1826], p.22. XErmittelung derAbsoluten Storungen inEllipsen vonbeliebiger Excentricitdt undNeigung, I.theil, [SchrifteuderSternwarte Seeburg:Gotha, 1843], p.106. SeealsotheFrench transla- tion,Memoire surladetermination desperturbations absolues(Paris, 1845), p.100,andLeipziger Abh. II.(1855), pp.250—251. 2*1,2-11] THEBESSEL COEFFICIENTS 15 Ifin(1)wewrite—1/tfort,weget 1t=-Xi 00=Xi-trj.niz),n=—:c onreplacingnby—n.Since theLaurentexpansionofafunction isunique*, acomparisonofthisformula with(1)shews that (2) /-„ {2)=(-rJn{Z), where nisanyinteger—aformula derived byBessel from hisdefinition of J,j(0)asanintegral. From(2)itisevident that (1)maybewritten intheform (3)e^zit-llt) =J^(^)+V l^n+(_Y^-n^^ J^^(^)_n=X Asuinmaryofelementaryresults concerning J,^(z)hasbeengiven byHall, TheAnalyst, I.(1874), pp.81—84,andanaccount ofelementary applications ofthese functions to problemsofMathematical Physics hasbeen compiled byHarris, American Journal of Math. XXXIV. (1912), pp.391—420. Thefunction oforder unityhasbeenencountered byTnwieYe, Notiv. Ann. deMath.(4) IX.(1909), pp.433—441,inconnexion with thesteepestcurves onthesurftice z=y {bx--y^). 2"11. Theascendingseries forJniz). I Anexplicit expressionfor/„{z)intheform ofanascendingseries ofpowers ofzisobtainable byconsideringtheseries forexp{hzt)andexp (—^2/^), thus exp11^(^-1/0}=S^-2-^VL_I^Z_^. When nisapositive integerorzero, theonlyterm ofthefirst series onthe right which, when associated with thegeneralterm ofthesecond seriesgives risetoaterminvolvingt^istheterm forwhich r=/;+m;and, since n^0, there isalwaysoneterm forwhich rhasthis value. Onassociatingthese terms forallthevalues ofm,weseethatthecoefficient ofV^intheproductis I{\zf^ {-\zr m={i{n-\-m)\ni\ Wetherefore have theresult (1) Jn{z)= S^/rM' *For,ifnot, zero could beexpanded into aLaurent series int,inwhich some ofthe coefficients (say,inparticular, that of?"')were notzero. IfwethenmultipUed theexpansion by (-m-i ajj^integrateditround acircle with centre attheorigin, weshould obtain acontradiction. This result wasnoticed byCauchy, Coniptes liendus, xiii. (1841), p.911. 16 THEORY OFBESSEL FUNCTIONS [CHAP. II where nisapositive integerorzero. The firstfewterms oftheseries are given bytheformula z'' (z^ z^) (2)Jn{z)= 2,7-;-,11-2M .(n+1)"^2M .2 .(n+l){n+ 2)••}' Inparticular (3) ^0(2)-I 22^2^ 4''2^42 .6- Toobtain theBessel coefficients ofnegative order,weselect theterms in- volvingt'^intheproductoftheseriesrepresenting exp(^zt)andexp (—hz/t), where nisstillapositive integer.Theterm ofthesecond series which, when associated with thegeneralterm ofthe first seriesgivesrisetoaterm int~"' istheterm forwhichm=n+r ;andsowehave whence weevidentlyobtain anew theformula|2"!(2),namely J.n{z)=(-rJ,,(z). Itistobeobserved that, intheseries(1),theratio ofthe(m+l)thterm tothemthterm is—jz^l{7n (n+m)],andthistends tozero as711^cc,forall values ofzand n.ByD'Alembert's ratio test forconvergence,itfollows that theseries representing J„(z)isconvergentforallvalues ofzand n,andsoit isanintegral function oizwhenw= 0,+1,+2,±3, Itwillappearlater(§4"73)thatJn{z)isnotanalgebraicfunction ofz and soitisatranscendental function;moreover, itisnotanelementary transcendent, that istosayitisnotexpressibleasafinite combination of exponential, logarithmicandalgebraicfunctionsoperatedonbysignsof indefiniteintegration. From (1)wecanobtain twousefulinequalities, which areofsomeimport- ance (cfChapter xvi)inthediscussion ofseries whosegeneral term isa multipleofaBessel coefficient. Whether zberealorcomplex, wehave 00 \Jn{z)\^\lz\^ X _2jf_l_ •^IjZ I nl,„=ow!(w-|-l)'«'andso,when n^0,wehave (4) 1/„(.) I,^exp(i^'),1ii^"exp(J I.n. This result wasgiveninsubstancebyCauchy, Comptes Rendus, xiii.(1841), pp.687,854;asimilar butweakerinequality, namely h Jn{z)\^^~'--exp{\z\-^),n wasgiven byNeumann, Theorie derBesseVscken Functionen(Leipzig, 1867), p.27. 2-12] THEBESSEL COEFFICIENTS 17 Byconsideringalltheterms oftheseries forJ^{z)exceptthe first, itis found that (5) J„W= <fe^'(l+«). where |ff |^exp(Ji^VU"'^P^^'"P-\ Itshould beobserved thattheseries ontherightin§2'1(1)converges uniformlyin anybounded domain ofthevariables zand twhich does notcontain theorigininthe f-})lane.For ifS,AandRarepositiveconstants and if theterms intheexpansionofexp {\zi)exp {^zjt)donotexceed inabsolute value thecorre- sponding terms oftheproduct exp(^/2a) exp{hR/8), andtheuniformityoftheconvergence follows from thetest ofWeierstrass. Similar considerationsapplytotheseries obtained byterm-by-termdifterentiations oftheexpansion 2^"/„ {z),whether thedifferentiations be performedwithrespecttozortorboth zand t. 2"12.Therecurrence formulae. Theequations* 2n (1) Jn-i (Z)+Jn+l (Z)=--Jn{z),z (2) Jn-Az)-Jn+dz)='^Jn(z\ which connect threecontiguousfunctions areuseful inconstructingTables of Bessel coefficients; theyareknown asrecurrenceformulae. Toprovetheformer, differentiate thefundamentalexpansionof§2-1, namely ,withrespecttot;weget h^z{l+llt^)e'=''~"'^ =int'^-'Jniz), n=-^ SOthat 1^(1+1/^^)iV'J.Xz)^ inV'-^Jn{z). )l=—Xl »=—X Iftheexpressionontheleft isarrangedinpowersoftand coefficients ofP~i areequatedinthetwoLaurent series, which areidentically equal,itisevident that ^Z{Jn-, (z)+Jn^ 1(z)}=nJn (z), which isthe first oftheformulaef. *Tlnoughout thework primes areused todenote thederivate ofafunction with respectto itsargument. fDifferentiations arepermissible because(g2-11)theresultingseries areuniformly convergent. Theequating ofcoefticients ispermissible because Laurent expansionsareunique. W.B.F. 2 18 THEORY OFBESSEL FUNCTIONS [CHAP.II Again,differentiate thefundamental expansionwithrespecttoz;andthen M=-00 sothat \{t-\\t) 2PJn{z)^ it^Jniz). n=-CO >i=-X Byequatingcoefficients offoneither sideofthisidentityweobtain formula (2)immediately. The results ofaddingandsubtracting (1)and(2)are (3)ZJn(z)+nJn(Z)=ZJn-, (z), (4) zJn (z)-nJn{Z)=-ZJn+i (^). These areequivalentto (5)^^{z-Jn(z)}=Z-J,,_,{z), (6)^Jz-J^{z)}=-z-Jn+d^). Inthecasen=0, (1)istrivial while theother formulae reduce to (7) Jo{z)=-./,(z). Theformulae(1)and(4)from which theothers maybederived were discovered by Bessel, Berliner Abh. 1824, [1826], pp.31,35.Themethod ofproof given here isdueto Schlomilch, Zeitschrift furMath, undPhys.II.(1857), p.138. Schlomilchproved (1)in thismanner, butheobtained(2)bydirect differentiation oftheseries forJ„{z). Aformula which Schlomilch derived {ibid, p.143)from(2)is (8)•^'~ip=^(-)'"> c../«-.^.„(.X where ,.C„, isabinomial coefficient. Byobvious inductions from(5)and(6),wehave (10)[jj-^{z-'''Jn{z)]={-r^Z—^-Jn^,,,{z), where nisanyinteger andmisanypositive integer. Theformula(10)isdue toBessel{ibid. p.34). Asanexampleoftheresults ofthissection observe that zJ^{z)=^Jo(z)-zJ^{z) =U,{z)-8J,(z) +zJ,(z) N 42{-Y-'nJ^,(z) +{-)^'zJ^^^,{z)n=l =4S(-r-'nJ.^,{z),n=l smce zJ^K+i (z)-*0asN-*cc,hy^211(4). 2'13, 2-2] THEBESSEL COEFFICIENTS 19 Theexpansionthus obtained, (11) zJAz)=^^ i{-r-'nJ,„{z), n=l isuseful inthedevelopmentsofNeumann'stheoryofBessel functions(§3".57). 2"13.Thedifferential equation satisfied hyJn{z)- When theformulae§2'12(5)and(6)arewritten intheforms theresult ofeliminating- J-,^_, (z)isseen tobe d' .d that istosaydz di'Z^--J,,{Z), d_ l^,_.dJ^^^^^^_^^J^^ ^^^^_^^_^^J^^ ^^^^ andsowehave Bessel 'sdifferential equation* Theanalysisissimplified byusingtheoperator'^defined asz(d/'dz). Thus therecurrence formulae are (^+n)J,,[Z)=Zjn-, {Z),(^-n+1)Jn_, {Z)=-zJn{z\ andso i^^^n +l)[z-^(^+n)J,{z)]=-zJn {z), that is ^-'(^-n)(^+n)J,,{z)=-zJn {z). andtheequation {'^'-iv)JJz)=-z\h{z) reduces atonce toBessel'sequation. Corollary. Thesame differentialequationisobtained '^1Jn+\{-)i^eliminated from the formulae (5+n+1)./„,1Kz)=zJ,, (z), (5-n)J„(z)=-zJ,^,(z). 2'2. Bessel'sintegral fortheBesselcoefficients. Weshallnowprovethat 1 r-'' (1) Jn{z)= ^^ \COS{ne-zs,\n 6)dO. Thisequation wastakenbyBessel fasthedefinition ofJn{z),f^ndhe derived theotherpropertiesofthefunctions from this definition. *Berliner Abh.1824[published 1826], p.34;seealsoFrullani, 3Iem. Svc. Ital.{Modena),xviii. (1820), p.504. tIbid. pp.22and35. 2—2 20 THEORY OFBESSEL FUNCTIONS [CHAP.II Itisfrequentlyconvenient tomodify (1)bybisectingtherangeofin- tegrationandwriting2'7r—Bfor inthelatterpart.Thisprocedure gives 1/'" (2) J"„(0)=- cos(nd-zsine)de. Since theintegrandhasperiod 2'rr,the firstequation maybetransformed into (3) Jn{z)=^[^'"'^cos{ne-2sind)de, ^"n-Ja. where aisanyangle. Toprove (1),multiplythefundamental expansionof§2'1(1)by^~"~^and integrate*round acontour which encircles theoriginonce counterclockwise. Wethusget J ,„=-x 27n Jm—n—idt. Theintegralsontherightallvanishexcepttheoneforwhichm=n;and soweobtain theformula 27n,\ Take thecontour tobeacircle ofunitradius andwrite t=e~'^,sothat maybetaken todecrease from 27r -t-atoa.Itisthusfound that •Zv+c(4) ^(^)=^(t—^e^'^'-''''dt (5) ^n(z)=^-I"^%^"(««-^sin^)«(^,"7* Ja aresult given byHansen finthecase q=0. Inthisequationtake a=—tt,bisect therangeofintegration and, inthe formerpart, replace by—0.Thisprocedure gives J",^(^)=i-r[e'^ne-zsmO) _^g-line- ^sine)j^0^ andequation (2),fromwhich(1)maybededuced, isnowobvious. Various modifications ofBessel'sintegralareobtainablebywriting Jn{z)—- COSnOCOS(zsin0)d0 A—sin»0sin(^sin ^)(Z^. TTJ'TTj|> ^ Ifbereplaced bytt—^inthese twointegrals,theformerchanges signwhen nisodd,thelatterwhen niseven, theother being unaffected ineach case; andtherefore 1l"'^. Jn (•s^)—~sinnQsm{zsm^)d0 =- Isinn0sin{zsin0')d0(nodd). *Terni-byterm integrationispermitted because theexpansionisuniformlj' convergent onthe contour. Itisconvenient tousethesymbol J'""*"'todenote integration round acontourencircling thepoint aonce counterclockwise. tErmittelung derabioluten Storungen (Gotha, 1843), p.105. 2-21] THEBESSEL COEFFICIENTS 21 (71even).1I'"" Jn(z)=- IcosnOcos(zsin6)dd (7)^-'^ 2/'^'^=-cos??^cosC^'sin^)c?(9ITJo^ If^bereplaced hyh-rr- i]inthelatterpartsof(6)and(7),itisfound that 2 r?" (8) J"n{z)=-(-)^*""^^ COSnT?sin(2cos77)dri {nodd),it111 (9) '/„(e)=- (-)*" cos *(7cos(zCOS7;)c^t; (neven)."^ .«j The lasttworesults areduesubstantially toJacobi*. [Note.ItwasshewnbyParseval, Mem. dessavansetrangers,i.(1805), pp.639—648, that ^~lP>+^y—r5~T'—r^r-^,+...= COS(asuiA')a.r,^- Z- .4- z- .4- .D- TVj andso,inthespecial case inwhich 5i=0,(2)willbedescribed asParseval'sintegral.It willbeseen in§2-3thattwointegral representations ofJ^(2),namelyBessel'sintegral and Poisson'sintegral become identical when «=0, soaspecial name forthiscase is ju.stified.] Thereader will find itinterestingtooljtain(after Bessel) theformulae§2-12(1)and §2-12(4)from Bessel'sintegral. 2•21. Modifications ofParsevaVsintegral. Twoformulaeinvolvingdefiniteintegralswhich areclosely connected with ParsevaPs integral formula areworth notice. The first,namely (1) Jo{s'{2^-f)\=-r'^-''°'^cos{zsin6)dd,TJ isduetoBessel +.Thesimplest method ofprovingitistowrite theexpression onthe rightintheform 1/"• SttJ—tt expandinpowers of^cos6+izsin6andusetheformulae 2r(n+|)r(^),., T{it+\) theformula then follows withoutdifficulty. Theother definiteintegral, duetoCatalani,namely (2)J"„(2^•v'2)=-rc(i+-)cos cos{(1-z)sin6)dd, •n-J isaspecial case of(1)obtained bysubstitutingI-zand 1+zforzand ?/respectively. *Journal furMath. xv.(1836), pp.12—13. [Ges. Math. IVerke, vi.(1891), pp.100—102];the integrals actually given byJacobi hadlimits andwwith factorsI/ttreplacingthefactors 2/V. See alsoAnger, Neiwste Schriftcn derNaturf. Ges. inDanzig,v.(1855), p.1,audCaucby, Comptes Jte7idns, xxxviii.(1854), pp.910—913. tBerliner Ahh., 1824 [published 1826], p.37.See al'ioAn^er, Keiieste Schriften derXatiirf. Ges. inDanzig,v.(1855), p.10,andLominel, Zeitschrifi filr^Math. iind P/ji/.v. xv.(1870), p.151. tBulletin deVAcad. 11.deBelgique, (2)xli.(1876), p.938.f:(ycose+izsin 6)''"+^de=0, Q/cos6+izsin^)2"dB=Z'—^ if"^)" ; /-n-'r(??+ 1) 22 THEORY OFBESSEL FUNCTIONS [CHAP.II Catalairs integral maybeestablished independently byusingtheformula 1 1z'W-^) m !27ri] sothat 00 ^.mIX ^Jn.'(0+ 1/("+)00ytti1X ^Jn r' ./o(2i\/~-)= 2-1—=—. 2-,t-"'~'e'dt m-o(w!)2271-?-,„=o»2! ./ exp '^+?^^=-L /" exp {e^-9+^6-'^lde, bytaking thecontour tobeaunit circle ;theresult then follows bybisectingtherangeof nitegration. 2"22. Jacohi'sexpansionsinseriesofBesselcoefficients. Two series, which arecloselyconnected with Bessel'sintegral,were dis- covered byJacobi*. Thesimplestmethod ofobtaining them istowrite ^=±e'^inthefundamental expansion §21(3).Wethusget =J,(z)+2iJ,n(^)cos 2nd±2i2Jo„+i {z)sin{2n+\)6. n=\ »=0 Onaddingandsubtractingthetworesults which arecombined inthisformula, wefind 00 (1)cos(^sin^)=/o(^)+2SJ^n(z)cos2n6, (2) sin(^sin^)=2SJo„+i(^)sin(2?i +1)^. 71=0 Write^TT— 77for6,andweget 00 (3)cos{zcos77)=Jq{z)+22(—)"J^n{z)cos2nr], n=\ (4)sin(2cos77)=2i(-fJ"2^+i(^)cos(2?i +l)77. The results(3)arid(4)weregiven byJacobi, while theothers were obtained laterby Angert.Jacobi's procedure wastoexpand cosucosj;) and sin(scosr;)intoaseries of cosines ofmultiplesof?;,anduseFourier's rule toobtain thecoefficients intheform of integrals which areseen tobeassociated with Bessel'sintegrals. Inview ofthe factthat the firstterms in(1)and(3)arenotformed accordingtothesame lawastheother terms, itisconvenient tointroduce Neumannsfactor \€„,which isdefined tobeequalto2when nisnotzero, andtobeequalto1when niszero.Theemploymentofthis factor, which *Journal furMath. xv.(1836), p.12.[Ges.Math. Werke, vi.(1891), p.101.] tNeueste Schrifteu derNaturf. Ges. hiDanzig,v.(1855), p.2. XNeumann, Theorie derBesseVschen Funcilonen(Leipzig, 1867), p.7. 2-22] THEBESSEL COEFFICIENTS 23 willbeoffrequentoccurrence inthesequel,enables ustowrite(1)and(2)in thecompactforms: 00 (5)cos{zsin6)=Se^nJ-m{z)cos2nd, 00 (6)sin(2^sin^)=1,eon+iJm+i{z)^v^{'^n +l)d. H= Ifweput^=in(0),wefind (7) 1=5 e,,J,n(2). 71= Ifwedifferentiate(5)and(6)anynumber oftimes beforeputting^=0,we obtainexpressionsforvariouspolynomialsasseries ofBessel coefficients. We shall, however, useaslightlydifferent methodsubsequently (§2'7)toprove that z'^isexpansibleintoaseries ofBessel coefficients whenmisanypositive integer.Itisthen obvious thatanypolynomialisthusexpansible.This isa specialcase ofanexpansion theorem, duetoNeumann, which willbeinvesti- gatedinChapterxvi. Forthepresent,wewillmerelynotice that, if(6)bedifferentiated once before disputequalto0,there results (8) z=i6,«+i(2n+1)Jsn+i (^). n= while, if6beputequalto^ttaftertwodifferentiations of(5)and(6),then (9)zsin2^2[22J,(z)-4^J,(z)+6'J,(z)- ...], (10)2cos^=2 11-J,(z)-3-^Js(z)+5-J,i^)-•••}• These results areduetoLommel*. Note. Theexpression exp{^^(i- 1/0}introduced in^^-l isnotageneratingfunction inbestrict sense. Thegeneratingfunction tassociated with(,iJn (z)is2e,i^"*A {^)- (1=0 Ifthisexpressionbecalled-S',byusingtherecurrence formula§2-12(2),wehave Ifwesolve this differential equation weget (11)s=ei-(t-iin+l(^+^e^^(t-vn jyh^it-mj^^^d,. Aresultequivalenttothiswasgiven byBrenke, Bull.American Math. Soc.xvi. (1910), pp.225—230. *Studien ilher dieBessel'schen Fiiitctionen (Leipzig, 1868), p.11. +Itwillbeseen inChapterxvi.that this isaform of"Lommel's function oftwovariables." 24 THEORY OFBESSEL FUNCTIONS [CHAP.TI 2"3. Poissoiisintegral fortheBesselcoefficients. Shortl}'before theappearanceofBessel's memoironplanetary perturbations, Poisson hadpublishedanimportantwork ontheConduction ofHeat*, inthe course ofwhich heinvestigated integralsofthetypes f rcos(zcos6)sin-'^+i ddS,jcos(zcos6)sin^'* dcie, Jo -JO where nisapositive integerorzero.Heprovedthat theseintegralsare solutions ofcertain differentialequations!andgavetheinvestigation,which hasalready beenreproducedin§1"6,todetermine anapproximationtothe latterintegral when zislargeandpositive,inthespecialcase >?=0. Weshallnowprovethat and, inview oftheimportanceofPoisson's researches, itseemsappropriateto describe theexpressionsontheright §asPoisson sintegralsforJ^i^).Inthe casen=0,Poisson'sintegralreduces toParseval'sintegral (§2"2). Itiseasytoprovethattheexpressionsunder consideration areequalto Jn{z)', for, ifweexpandtheintegrandinpowersofzandthenintegrate term-by-term||, wehave -cos(zCOS6)sin2« 66.6=-t\^ ,,cos^'" dsin^*^OcW T^.'o 7r,,,=o (2m)! Jo =2(_)m^2m 1.3.5... (2n-1).1 .3 .5...(2m-1) „,=o(2w)!• 2.4.6...(2?? +2m) =1.3. 5...(2/1-1) 2^,^,,1 \,^ .,,m=o2"^+^'"'m ;{n+ni)I andtheresult isobvious. *Journal deI'Ecole E.Polytechnique,xii.(cahier 19), (1823), pp.249— i03. tIbid.p.293, etseq. ;p.340, etseq. Integrals equivalent tothem hadpreviously been examined byEuler, Inst. Calc. Int. ii.(Petersburg, 1769), Ch. x.§1036, butPoisson's forms are moreelegant, andhisstudy ofthem ismore systematic. Seealso§3-3. +E.g.onp.300,heproved that, if Jo thenEsatisfies thedifferential equation <PE n{n+l)^ §Nielsen, Handhuch derTheorie derCylinderfunktionen (Leipzig, 1904), p.51,callsthem BesseVs secondintegral, buttheabove nomenclature seems preferable. IIThe series tobeintegrated isobviously uniformly convergent;theprocedure adoptedisdue toPoisson, ibid.pp.314, 340. 2-3,2-31] THEBESSEL COEFFICIENTS 25 Poisson alsoobserved* that gizcose^^^^n0dd=r COS(zCOS6)siri""ddO; .' this isevident whenweconsider thearithmetic mean oftheintegral onthe leftandtheintegralderived from itbyreplacing^bytt—0. Wethusget (2) /„(.)= j.^;ii|£j^/%-.'sin»^dft Aslightmodification ofthisformula, namely hassuggested important developments (cf. §6'1)inthetheoryofBessel functions. Itshould alsobenoticed that (4) jcos{zcos9)sin2«dd0=2 jcos(zcos$)sin-'*Odd Jo Jo =2['"cos (zsin6)cos-'*0d6, Jo andeach oftheseexpressions givesrisetoamodified form ofPoisson 'sintegral. Aninteresting applicationofBessel's andPoisson'sintegralswasobtained byLommelf whomultipliedtheformula « .,4n-[47i2-22|...(4w2-(2m-2)-}., ,,,cos2n^= :i(-)"*^-Tii^,^ ^^sm-"*^ bycos{zcos6)andintegi-ated.Itthus follows that ,« , ,4??-[4n--2-}... [4n--(2m-2)-|j;„(^) m=0 Z .lU: Z 2"31. Bessel's investigation ofPoisson sintegral. Theproof,that ./„{z)isequaltoPoisson'sintegral,which wasgiven by BesselJ,issomewhat elaborate; itissubstantiallyasfollows : Itisseenondifferentiation that dd2' cos6sin-'*~i 6cos{zcos6)— ^sin^'*+^ 6sin{zcos6)^ft~\X (2n-1)sin-"-- d-2wsin^**6+;^-^sin-^«+-^ ^2n+1cos(zcos6), *Poisson actually made thestatement(p.293) concerning theintegral which contains sin'-"-!-!.but,ashepointsoutonp.340,oddpowers maybereplaced byevenpowers throughout hisanalysis. fStudien ilher dieBesseVschen Functionen (Leipzig, 1868), p.30. +Berliner Ahh. 1824[published 1820], pp.36—37. Jacobi, Journal fiir3Iath. xv.(1836), p.13. [Ges.Math. Werke, vi.(1891), p.102],when givinghisproof (§2-32)ofPoisson's integralformula, objectedtotheartificial character ofBessel's demonstration. 26 THEORY OFBESSEL FUNCTIONS [CHAP.II andhence, onintegration,when n^1, (2n-1)f" cos(zcos6)sin^'^-- ed0-.2njcos(zcos0)sin^"OcW +-^^^--["cos (zcos0)sin-"+- OdO=0. Ifnowwewrite T^-rrv-w-r.I"cos(^cos6)sin- ^fZ^^c^{n), thelastformula shews that z(j)(n-1)-2n(f>(n)+Z(f)(n4-1)=0, sothat(f)(n)andJ^(z)satisfythesame recurrence formula. But,byusingBessel'sintegral,itisevident that (f>{0)=Jo(z), ^r-TT^f^d ( .) ^(1)=-cos{zcos6)sin-6d0=— 7^1^^"^^c*^^^H^^^^^^^ =-sin(^^cos6)cosOdd=—J^{£)=J^{z), and so,byinduction from therecurrence formula, wehave when 11=0,1,2,3,.... 232. Jacohis investigation ofPoissonsintegral. Theproblemofthedirect transformation ofPoisson'sintegralintoBessel's integralwassuccessfullyattackedbyJacobi*; thismethod necessitates theuse ofJacobi's transformation formula c?'»-i sin-»-i ^ 1.3.5...(2/1-1). „ ———=(-)"-i^ ^smnd, wherefjb=cos6.Weshallassume thisformula forthemoment, and,sinceno simpledirectproofofitseems tohavebeenpreviously published, weshall giveanaccount ofvariousproofsin§§2'321—2'323. Ifweobserve thatthe lirstn—1derivates of(1—/a-)'*"*, withrespectto jx,vanish when/i=+1,itisevident that,bynpartial integrations, wehave z'^ \COS{zcosB)sin2«ddd=2" cos{zyC).(1- yu,-)'*-* fZ/x =(-r I_^cos{Zi,-Invr)^ ^\^^^dfx. *JournalfiirMath. xv.(1836), pp.12—13. [Ges.Math. Werke,vi.(1891), pp.101—102.] See alsoJom-nal deMath. 1.(1836), pp.195—196. 2-32, 2-321] THEBESSEL COEFFICIENTS 27 IfwenowuseJacobi's formula, thisbecomes 1.3.5 ...(2/i-l)fi , ,,dsinnd ,C0B(za —i;n7r) -^da n J-I^'" - / (i^ =1.3.0 ...(2?i-l)Icos(zcos0 -i.n'Tr)cosnddO J(J =l.S.5...{2n-l)'7rJn{z), byJacobi's modification of§2*2(8)and(9),since cos(2cos^—^nrr)isequal to(—)^"cos(2cos6)or(—)i("-i)sin(zcos6)accordingasniseven orodd;and thisestablishes thetransformation. 2'321.Proofs ofJacobi'stransformation. Jacobi'sproofofthetransformation formula used in§2-32 consisted inderivingit asaspecial case ofaformula due toLacroix*;buttheproof which Lacroix gaveof hisformula isopentoobjectioninthat itinvolves theuseofinfinite series toobtain aresult ofanelementarycharacter. Aproof, based onthetheoryoflinear differential equations, wasdiscoveredl)yLiouville, Journal deMath. vi.(1841), pp.69—73; this proofwillbegivenin§2-322. Twoyearsafter Liouville, aninteresting symbolic proof waspublished byBoole, Camh. Math. Journal, iii.(1843), pp.216—224. Anelementary proof byinduction wasgiven byGrunert, Archie derMath, und Phi/s. iv.(1844), pp.104— 109. Thisproofconsists inshewing that,if t/»-l(l-^-)"-i©«= then- --'^®"©«+1= 11-m")^'- 2>i/ie„-n{n-\)Ie„dfi, andthat(-)"~^1.3.5...(2?i—1)(sinnd)lnsatisfies thesame recurrence formula. Other proofsofthischaracter have beengiven byTodhiuiter, DifferentialCalculus (London 1871), Ch. xxviii., andCrawford+,Proc. EdinburghMath. Soc. xx.(1902), pp.11—15,but allthese proofsinvolve complicated algebra. Aproof dependingontheuseofcontourintegrationisduetoSchliifli, xinn. diMat.(2) V.(1873), pp.201—202. Thecontour integrals areofthetypeused inestablishing Lagrange's expansion;andin§2-323weshallgivethemodification ofSchlafli's proof, inwhich theuseofcontourintegralsisreplaced byauseofLagrange's expansion. ToproveJacobi's formula, differentiate byLeibniz' theorem, thus : 1.3.5...(2n-l)(^/x«-i'^^>"> ^^^^^ ' n .•2'(-)-,.-fi,. <"-*'i":»;:±.:'"-^* 'a-Mr+'(i+ri'^^ ))}= 2• '-2•••V"<'+2 =' 2\- )'".„Co,„ +1(sinhdf^-+'(COSidf--•-'--1 OT= =sin{2nxid), andthis isthetransformation required J. *Traite duGale.Diff.i.(Paris, 1810, 2ndedition), pp.182—183. Seealso anotewritten by Catalan in1868,Mem. delaSoc. 11.desSci.deLiege, (2)xii.(1885), pp.312—316. tCrawford attributes tlieformula toRodrigues, possiblyinconsequenceofanincorrect state- ment byFrenet, Eecueil d'K.vereices (Paris, 1866), p.93,that itisgiveninRodrigues' dissertation, Corresp. surVEcole R.Folytechnique,iii.(1814—1816), pp.361—385. XIowethisproof toMrC.T.Preece. 28 THEORY OFBESSEL FUNCTIONS [CHAP.II 2*322. Liouville's proof ofJacohVs transformation. Theproof givenbyLiouville ofJacobi's formula isasfollows : Lety={\-^-}^~-and letI)bewritten ford/dii;then obviously Difi'erentiate thisequation ntimes;andthen buta-'^^)^-^-M^=«-^^X^S+'''^^=<^^^' sothatf-^,+nAD"-^7/=0. Hence Z>"~iy=/1sinw^+^cos??^, whereAandBareconstants;sinceI)"-'^yisobviouslyanoddfunction ofB,Biszero. Todetermine Acomparethecoefficients of6intheexpansionsofD"-'^y andAsin??(9 in ascending powersof6.Theterm involving6inD"-^yiseasilyseen tobe sothat «4=(-)"^i 1.3. 5...(2/1-1), andthence wehave theresult, namely c?"-isin2"-i(9, , ,1.3.5. ..(2»-l). . = ;=(_)"•-!^ ^suin6. 2•323.SchUiJii's proof ofJacobi's transformation. We first recallLagrange's expansion, which isthat,ifs= /x+A/'(s),then ceAnfJn—1 subjecttotheusual conditions ofconvergence*. Now take/(2)=_i (1-^2)^ <^'(2)=V(1-A itbeing supposed that^'{z)reduces tov^(l-m^),i.e.tosin6whenA^-0. Thesingularities ofzqaafunction ofhareatA=e^'^;andso,when 6isreal,theex- pansion ofJ{\— z^)inpowersofhisconvergent when both |h \and |z \arelessthanunity. Now 3=(1_^/(1_2^/i+k^)]ih^ ^"""^ ^-^(1-2^/^+/.^-)'^^^-'^A?-• (\)t-icZ"~^ sin^""^^Hence itfollows that^„\/-—. isthecoefficient ofA"-" intheex- 2"~i.(?i-l)!o?/i""i pansionofJ{1-z-).{czjOjj.)inpowersofh.But itisevident that ,2cz_{\-he'')---{\-he-'\'- _I1.3.5...(2>.-1) e^»^-e-"'^„_, ^^^'3m/« „=i2.4.6...(2?t}I^' andaconsideration ofthecoefficient of/i"~^inthelastexpression establishes thetruth of Jacobi's formula. *Cf.ModernAnalysis, §7-32. 2'322-2-33] THEBESSEL COEFFICIENTS 29 2'33.Anapplication ofJacobi'stransformation. Theformalexpansion /(cos cc)cosnxdx=IS(—)"*a.w/"^'+-'"' (cosx)dx, } .m= inwhich a,„isthecoefficient ofp^^-'"^ intheexpansionofJn{t)/Jo(t)inas- cending powersoft,hasbeen studied byJacobi*. Toestablishit,integrate theexpressiononthe leftntimes byparts;ittransforms(§2'32) into 1f""—j^ rx I/"^'(cosx)sin-^xdx, ,..(2?i-l)Jo 1.3.5...(2/1- 1) and,when sin-"^ isreplaced byaseries ofcosines ofmultiples ofx, thisbecomes 2.4.(/...(2.) /o>'^^^^">2/1 . 2n(n-\)1 r-COSZX+z ^rr-.^COS\Xn-^\ (?i-fl)(w +2)dx. Wenowintegrate /""(cos a?)COS 2.r,/'"'(cos a.')cos 4a;, ...byparts,andby continualrepetitionsofthisprocess, weevidentlyarrive ataformalexpansion ofthetypestated. When/(cos a;) isapolynomialincosa?,theprocess obviouslyterminates andthetransformation iscertainlyvalid. Todetermine thevalues ofthecoefficients a,„intheexpansion If(cosx)cosnxdx= (2(-)'" am/"'"^'"" (cosx)dx thus obtained, write /(cos x)=(—)-'* cos(tcosx),(—)i<«-i'sin(tcosx), accordingasniseven orodd,andwededuce from§2-2(8)and(9)that Jn{t)=i(-)'" a,,^"+-^'« {(-r /o(t)], SOthatamhasthevalue stated. Ithasbeen stated that theexpansionisvalidwhen/(cos x)isapoly- nomial incosX;itcan,however, beestablished when/(cos x)ismerelyre- stricted tobeanintegralfunction ofcosx,say ..56«co.s"a; providedthatlima/j6„ jislessthan thesmallestpositiveroot oftheequation J^(0=0;theinvestigationofthis willnotbegivensince itseems tobeof nopractical importance. .*JournalJilrMath. xv.(1836), pp.25—26[Ges.Math. Werke, vi.(1891), pp.117—118]. See alsoJacobi, Astr.Nach. xxviii. (1849),col.94[Ges. Math. Werke,vii.(1891), p.174]. 30 THEORY OFBESSEL FUNCTIONS [CHAP.II 24.Theaddition formula fortheBesselcoefficients. The Bessel coefficients possessanaddition formula bywhich Jn{y+z) maybeexpressedinterms ofBessel coefficients ofyand z.This formula, which was firstgiven byNeumann* andLommelf, is (1) J,,{y+z)=2J„,(y)Jn-,n{z). in=-CO Thesimplest wayofprovingthisresult isfromtheformula§2*2(4),which gives /„{y+z)=^-~.r"-1e^'2'+^' "-"'«'dt r 1 rto+) CO1SV^-''-'J,n{y)e^'^^-'!*^dtV.Trl' ... _'Ztti m=-00 1 oo /•(0+) 27nm=-00 00=SJ,a(y)Jn-m (z), 1)1=-ao onchangingtheorder ofsummation andintegrationinthethird lineofthe analysis;andthis istheresult tobeestablished. NumerousgeneralisationsofthisexpansionwillbegiveninChapterxi. 2'5.Hansen's seriesofsquares andproducts ofBesselcoefficients. Specialcases ofNeumann's addition formula weregiven byHansenJas earlyas184-3. The firstsystemofformulae isobtainablebysquaringthe fundamentalexpansion §2'1(1),sothat (,r=-00jI)?(=-X ByexpressingtheproductontherightasaLaurent series int,andequating thecoefficient of("intheresult tothecoefficient ofPintheLaurent ex- pansionoftheexpressiononthe left,wefindthat 00 Jn{2z)= 1Jr{z)Jn-r{z). Inparticular, takingn=0,wehave§ (1) J,(2^)=Jo^(^)+25(-)'• J;^(z)=i(-)'• e,J,'(2). *Theorie derBesseVschen Functionen(Leipzig, 1867), p.40. tStiidien Uber dieBesseVschen Functionen(Leipzig, 1868), pp.26—27;seealso Schlafli, Math. Ann. in.(1871), pp.135—137. +Ermittelung derabsoluten Storungen (Gotha, 1843), p.107etseq.Hansen didnotgive (4), andhegave only thespecial case of(2)inwhich n=\.Themore general formulae aredue to Loramel, Stiidien ilber dieBesseVschen Functionen(Leipzig, 1868), p.33. §Forbrevity, J,f {z)iswritten inplaceof{J^^ (z)}-. 2-4-2-6] THEBESSEL COEFFICIENTS 31 From thegeneralformula wefindthat (2) Jni^z)=1 Jr{Z)Jn-r (^)+21(-)'• ./,(z)J,^, (z), )•=() J'=l when theBessel coefficients ofnegativeorder areremovedbyusing §21(2). Similarly,since =exp{^z (t-1/01exp[iz(-t+1/01 =1, itfollows that (3) J,H^)+22/.n^)=l. r=l incc (4) S(-)'-j:,(^)J,„_,.(^) +22J,(^)/,„+,. (^)=0. r= !•=! Equation (4)isderived byconsideringthecoefficient ofP*intheLa expansion;theresult ofconsideringthecoefficient off-'^+i isnugatory. Averyimportant consequenceof(3),namely that,when xisreal, (5) |Jo(^)kl, \Jr{x)\^lls/% where ?•=1,2,3,...,wasnoticed byHansen. 2'6.Neumanns integral forJn"(z). Itisevident from§2-2(5)that J,,(2)=J-re^ine-zsm9>^0^ andso Toreduce thisdoubleintegraltoasingle integraltakenew variables defined bytheequations 6-cf>=2x,e+<t>=2f, sothat Itfollows that J,;-;(2)=^^,iie-'"'^ f,-2usin^cosx(^;^c?x/r, where thefield ofintegrationisthesquareforwhich Since theintegrandisunaffected ifbothx^^^^"^^^'®increased by tt,orifx isincreased byttwhile\/rissimultaneouslydecreased by tt,thefield ofinte- gration mayevidentlybetaken tobetherectangleforwhich 32 THEORY OFBESSEL FUNCTIONS [CHAP.II Hence 1 J^n(22COS;j;)C^X. •n".' Ifwereplace %by^tt+^,accordingas;^isacute orobtuse, weobtain the result (1) J,-^{z)=-I'"J,n{2z sine)dO. ITIfi•in- 'hisformula mayobviouslybewritten intheform istheresultactually given byNeumann*. Itwasderivedbyhimby 3laborate transformations from theaddition-theorem which willbegiven 2.Theproofwhich hasjustbeengivenissuggested bytheproofof jdition-theorem which waspublished byGrafandGublerf. 3obtain adifferent form oftheintegralifweperformtheintegration aspecttoXinstead ofwithrespecttoyfr.Thisprocedure gives Jn'(^)=^f" ^0(22sinf)e^"'l'dyjr, t 1f"Jn(^)=TT- ^0(22sini/r)COS2n\lrdyfr f 1f"=—Jo{'22sinylr)COS2)i\lr dylr, •"^Jo aresult whichSchlafli;|: attributed toNeumann. 2"61.Neumanns seriesforJ^(2). Bytakingtheformula§2'6(1), expandingtheBessel coefficient onthe rightinpowersof2andthenintegrating term-by-term, Neumann§shewed that 1TttX/\)n«2H+2)n ciri 2n+2*;i O Jn'(z)=-\S^ ^ \,^''" „"dd TTJo;«=o m\{2n -fm). ^(-Y{2n +2my.{^2Y"+-'^ ~ m=om\{tn+ m)\{{n+m)\\-' 'Theorie derBesseVsehen Fiaictionen(Leipzijj, 1867), p.70. tEinleitung indieTheorie derBesseVsehen Funktioiien, 11.(Bera, 1900), pp.81—85. JTheformula isanimmediate consequence ofequation 16onp.69ofNeumann's treatise. §2Iath. Ann. iir.(1871), p.603.Thememoir, inwhich this result vv-asgiven, was firstpub- lished intheLeipziger Berichte, xsi.(1869), pp.221—256. 2-61, 2-7] THEBESSEL COEFFICIENTS 33 This result waswritten byNeumann intheform (1) Jn'i^)Toz^+T,^ where (2)1{2n+1)1.2. (2w+1)(2/1+2) 2/1+1 2/rr2' (2/1+1) (2/1+3) (271+2)(2/1+4)' (2//+1)(2/^+3)(2/^+5) ' (2/i+2)(2/«+4)(2;z +6)' Thisexpansionisaspecialcase ofamoregeneral expansion (due to Schlafli)fortheproductofanytwoBessel functions asaseries ofpowers with comparatively simplecoefficients(§5"41). 2'7. ScJddinilch'sexpansion ofz^inaseriesofBesselcoefficients. Weshallnow obtain theresult which wasforeshadowed in§2*22con- cerningtheexpansibilityofz"^inaseries ofBessel coefficients, wheremisany positive integer.Theresult form= hasalreadybeengivenin§2"22 (7). Intheresults§2-22(1)and(2)substitute forcos2//^and sin(2/i+1)^ theirexpansionsinpowersofsin- 6.Theseexpansions are* cos2„.=i^(_).M^+--;^f(2,,„,). Sin1"('2??+1)(n+sV The results ofsubstitution are (cos(zsin6)=.J,(z)+2IJ,„(z)\I(-fy*/—t^^'(2sin^)4 , 1 Ifwerearrangetheseries ontherightaspowerseries insin(assuming that itispermissibletodoso),wehave '/•.X (r/X^Sr .x)^(-)H2sin^y^^i ^2//..(/i+.9-l)! ^,J cos(.sm^)^|/.(.)+ 2JJ..(.)j-+J^^-^A^^i £ ^^^_^^,^ J.M\, , ns^(-y(2sindr+' {^,(2//+l).(/i+5)! sin(2'sni6')= S--^ rvi— 1-^7 ^n^ ^ ^ ,=0 (2.5+1)! {n=s {n-sy."271+1 i^J^• Cf.Hobson, Plane Trigonometry (1918), §§80,82. W.B.F. 34 THEORY OFBESSEL FUNCTIONS [CHAP.II Ifweexpandtheleft-hand sides inpowersofsin6andequate coefficients, wefindthat n=l (i^>"=i'^4S^-^'"<^)'(^=1-2.3.-) (,,).«=iitn+l).(n +sy. j^^^^ ^^^^ (.=0,1,2,...) n=s v'' ^7- The firstofthese istheresultalreadyobtained;theothers maybecom- bined intothesingleformula Theparticularcases of(1)forwhichm=1,2,3,weregivenbySchlomilch*. Healsoshewed how toobtain thegeneralformula which wasgiven explicitly someyearslaterbyNeumannfandLommel:|:. Therearrangemeutofthedouble series nowneedsjustification;therearrangementis permissibleifwecanestablish theabsolute convergenceofthedouble series. Ifwemake'use oftheinequalities I'^2,,+1{z)!< ^2nJrl)\^^'^^^'^''^'72^"^^'^"^^^' inconnexion withtheseries forsin{zsin6)weseethat °° I2sin6|2»+i—5I°'" "^ I II,128+1pxn ('A I2|2'l"i (2.+ 1)!I^'l exp(^|2|) =sinh(Izsin6\)exp (|- 1z|2), andsotheseries ofmoduli isconvergent. The series forcos(2sin6)maybetreated in asimilar manner. Thesomewhat elaborateanalysis which hasjustbeengivenisavoided in Lommel'sproof byinduction, butthisproofsuffers from thefactthat itis supposedthattheform oftheexpansionisknown andmerely needs verifica- tion. If,following Lommel, weassume that az\m- y(m+2n).(m +n-l)l *Zeitschrift filrMath, undPhys.ii.(1857), pp.140—141. j-Theorie derBesseV schen Functionen(Leipzig, 1867), p.38. XStudien iiber dieBessel'schen Functionen(Leipzig, 1868), pp.35—36. Lommel's investigation ISgivenlater inthis section. 2-71] THEBESSEL COEFFICIENTS [which hasbeenprovedin§2-22(8)inthespecialcasem=1],wehave /I,^r«+^_y(m+2n).(m +n-l)l M= 't- 1 Ir/N ,V{(m+n)l(m+w-1)!) ,^35 H=0»i ^.(m+l +2n).(m +ny. *"„ ^I''jd-t-i+an (^)-M=0 Since(?H+?0!^m+on (i^)/>i!^ as ?i^co,therearrangementinthethird lineoftheanalysisispermissible.Itisobvious from thisresult thatthein- duction holds form=2,3,4, Allextremely elegant proofoftheexpansion, duetoA.C.Dixon* i.sasfollows:— Let tbeacomplex variable and let ?/bedefinedbytheequation?/,=^^,, sothatwhen tdescribes asmall circuit round theorigin (inside thecircle |;: |=1),lidoesthesame. Wethenhave ml /«'+) =^r—. exp{-iz{t- 1/0}2^ ,t'"*-"~1dt ~'^>-J~ ?i=o n ! _-(w+2/0(/«+/>-!)! whenwecalculate thesum oftheresidues attheoriginforthelastintegral;theinter- changeoftheorder ofsummation andintegrationispermitted because theseriesconverges uniformly onthecontour;andtherequiredresult isobtained. Note. Whenmiszero, =-hastobereplaced by—^—,mdt^ -^dt I'll. Schlbmildi'sexpansions ofthetype2n''JH (s). Tlieformulae (1) 2{27ir-yJ,„{z)= ipZ^z"--,n=l )H=0"'* (2) ii2n+iy-"+W,„^,{z)= 2P£:t^"""'' inwhich;jisanypositive integer [zeroincluded in(2)butnotin(1)]and P'' isanumeri- calcoefhcient, areevidently very closely connected withtheresults of§27.Thefornnilae * Messenger, xxxii.(1903), p.8;aproof onthesame lines forthecase vi—1hadboon pre- viously given byKapteyn, Nietiw Archief voor U'iskunde, xx.(IS'JS), p.120. 3—2 36 THEORY OFBESSEL FU^'CTIOXS [CHAP.U were obtained bySchlomilch, ZeiiscArift fiirMath, undPhyi.n.(1857), p.141,andhe gave,asthevalue ofi*^, where^C'tisabinomial coefficient andthelastterm ofthesummation isthat forwhich k is^rfi-1ori^(m— 1).Toprovethe firstformula, taketheequation §2-22(1),differentiate 2ptimes withrespectto6,andthenmake 6equaltozero. Itisthusfound that Theterms oftheseries forwhich jn>p,when expandedinasc-ending powersof0, contain noterm in$^,andsoitissufficient t<3evaluate =-^2^1^ 2-•'Ci'2//i-2X-)^ Di=o(2m;!t=,>- =2(-V' 2z^I^S!, sincetermsequidistant from thebeginning andtheendofthesummation withrespect to kareequal Thetruth ofequation (1)isnowe«dent, andequation ,2}isproved ina similar manner from§2-22(2). The reader willeasilyestablish thefollowing special cases, which were stated by Schlomilch : |13J,(^)+33 Ja(z)+bKJ., (.)+...=! (5+^3)^ (4) -,22./,(2)+42.7,(2}-5-6^Jg(£)+...=l^, l2.3.4J3(2;^4.5.6J5(2:-i-6.7.8^-(2) +...=M 2-72.Xeumannsexpansion ofz^asaseriesofsquares ofBesselcoejjicierits. From Schlomilch'sexpansion (§2-7)of2^asaseries ofBessel coefficients ofeven order, itiseasytoderive anexpansionof2*^asaseries ofsquares of Bessel coefficients, byusingNeumann'sintegral givenin§2"6. Thus, ifwetaketheexpansion /•Q\'^ ^(2m+2n).(27n +/(-!): ...^.^ andintegratewithrespectto6,wefindthat ^(2m-2n).{2m-\-n-l^ sothat(whenrn>0) (1) (i^)-=i^V(2m+9n).(2rn^n-l): •2-72] THEBESSEL COEFFICIENTS 37 This result wasgiven byNeumann*. Analternative form is andthis istruewhenm=0,for itthen reduces toHansen's formula of§2"5. Asspecial cases, wehave 2-=52f„.^n-J^-'Z), (3) 4.o .bit=3 Ifwedifferentiate(1),use§2-12(2;andthen rearrange,itisreadily found that „ , ,m\{rfL-V)\--"(2«i+2n-l). (2rft+«-2)! ^ ,.j,, anexpansion whose existence wasindicated byNeumann. 'Leipziger Berichte,xxi.(1869;, p.226. [Math. Ann. iii.(1871), p.585.] CHAPTER III BESSEL FUNCTIONS 3'1. TJiegeneralisation ofBesseVsdifferential equation. TheBessel coefficients, which were discussed inChapterii,arefunctions oftwovariables, zand n,ofwhich zisunrestricted butnhashitherto been requiredtobeaninteger. Weshallnowgeneralisethese functions soasto have functions oftwounrestricted (complex)variables. Thisgeneralisationwaseffected byLommel*, whose definition ofaBessel function waseffectedbyageneralisationofPoisson'sintegral;inthecourse ofhisanalysis heshewed that thefunction, sodefined, isasolution ofthe linear differentialequationwhich istobediscussed inthissection. Lommel's definition oftheBessel function Jv{z)ofargumentzandorder vwasf J.{z)= r(^+1)^(1) C'^''"^'''''"^^''''"' ^'^^' andtheintegralontherightisconvergentforgeneral complexvalues ofv forwhich R{v) exceeds —\.Lommelapparently contemplated onlyreal values ofv,theextension tocomplexvalues beingeffected byHankeliJ: ; functions oforder lessthan—\were defined byLommelbymeans ofanex- tension oftherecurrence formulae of§2"12. Thereader willobserve, oncomparing §3'8with§1'6thatPlana and Poisson hadinvestigatedBessel functions whose order ishalfofanoddinteger nearlyhalfacenturybefore thepublicationofLommel's treatise. Weshallnowreplacetheintegernwhich occurs inBessel's differential equation byanunrestricted (real orcomplex) number§ v,andthen define a Besselfinctionoforder i^tobeacertain solution ofthisequation;itisof course desirable toselect such asolution asreduces toJn(z)when vassumes theintegralvalue n. Weshall therefore discuss solutions ofthedifferentialequation (1).^g+^g+(.^-.-^)y=0, which willbecalled Bessel'sequation forfunctions oforder v. *Studien iiher dieBesseVschen Functionen (Leipzig, 1868), p.1. tIntegrals resemblingthis(withvnotnecessarily aninteger) were studied byDuhamel, Coum d'Analyse,ii.(Paris, 1840), pp.118—121. XMath. Ann. i.(1869), p.469. §Following Lommel, weusethesymbols v, fj.todenote unrestricted numbers, thesymbols n,mbeing reserved forintegers. This distinction iscustomary ontheContinent, thoughithas notyetcome intogeneral useinthiscountry.Itliastheobvious advantage ofshewing ata glance whether aresult istrue forunrestricted functions orforfunctions ofintegral orderonly. 3-1] BESSEL FUNCTIONS 39 Letusnowconstruct asolution of(1)which isvalid neartheorigin;the formassumed forsuch asolution isaseries ofascending powersofz,say 00 y——^111-^> where theindex aandthecoefficientsc,naretobedetermined, with thepro- visothat Coisnotzero. Forbrevitythedifferentialoperatorwhich occurs in(1)willbecalled V^, sothat (2) ^.,,=*^,^^U,=_,, Itiseasytoseethat* w= 111=111=0 Theexpressionontherightreduces tothe firstterm ofthe first series, namely Co(a-— i/-)2",ifwechoose thecoefficients Cmsothatthecoefficients of corresponding powersofzinthetwoseries ontherightcancel. This choicegivesthesystemofequations c,{{a+ly- V-] =0 c,\{a+2)--v~\+Co =0 (3) 1 c,n[(a+mf- v"}+C,n_->= If,then, theseequationsaresatisfied, wehave (4) V,2CmZ''+"'=Co{or-v')z'^. From this result, itisevident thatthepostulatedseries canbeasolution of<'l)onlyifa=+i^;forCqisnot zero,and ^"vanishesonlyforexceptional values ofz. Now consider thentthequationinthesystem (3)when ui> 1.Itcanbe written intheform c,a(a—v+III)(a+f+III)+c„i_o=0, andsoitdeterminesc,„,interms ofc„i_o forallvalues ofmgreaterthan 1 unless a—?/ora-Yvisanegative integer,thatis,unless—2visanegative integer (whena=—v)orunless 2visanegative integer (whena=i>). Wedisregardtheseexceptionalvalues ofvforthemoment(see §§8'11, 3-5),andthen{a+mf—v-does notvanish when ??i=l,2,3,.... Itnow *When theconstants aandc,,^have been determined bythefollowing analysis,theseries obtained byformal processesiseasily seen tobeconvergent anddifferentiable, sothattheformal procedure actually producesasolution ofthedifferential equation. 40 THEORY OFBESSEL FUNCTIONS [CHAP.Ill follows from theequations (3)that 01=03-65=...=0,andthat c^misex- pressibleinterms ofCobytheequation _ (-rco ^. Thesystemofequations (3)isnow satisfied;and, ifwetake a=v,wesee from(4)that (5) Co2- "^ „,r 1ml{v+l){v+2)...{v +m) isaformal solution ofequation (1).Ifwetake a=~v,weobtain asecond formal solution (6) Co'2-1+2 ,riml{-v +l){-v +2)...{-v+m)_' Inthelatter, cjhasbeen written inplaceofCq,because theprocedureof obtaining (6)canevidentlybecarried outwithout reference totheexistence of(5),sothattheconstants Cqand Cqareindependent. Anyvalues independentofzmaybeassignedtotheconstants CqandCq' ; but,inview ofthedesirabilityofobtainingsolutions reducible toJn{z)w^hen V-^n,wedefine thembytheformulae* The series (5)and(6)maynowbew-ritten »(-y'^(l^)''+2m ^(-)'" (i^)""^'"' ,«tomir{v+m+l)' ,„romir{-v +m+l)' Inthecircumstances considered, namely when 2visnotaninteger,these series ofpowers convergeforallvalues ofz,{z=excepted)andsoterm-by-term differentiations arepermissible. Theoperationsinvolved intheanalysis fby whichtheywereobtained areconsequently legitimate,andsowehaveobtained twosolutions ofequation (1). The first ofthetwoseries defines afunction called aBesselfunctionof order vandargument z,ofthefirstkindX; andthefunction isdenotedby thesymbol /^{z).Since visunrestricted(apartfrom theconditionthat, for thepresent,2i^isnotaninteger),thesecond series isevidently JL.^ {z). Accordingly,thefunction Jt.{z)isdefined hytheequation Itisevident from§2-11that thisdefinition continues toholdwhen i^isa positive integer (zero included), aBessel function ofintegralorderbeing identical withaBessel coefficient. *ForpropertiesoftheGamma-function, seeModernAnalysis,ch.xii. +Which, uptothepresent, hasbeenpurely formah XFunctions ofthesecond andthird kinds aredefined in§§3-5, 354,3-57, 3*6. 3-11] BESSEL FUNCTIONS 41 Aninteresting symbolicsolution ofBessel's equation hasbeengivenbyCotter* inthe form [l+z"D-'^ z-'^"-^ B-'^2"+^]-^ {Az'+Bz-"), whereD=djdzwhileAandBareconstants. Thismaybederived bywriting successively [D{zB-2v)+z]z''^=0, [zD-2v +L>-^z]z''i/=-2vB, zD{z-''y) +z--''D-h''^\j= -2vBz-~\ which givesCotter's result. 3'11. Functions whose order ishalfofanoddinteger. In§o\,twocases ofBessel'sgeneralised equation weretemporarilyomitted from consideration, namely (i)when vishalfofanoddinteger, (ii)when vis anintegerf.Itwillnowbeshewn thatcase(i)maybeincluded inthegeneral theoryforunrestricted values ofv. When Vishalfofanoddinteger,let ir=(r+i)^ where risapositive integerorzero. Ifwetake a=r+-^intheanalysisof§3'1,wefindthat ,,. [Cx.l(2r+2) =0, ^' \c,n..ni('>n+2r+1)+c„,_,=0,{m>i) andso (2) c,..--<-*""« 2.4... (2m).(2r+3)(2?-+5)...(2r+2ni+1)' which isthevalue ofComgiven by§3'1when aandvarereplaced byr+^, Ifwetake " 2''+*r(r +|)' weobtain thesolution ,,Zomir (r+m+^)' which isnaturallydenotedbythesymbol Jr+ki^),sothatthedefinition of §3-1(S)isstill valid. If,however, wetake a=—r— |,theequationswhich determine Cmbecome (3)(o..l(-2;) =0.(„,^j^ [Cmm(m—1—2r)+c,„_2=0. /Asbefore, Ci, c-^,...,c.2,_i areallzero,buttheequationtodetermine c.>,-~i is .&>,.+!+Cor-i=0, and thisequationissatisfied byanarbitraryvalueofc.i,-+i',whenm>r,c..»n+i isdefined bytheequation f\in—rf^ -'"+'~ (2r+3)(2?-+5)...(27/i+i).2 .4 ...(2m-2/-)" *Froc. E.Irish. Acad. xxvn.(A),(1909), pp.157—161. tThecasescombine toform thecase inwhich '2visauinteger. 42 THEORY OFBESSEL FUNCTIONS [CHAP.Ill IfJy{z) bedefined by§3-1(8)whenv=-r-l,thesolution nowcon- structed is* Co2-'--^r(1-r)J_r-i {z)+C2.+12'-+^r(r+|)/,+ (^). Itfollows thatnomodification inthedefinition ofJ^{z)isnecessarywhen v=±{r+\);therealpeculiarityofthesolution inthis case isthat the negativeroot oftheindicialequation givesrise toaseriescontainingtwo arbitrary constants, CqandCar+i,i.e.tothegeneralsolution ofthedifferential equation, 3"12.Afundamental system ofsolutionsofBessel'sequation. Itiswellknown that, ifyiand3/2aretwosolutions ofalinear differential equationofthesecond order, and if?//and2/2'denote their derivates with respecttotheindependent variable, then thesolutions arelinearlyinde- pendentiftheWronskian determinant^ does notvanishidentically;and iftheWronskian does vanishidentically, then, either oneofthetwosolutions vanishesidentically,orelsetheratio of thetwosolutions isaconstant. IftheWronskian does notvanishidentically,thenanysolution ofthe differentialequationisexpressibleintheformc,2/1+c^y-zwhereCyandCgare constantsdependingontheparticularsolution under consideration;the solutions2/1and3/2arethen said toform afundamentalsystem. ForbrevitytheWronskian oft/iandy^willbewi'itten intheforms m^z[y„yo], M|yi,i/.}, theformerbeingusedwhen itisnecessarytospecifytheindependentvariable. Wenowproceedtoevaluate im[j.{z), j_.(^)}. Ifwemultiplytheequations V,/__,{z)=0, V,J,{z)= byJ^(z),J"_^(z)respectivelyandsubtract theresults, weobtain anequation whichmaybewritten intheform j^[zm{JAz),J-A^)]]=o, *luconnexion with seriesrepresentingthis solution, seePlana, 2Iem. della E.Accad. delle Sci.diTorino, xxvi.(1821), pp.519—538. tForreferences totheorems concerning Wronskians, seeEncyclopedle desSci.Math. u.16 (§23), p.109. Proofs ofthetheorems quoted inthetextaregiven byForsyth, Treatise on Differential Equations (1914), §§72—74. 3-12] BESSEL FUNCTIONS 43 andhence, onintegration, z whereCisadeterminate constant. Toevaluate C,weobserve that, iiohen visnotaninteger, and^jissmall, wehave •^'<^>=T^)I'+^<^'>1' •^''<^>=1^)!i+<^")1' with similarexpressionsforJ-v{z) andJ'_^,{z);andhence J,(.)/_, (.)-/_.(.)./;(.)= ^1y^^~ri::r^- 1>)r^TTT)!+"^'^ 2sin v-n-^^.= +{z). Ifwecomparethis result with(1),itisevident that theexpressiononthe right which is0{z)must vanish, andso* TTZ Since sinvir isnotzero(becausevisnotaninteger),thefunctions J^{z), /_^(^)form afundamental systemofsolutions ofequation §3"1(1). When Visaninteger, n,wehave seen that,with thedefinition of§2'1(2), andwhen vismadeequalto—yiin§3"1(8),wefindthat '^-''^'^^Jo^mTr{-n+m+T)' Since the firstnterms ofthelastseries vanish, theseries iseasilyreduced to {—y^Jn{z),SOthat thetwo definitions ofJ^n{z)areequivalent,andthe functions Jn{z),J_n {z)donotform afundamentalsystemofsolutions of Bessel'sequationforfunctions oforder n.Thedetermination ofafundamental systeminthiscase willbeinvestigatedin§3'63. Tosumup,thefunction Ji,{z)isdefined, for allvalues ofv,bythe expansionof§3'1(8);and J^,{z),sodefined,isalwaysasolution oftheequation V^2/—0.When visnotaninteger,afundamentalsystemofsolutions ofthis equationisformedbythefunctions J^,{z)and ./_^ {z). ^Ageneralisation oftheBesael function hasbeen effected byF.II.Jackson inhis »"-1 researches on"basic numbers."Briefly,abasicnumber[?*]isdefined as^_,where/l>is thebase,andthebasicGamma functionT^,{v)isdefined tosatisfytherecurrence formiUa T^{v+l)=[v].Y,{v). Thebasic Bessel function isthen defined byrepkcingthenumbers which occur inthe series fortheBessel function bybasic numbers. Ithasbeenshewn thatverymany theorems *This result isduetoLommel, Matli. Ann. iv.(1871), p.101.Hederived thevalue ofCby making2;^-ocandusing theapproximate formulae which willbeinvestigatedinChaptervii. 44 THEORY OFBESSEL FUNCTIONS [CHAP.Ill concerning Bessel functions have their analoguesinthetheoryofbasic Bessel functions, butthediscussion ofthese analoguesisoutside thescopeofthiswork. Jackson's main results aretobefound inaseries ofpapers,Proc. EdinhurghMath. Soc.xxi.(1903), pp. 65—72;XXII.(1904), pp.80—85;Proc.RoyalSoc.Edinburgh,xxv.(1904), pp.273—276; Trans.RoyalSoc.Edinhurgh,xli. (1905), pp.1—28, 105—118, 399-408; Proc.London Math. Soc.(2)I.(1904), pp.361—366; (2)ll.(1905), pp.192—220; (2)III.(1905), pp.1—23. Themore obvious generalisationoftheBessel function, obtained byincreasingthe number ofsetsoffactors inthedeuominators oftheterms oftheseries, willbedealt with in§4"4. Inconnexion with thisgeneralisationseeCailler, Mem. delaSoc.dePhys.de Oeneve, xxxiv, (1905), p.354;anothergeneralisation,intheshapeofBessel functions oftwo variables, hasbeen dealt withbyWhittaker, Math. Ann. LVii.(1903), p.351,and Perfes, Comptes Rendus,CLXi.(1915), pp.168—170. 3'13. Generalproperties ofJ^,{z). The series which defines Jy{z) converges absolutelyanduniformly*inany closed domain ofvalues ofz[theoriginnotbeingapointofthedomain when R(v)<0],andinanybounded domain ofvalues ofv. For,when\v\^Nand\z\•$A,thetestratio forthisseries is 4Z m{v+m) in{m—N) whenever mistaken tobegreaterthan thepositiverootoftheequation m--mN-l^-'=Q. This choice ofmbeing independentofvand z,theresult stated follows from thetestofWeierstrass. Hencej- J^(z)isananalytic function ofzforallvaluesofz{z=possibly being excepted) and itisananalytic function ofvforallvalues ofv. Animportant consequenceofthistheorem isthatterm-by-termdifferen- tiations andintegrations (with respecttozorv)oftheseries forJ^{z)are permissible. Aninequality duetoNielsen%should benoticed here,namely where I^ i<expl,-^'^!, 1-1,Iko+l\} and 11/0+1 1isthesmallest ofthenumbers|i/+l|,|i'+2|,|i/+3j,— This resultmaybeprovedinexactlythesamewayas§2*11(5) ;itshould becom- paredwith theinequalities which willbegivenin§3*3. Finally,thefunctionz",which isafactor ofJ^{z),needsprecise specifica- *Bromwich, Theory ofInfinite Series, §82. tModern Analysis, §5"3. J3Iath. Ann. Lii.(1899), p.230;NytTidsskrift,is.B(1898), p.73;seealsoMath. Ann. lv. (1902), p.494. 3-13, 3-2] BESSEL FUNCTIONS 45 tion.Wedefine ittobeexp(vlogz)where thephase (orargument) ofzis givenitsprincipalvalue sothat —TT<argz^w. When itisnecessaryto"continue" thefunctionJ^{z) outside thisrangeof values ofarg z,explicit mention willbemade oftheprocesstobecarried out. 3'2. TlierecurrenceformulaeforJ^{z). Lommel'sgeneralisations*oftherecurrence formulae fortheBessel co- efficients(§2'12)areasfollows: (1) J.^,{z)-vJ.^Az)=^JAz)z (2) J".-: {z)-J.+, {z)=2J; (z), (3) zJ;(z) +vJ..(^)=^J.-A^\ (4) zJJ (z)-vJ,{z)=-zJ,.+, (z). These areofpreciselythesame form astheresults of§2"12, theonlydifference beingthesubstitution oftheunrestricted number vfortheintegern. Toprove them, weobserve firstthat dz'"^^'dz,Zo^"^"^m\V{v+ni-rl) 00 /\iti^iv—l-\-2m^ ,„=o2-^+^'«.m!r(i. +m) When wedifferentiate outtheproductonthe left,weatonce obtain(3). Inlikemanner, J J CO(—Vn,2W dz^"^^^dz,r=o^^^'"'.m\V{v^m +l) V C_yft^2 -^2"+"'"-' .{in-\)\V{v+111+1) 00C_y«+i 2'-'«+i „r=^2''+'^'»+i .m!r(z/+m+2) =-z-^J,^,{z), whence(4)isobvious; and(2)and(1)maybeobtained byaddingandsub- tracting (8)and(4). *Studien ilher dieBesseVschen Functionen (Leipzig, 1868), pp.2,6,7.Formula(3)wasgiven when fishalf ofanoddinteger byPlana, Mem. della R.Accad. ddle Sci. diTorino, xxvi.(1821), p.533. 46 THEORY OFBESSEL FUNCTIONS [CHAP.Ill Wecannowobtain thegeneralisedformulae (5)(^)"[Z^J. {Z)\=Z^-^-J.-m (Z), (6)(^)"[z-^J. (z)]=(-)-^-'-/.+«. (^) byrepeated differentiations, whenmisanypositive integer. Lommel obtained allthese results from hisgeneralisationofPoisson's integralwhich hasbeen described in§3'1. Theformula (1)hasbeen extensivelyused* intheconstruction ofTables ofBessel functions. Byexpressing /^_i (z)andJi_^ (z)interms ofJ±„(z)and J'±^, {z)by(3) and(4),wecanderive Lommel's formulaf (7) /,(z)/i_, iz)+/_,(z)./,_, (z)=^~^^f^ from formula(2)of§3-12. Auinteresting consequenceof(1)and(2)isthat,ifQ,,(z)=J^[z\then (8) ^^.-i(^)-<?.+i(2)=y«;(^); thisformula wasdiscovered byLommel, whoderived various consequencesofit,Studien iiher dieBesseVschen Functionen(Leipzig, 1868), pp.48etseq.SeealsoNeumann, Math. Ann. III.(1871), p.600. 3'21. Besselfunctions ofcomplexorder. The realandimaginary partsofthefunctionJ„^ifj,(x), wherev,fxandx arereal,havebeen discussed insome detailbyLommelJ,andhisresults were subsequentlyextended byB6cher§. Inparticular,afterdefiningtherealfunctionsK^^^{x)andSi,^^(x) bythe equation11 Lommel obtained theresults (1)^,{K,,^(x) ±iS„,A^)] +{^^^M (^)±iS,,^(x)} 2(v±ifi) +ld,j.,X ,-o /MA X dx (2) A",+i,^ (x)=K,^^ (x)+K'\^^{x), (3) 'S^^+i,K (^)= 'S^.-.M(^)+-S^'Vm (^)' *See, e.g.Lommel, Milnchener Ahh. xv.(1884—1886), pp.644—647. tMath. Ann. iv.(1871), p.105.Some associated formulae aregiven in§3'G3. +Math. Ann. ra.(1871), pp.481—486. §AnnalsofMath. vi.(1892), pp.137—160. IIThereason forinserting thefactor ontherightisapparent from formulae which willbe established in§3'3. 3-21, 3-3] BESSEL FUNCTIONS 47 withnumerous other formulae oflikecharacter. These results seem tobeof nogreat importance,andconsequently wemerelyrefer the I'eader tothe memoirs inwhichtheywerepublished. Inthespecialcase inwhich v=0,Bessel'sequation becomes solutions ofthisequationintheform ofseries weregiven byBoole* many years ago, 3'3.Lormnel's expression ofJ^{z) hyanintegral ofPoissonstype. Weshallnowshew that,whenR{v)>-^,then (1) J.{z)=f>TmvT)//os(^cos6)sin'^'6dO. Itwasproved byPoissonf that,when 2visapositive integer (zeroin- cluded), theexpressionontherightisasolution ofBessel'sequation;and thisexpressionwasadopted byLommel;]:asthedelinition ofJ^(2)forpositive values ofV+^. Lommel subsequently provedthat thefunction,sodefined,isasolution ofBessel's generalised equation andthat itsatisfies therecurrence formulae of5^3"2;andhethen defined /„(i)forvalues ofvintheintervals{—h,-f),(—f,—#),(-#, -f),...hysuc- cessive applicationsof§3'2(1). Todeduce(1)from thedefinition ofJt,{z) adoptedinthiswork,wetrans- form thegeneralterm oftheseries forJyiz)inthefollowing manner: (-y^(^zy+'"' ^(-yjhzy f^r(i;+-|)r(m +^) m\V{v +m+\)V{v+h)V{hy{'2.my: T{v+m+\) providedthatR(v)> —^. NowwhenR(v)^^,theseries s-l^p'-Hi-ty-in>=\ (2r/i)! ^eoiiverges uniformlywithrespecttotthroughouttheinterval(0,1),andsoit maybeintegrated term-by-term;onaddingtotheresult theterm forwhich ^Pliil. Trans,oftheRoyalSoc. 1844, jj.239. SeealsoaquestionsetintheMathematical Tripos, 1894. fJournal deVEcole B.Pohjtechnique,xii.(cahier 19), (1823), pp.300 etseq.,340 ctseq. Strictly speaking, Poisson shewed that,when 21/isanoddinteger, theexpression ontheright multiplied byijzisasolution oftheequation derived from Bessel's equation bytheappropriate change ofdependent variable. JStudien iiher dieBoi^eVscheu Fuiictioneii(Leipzig, 18G8), jip.1etseq. 48 THEORY OFBESSEL FUNCTIONS [CHAP.Ill m=0,namelyIf-^il—ty^dt,which isconvergent,Avefind that,when Jo '^(">-r(. +i)r(i)Jo' i„^o (2m)! r^' whence theresult stated follows bymakingthesubstitution f=sin-^ and usingthefactthattheintegrandisunaffected bywritingtt—^inplaceof0. When—^<R(v)<h,theanalysis necessarytoestablish the lastequationisalittle more elaborate. Thesimplest procedureseems tobetotake theseries with the firsttwo terms omitted andintegrate byparts,thus (2m)! jo m=2v+h(2m)! J^ m=2 Vi^2^ /^ ;,{i-ty"-Kdtov-^^dti,„=2 (2m)! j onintegrating bypartsasecond time. Theinterchangeoftheorder ofsummation and integrationinthesecond line ofanalysisispermissible onaccount oftheuniformityof convergenceoftheseries. Onadding theintegrals correspondingtothetermsm=0,m=\ (which areconvergent), weobtain thedesired result. Itfollows that,when R{i>)>— ^,then -^^^'^= r(z.l'i)r(|) C^"'*^^~^'''''''^^~ ^^'^"^^^ Obvious transformations ofthis result, inaddition to(1),arethefollowing: (2) J,(z)= p.,^^.^!fpp.-r(1-n-^ cos(zt)dt, ^^^ '^^^^^= r(.+Ijr(i) Wi:'^"^'^""^^"^^^'^^ ^*' (5) •-^-(^)=rA^t)r(i )IJ''cos(^cos^)sin-^d^, (6) J.(^)=^^*^^''g,>cosegin2''^(;?6'. r(/^+i)r(^) Theformula obtained byapartial integrationof(5),namely (7) J,(2)=^ :;^^;;^ YTIsin(2cos(9)sin^''-^5'cos^rf^,1(i;+^)1(^). issometimes useful;itisvalidonlywhenR(v)> ^. 3-31] BESSEL FUNCTIONS 49 Anexpansion involving BernouUianpolyn(juiials hasbeen obtained from(4)byNielsen* with thehelpoftheexpansion inwhich(^„(^)denotes the?ithBernouUianpolynomial anda=izt. [Note. Integralsofthetype (3)were studied before Poisson byPlana, Mem. della R. Accad. delle Sci.diTorino, xxvi.(1821), pp.519—538,andsubsequently byKummer, JournalfurMath. xii.(1834), pp.144—147; Lobatto, JournalfilrMath. xvii.(1837), pp. 363—371; andDuhamel, Cours d'Analyse,li.(Paris, 1840), pp.118—121. Afunction, substantially equivalenttoJ^,(z),defined bytheequation J(jx,x)=1(1- v^)'*cos V.V .do, J WASinvestigated byLommel, Archiv derMath, imdPhys. xxxvii.(1861), pp.349—3G0. Theconvei'se problemofobtainingthedifferentialequation satisfied by z^[^<f''i,v-af-^ iv-^J-^ dv was alsodiscussed byLommel, Archiv derMath,undPhys.xl.(1863), pp.101—126. In connexion with this integralseealsoEuler,Inst. Calc. Int. li.(Petersburg, 1769), §1036, andPetzval, IntegrationderlinearenDijferentialgleichungen (Vienna, 1851), p.48.] 3'31.Inequalitiesderived fromPoisson'sintegral. From§33((3)itfollows that, ifvberealandgreaterthan—i,then (1)I/.(2) I^pj !ff|p(X) j^^exp\I(z)\sin^^edd Byusingtherecurrence formulae§3*2(1)and(4),wededuce inasimilar manner that (3)l-A'(^)K^||l+|^i)|}expr/Wl•(.>-i). Byusingtheexpressionf {2/(7r^)}2cos^for./_j(z)itmaybeshewn that (1)isvalidwhen v=—^. -^heseinequalitiesshould becomparedwith thelessstringent inequalities obtained in§3-13. When viscomplex, inequalitiesofamorecomplicated character canbeobtained inthesame manner, buttheyareofnogreatim- portance. *Math. Ann. lix.(1904), p.108.Thenotation used iuthetext isthatyiveniuModern Analysis, §7'2; Nielseia uses adifferent notation. tThereader should havenodifficultyinverifyingthis result. Aformal proofofamore general theorem willbegivenin§3'4. w.B.F.* 50 THEORY OFBESSEL FUNCTIONS [CHAP.Ill 3-32. Gegenhauers generalisation ofPoisson's integral. Theintegralformula inwhich C^"(t)isthecoefficient ofa"intheexpansionof(1-2at+a^)-"in ascending powersofa,isdue toGegenbauer*;theformula isvalidwhen R(v)>-^andnisanyoftheintegers 0,1,2,... .When n=0,itobviously reduces toPoisson'sintegral. Inthespecialcaseinwhich v=h,theintegralassumes theform * TV (2) Jn+i (z)={-iT(2^j' j ,^'''""''^»(^°^^)^^^^'^^' thisequationhasbeen thesubjectofdetailedstudy byWhittakerf. Toprove Gegenbauer'sformula, wetake Poisson'sintegralintheform andintegratentimes byparts;theresult is J^n {z)- (_2-y.V{v+n-\-\)V a)jJ I dt-\"*'• Now itisknownthat:|:^ Viv+hjri-lv+n)^^~^'^'^" ^^^' whence wehave («) '-*" (-)=^^^ v<^t )f.f('-''>^-^^' ft>"* andGegenbauer'sresult isevident. Asymbolicform ofGegenbauer's equationis thiswasgiven byRayleigh§inthespecialcase v=\. Thereader will find itinstructive toestablish(3)byinduction with theaidofthe recurrence formula (n+1)ci^^(o=(2.+.o «c/(0- (1-^2)^?^. *WienerSitzuvgsherichte, lxyii.(2),(1873), p.203; lxx.(2),(1875), p.15.SeealsoBauer, MilnchenerSitzungsberichte, v.(1875), p.262,and0.A.Smith, Giornale diMat.(2)xii.(1905), pp.365—373. ThefunctionC^" (t)hasbeen extensively studied byGegenbauer inaseries of memoirs intheWieiier Sitzungsberichte ;some ofthemore importantresults obtained byhimare giveninModernAnalysis, §15-8. tProc.London Math. Soc.xxxv.(1903), pp.198-206. See§§6-17, 10-5. tCf.ModernAnalysis, §15'8. §Proc.London Math. Soc. iv.(1873), pp.100, 263. 3-32, 3-33] BESSEL FUNCTIONS 51 Aformula which isakind ofconvex'se of(4),namely* inwhich P~'^denotes ageneralised Legendre function,isdue toFilon, Phil.Mug. (6)vr. (1903), p.198; theproofofthisformula islefttothereader. 3*33. Gecjenhauers doubleintegral ofPoissonstype. Ithasbeenshewn byGegenbaueri* that,whenR{v)>0, (1) ./^(ot)=-^r—T exp[iZcos6-iz(cos6cos+sm6sin6cos \/r)lTTi(Z^;Jo-/'^ -^ where ot-=.^- -\-z^-IZz cos(^andZ,z,^areunrestricted(complex)variables. This result wasoriginallyobtainedbyGegenbauer byapplyingelaborate in- tegraltransformations tocertain addition formulae which willbediscussed in Chapterxi.Itispossible, however, toobtain theformula inaquitenatural manner bymeans oftransformations ofatypeused inthegeometryofthe sphere |. After noticing that,when z—0,theformula reduces toaresult which is anobvious consequenceofPoisson'sintegral, namely JAZ)=^^^ ["e^^co^^sin-^"^. rsin'"-'ylrdylr, TTi{v)J Jo weproceedtoregard yjrand aslongitude andcolatitude ofapointona unitsphere;wedenote thedirection-cosines ofthevector from thecentre to thispoint by(I,m,n)andtheelement ofsurface atthepointbydco. Wethen transform Poisson'sintegral bymakingacyclical interchangeof thecoordinate axes inthefollowing manner§: TTfTT J^(ot)=--T^. I Ie'^cose gin-- sin^"-!^|rd0d^{r "" 0.' w>0 Ql-^l ^-f^V-l (I0)- TtF(v)J.'n^O =^^r\ f"["e'-°^*'"^«°^'^cos-''-i ^sin 0dylrd0. ttI{v)Jo Jo /Mt issupposed that tWiener Sitzuugsberichte, lxxiv.(2),(1877), pp.128—129. JThismethod iseffective inproving numerous formulae ofwhich analytical proofs were given byGegenbauer ;and itseems notunlikely thathediscovered these formulae bythemethod inquestion;cf.§§12-12, 12-14. Thedevice isused byBeltrami, Lombardo licndiconti, (2)xiii. (1880), p.828, forarather different purpose. §Thesymbol JJ,„>omeans thattheintegration extends over thesurface ofthehemisphere on whichmispositive. 4—2 52 TPIEORY OFBESSEL FUNCTIONS [CHAP.Ill NoAv theintegrandisanintegral periodicfunction ofyjr,andsothelimits of integrationwithrespecttoyjrmaybetaken tobeaanda+27r,where aisan arbitrary (complex)number. This follows fromCauchy'stheorem. Wethusget JJru)^-'^'I'^r^ e^'^^'^'^'^^^'^cos-''-^ 6sindd^lrdd 7rr(z^).'o Ja =ii^Z/"" I" e^^sinecosi^+a) cos^--! esinOdyfrdd. 7rr(i/)Jo .'o Wenowdefine abythepairofequations OTcosa=Z—zcoscf),IS-sina=zsincf), sothat Jv{^)= t,/. exp [i{Z—zcos(i>)sindcos-dr- izsin(^sin^sin^] ttI(j/)JoJo cos-"-! ^sin(9c^'v|rcZ^. Theonlydifference between thisformula andtheformula J^(tij-)=^\^Iexp[msin6cosi/r]cos^"-! ^sinOd-^dd isintheform oftheexponentialfactor;andwenowretrace thestepsofthe analysiswith themodified form oftheexponentialfactor. When thestepsare retraced thesuccessive exponentsare i{Z—zcos(j))l—izsin(^.m, i{Z—zcos<f))n—izsin</>.I, i{Z~ zcos<^)cos6—izsin<^cos-y^sin6. The lastexpressionis iZcos6—iz(cos^cos6+sin<^sin6cos>/r), sothattheresult ofretracingthestepsis \, exp\iZcos6—iz(cos (^cos+sin rf)sin6cos\|^)]ttI{v)joJ sin2''-ii/rsin-^''^rf>/rrf^, andconsequently Gegenbauer'sformula isestablished. [Note. Thedevice ofusing transformations ofpolar coordinates, after themanner of this section, toevaluate definiteintegrals seems tobeduetoLegendre, M^m. de I'Acad, des Sci., 1789, p.372,andPoisson,lle'vi. deI'Acad. desSci. iii.(1818), p.126.] 3"4.Theexpression ofJ±(„j^i) {z)infiniteterms. Weshallnowdeduce from Poisson'sintegraltheimportanttheorem that, when Vishalfofanoddinteger,thefunction J^iz)isexpressibleinfiniteterms hyineansofalgebraic andtrigonometrical functions ofz. Itwillappearlater(§4-74) that,when vhasnotsuch avalue, then J^,(z) isnotsoexpressible;butofcourse thisconverse theorem isofamuch more recondite character than thetheorem which isnowabout tobeproved. 3-4] BESSEL FUNCTIONS 53 [Note.Solutions infiniteterms ofdifferential equations associated with«/„i(2)were ob- tained byvariousearlywriters;itwasobserved byEuler, Misc. Taurinensia,iir.(1762— 1765), p.76thatasolution oftheequationfor e"«/^^j (2)isexpressibleinfinite terms; while theequationsatisfied by2- t/j,^.i.(2)wassolved infinite terms byLaplace, Conn, desTerns, 1823[1820], pp.245—257 andMecMnique Celeste,v.(Paris, 1825), pp.82—84;byPlana, J/em. della R.Accad. delle Sci.diTorino,x.x.vi.(1821), pp.533—534;byPaoli, Mem. diMat. e diFis.{Modena),xx.(1828), pp.183—188; andalsobyStokes in1850, Trans. Camh. Phil. Soc. IX.(1856), p.187[Math, andPhys. Papers,II.(1883), p.356]. Theinvestigation which willnowbegivenisbased onthework ofLommel, Sttulien ilber dieBessel'schen Functione7i(Leipzig, 1868), pp.51—56.] Itisconvenient torestrict ntobeapositive integer (zero included), and then,by§3-3(4), _(i^)'*+i n \s/tt. 54 THEORY OFBESSEL FUNCTIONS [CHAP.Ill Inparticularwehave (3) /j{z)= (^—jsm^,J|(^)= (^~j(^^cos 5J; theformer ofthese results isalsoobvious from thepowerseries forJjiz). Again,from therecurrence formula wehave andhence, from(1), .^njr'^A?i+ry^_ _.^4(-^•)'-'^ (7?+r)! ^ ,^0r!(n-r)!(2^)'-^ ,rorl{n-r)l(22y But, obviously, byinduction wecanexpress \zdzj z asapolynomialin1/zmultiplied bye*'^,and sowemust have ±,..^(±iY-''(n +r)\ ,,„„^,fd\»e±''2—12 z for, ifnot,thepreceding identitywould lead toaresult oftheform e'^<j),(z)-e-''4>o{z)=0, Avhere^j{z)and^o(•2)arepolynomialsinIfz;andsuchanidentityisobviously impossible*. Hence itfollows thatf It'-^.Q^ +r)!.^-(-tr-"-(n +r)! r=or!(«-r):(22)'- ,"0r!(n-r)!(25r)'- Consequently (4)J^n-l{z)=^ olZ= {-T{2'jryz-^^(^^^\z-iJ_,{z)] ={-y(2'jTz)KI_„_,(z). ^^'+'^(«+/); ,.»(-iy+'\{n +r)l' V(27r^) L,=0r !(n-r):(2^)'-^ ,.ror!(?i-r)!{2z) *Cf.Hobson, Squaring theCircle (Cambridge, 1913), p.51. tFrom theseries -i^'r(i),„tom!i.f...(m-i)' itisobvious that J,lz)=(— )cos z.-i\TrzJ 3-41] BESSEL FUNCTIONS 55 andhence (5) J_,,_i {z)= kTTZI lo(•2r)!(n-2r)!(2^ —sin(^+^/«7r)iV/\ r.=0 (2r+1)!(rt-2r-1)!{2zy^^\ .Inparticular, wehave (6)J_i(.)=(-Vcos^, ./_3,(^)=(iy(-^^^_sin^\7r^/ \7rzJ \ z Wehavenowexpressedinfinite termsanyBessel function, whose order is halfofanoddinteger, bymeans ofalgebraic andtrigonometricalfunctions. Theexplicit expressionofanumber ofthese functions canbewritten down from numerical results contained inaletter fromHermite toGordan, Journal furMath. Lxvi. (1873), pp.303—311. 3*41. Notations forfunctionswhose order ishalfofanoddinteger. Functions ofthetypes J±(n+h(z) occur with suchfrequencyinvarious branches ofMathematicalPhysicsthatvarious writers have found itdesirable todenote thembyaspecialfunctionalsymbol. Unfortunatelynocommon notation hasbeenagreed uponandnone ofthemany existingnotations can besaidtopredominateovertheothers.Consequently, apartfromthesummary which willnowbegiven,thenotations inquestionwillnotbeused inthiswork. Inhisresearches onvibrating sphei-es surrounded byagas,Stokes, Fhil. Trans,ofthe RoyalSac. CLViii.(1868), p.451[^Math. andPhys. Papers,iv.(1904), p.306],made useof theseries "*"2.imr 2.4.(mr)2+••' which isannihilated bytheopei'ator d'^ ^.d?i(n+l) dr^ dr ;-^ This series Stokes denoted bythesymbol f^(?•)andhewrote rv/.„=>S'„e- i'"'7„ (r)+>S';e"'"'/;, (- r), whereS,^and;SV/arezonal surftxce harmonics;sothat\/^„isannihilated bythetotal operator andbythepartial operatordr' 2d„n(n->r\) dr-' rdr ;- c2 2a 1 { ..7)} Inthisnotation Stokes wasfollowed byRayleigh,Proc.London Math. Soc. iv.(1873), pp.93—103, 253—283, andagainProc. RoyalSoc. LXXii. (19Q3), pp.40—41[Sckmtifc Papers,v.(1912), pp.112—114], apart from thecomparativelytrivial change thatRuyleigh would have written/„(ijnr)where Stokes wrotef^(r). 56 THEORY OFBESSEL FUNCTIONS [CHAP.Ill luorder toobtain asolution finite attheorigin, Rayleighfound itnecessarytotake Sn=(-)"*! Sninthecourse ofhisanalysis, andthen >/'„=(-0"^^m,SJ—jJ,,+^(mr). Itfollows from§3-4that^ "'^/'''^=(i^^"-twrj IXp—ir 5 andthatJ^^^(r)=-j^^[e'^^i"+1/„(ir)+e«- ^•-»"i/„ (- ir)]. Inorder tohaveasimjilenotation forthecombinations ofthetypes e+''"/„ (±t>)which arerequiredforsolutions finite attheorigin. Lamb found itconvenient towrite 2{271+3)2.4.(2?i +3)(2?t+5) inhisearlier papers, Froc. London Math. Soc. xiii. (1882), pp.51—66;189—212; xv. (1884), pp.139—149; xvi.(1885), pp.27—43;Phil. Trans, oftheRoyalSoc.CLXXiv. (1883), pp.519—549;andhewasfollowed byEayleigh, Proc. RoyalSoc. lxxvii. A,(1906), pp.486—499[Scientific Papers,v.(1912), pp.300—312], andbyLove*, Proc. London Math. Soc.xxx. (1899), pp.308—321. With thisnotation itisevident that T{n+%) f\ r\n, oK f» <-,\f'^Y^^^^ ^^»(^)= -^T^i-'^„+j(^)=(-)"i-3.5...(2,. + l).(^_^J^. Subsequently, however. Lamb found itconvenient tomodifythisnotation, andaccord- inglyinhistreatise onHydrodjaiamics andalsoProc. London Math. Soc.xxxii.(1901), pp.11—20, 120—150 heused thenotationf ^"^^^^1.3.5... (2n+l)L^~2(2?i +3)'''2.4(2?i +3)(2w+5)~ ""J' (d\"e~*- zdzjz SOthat ^„iz)= -^^^-— ,^„(2)=,7+r^' whileEayleigh,Phil. Trans, oftheRoyalSoc. CCIII. A,(1904), pp.87—110[Scientific Papers, V.(1912) pp.149—161jfound itconvenient torejjlacethesymbol f,i{z)byXn{^)- Love, Phil. Trans,oftheRoyalSoc.ccxv. A,(1915), p.112omitted thefactor(—)"andwrote whileyetanother notation hasbeenusedbySommerfeld, Ann. derPhysikiindChemie, (4) XXVIII.(1909), pp.665—736,andtwoofhispupils, namely March, Aim. derPhysik mid Chemie, (4)xxxvii.(1912), p.29andRybczyiiski, Ann. derPhysik undChemie, (4)XLI. (1913), p.191;thisnotation is V.„(2)= (Wj,+.(.)=.-i(^-^-^J— , Cn{^)={hTrZ)^ [J,,+,{^)+(-)»iJ_,_,(Z)], and itiscertainly thebestadaptedfortheinvestigation onelectric waves which wasthe subjectoftheir researches. (d\"e~^^ —J- )— ,but,as stated, hemodified thedefinition inbislater work. tThis isnearer thenotation usedbyHeine, Handbuch derKugelfimctionen,i.(Berlin, 1878), p.82;except thatHeine definedi^,j(z)tobetwice theexpression ontheright inhistreatise, but notinhismemoir, Journal fiirMath. lxix.(1869), pp.128—141. 3-5] BESSEL FUNCTIONS 57 Sommerfeld's notation isaslightly modified form ofthenotation usedbyL.Lorenz, who used Vnand v„+{-)"nv„inplaceof\/r„andf,, ;seehismemoir onreflexion andrefraction oflight, K.Danske Videnshahernes SelskahsSkrifter, (6)vi.(1890), [Oeuvres scientijiques,i. (1898), pp.405—502.] 3*5.Asecond solutionofBesseVsequation forfunctions ofintegralorder. Ithasbeen seen(§3']2)that,whenever visnotaninteger,afundamental systemofsolutions ofBessel'sequationforfunctiong^jof order visformedby thepairoffunctions ./^{z)andJ_^ {z).When visaninteger (=n),this isno longerthecase,onaccount oftherelation J_n{z)=(—)'*J^{z). Itisthereforenecessarytoobtain asolution ofBessel'sequation which is linearly independentofJ„,{z);andthecombination ofthissolution withJn{z) willgiveafundamentalsystemofsolutions. The solution which willnowbeconstructed wasobtainedbyHankel*; the fulldetails oftheanalysisinvolved intheconstruction were firstpublished byBocherf. Analternative method ofconstructing Hankel's solution wasdiscovered byForsj'th; hisjjrocedureisbased onthegeneral method ofFrobenius, JournalfiirMath, lxxvi.(1874), pp.214—235, fordealing withanylinear difterentialequation. Forsyth's solution was contained inhislectures ondifferential equations delivered inCambridgein1894,and it hassince beenpublishedinhisTheory ofDifferential Equations,iv.(Cambridge, 1902), pp.101—102,and inhisTreatise onDifferential Equations (London, 1903and1914), Chajjtervi.note 1. Itisevident that, ifvbeunrestricted, and ifnbeanyinteger (positive, negativeorzero), thefunction J,.{z)-{-YJ_^{z) isasolution ofBessel'sequationforfunctions oforder v;and thisfunction vanishes when v=n. Consequently,solongas i^^n,thefunction J,{z)-{-TJ_Az) V—n isalsoasolution ofBessel'sequationforfunctions oforder v;andthisfunction assumes anundetermined form| when v=n. Weshallnowevaluate^li„/,W-(-)»/-.(^)_ r-*n V-n andweshallshew that itisasolution ofBessel'sequationforfunctions of *Math. Ann. r.(1869), pp469—472. tAnnals ofMath. vi.(1892), pp.85—90. SeealsoNiemoller, Zeitschrift filrMath, undPhys. XXV. (1880), pp.(;5-71. XTheessence ofHankel's investigation istheconstruction ofanexpressionwhich satisfies theequation when pisnotaninteger, which assumes anundetermined formwhen visequalto theinteger nandwhich hasalimitwhen v-*-n. 58 THEORY OFBESSEL FUNCTIONS [chap.Ill order nandthat itislinearly independentofJn(2) ;sothat itmaybetaken tobethesecond solutionrequired*. Itisevident that V—n dv^^dv asv-^n, since both ofthedifferential coefficients existf. Hence Mz)-{-YJ..{z)lim V—n exists; itiscalled aBessel functionofthesecond kind oforder n. Todistinguishitfrom other functions which arealsocalled functions of thesecond kind itmaybedescribed asHankeVsfunction. Following Hankel, weshalldenote itbythesymbol J"^ni^)sothat (1) and also (2)Y„{z)=limn Y,(^)dJ,{z)_,_y,dJ-,{zy (/=?l dv^^dv Ithasnow tobeshewn that Y,;(z)isasolution ofBessel'sequation. Since thetwofunctions J±„(z)areanalyticfunctions ofboth zand v,the order ofperforming partialdifferentiations onJ^.^(z)withrespecttozand v isamatter ofindifference§.Hence theresult ofdifferentiatingthepairof equations withrespecttovmaybewritten zJ-;--o 1-^3 ^ h{z-—V-)—;r- IvJ±^(z)=U. dz^ dv dz dv^dv^^v / When wecombine theresults contained inthisformula, wefindthat dJA^) _^_y^^^J-A^ydv dv=2v{jA^)-{-rJ-A^)], *Thereader willrealise that, given asolution ofadifferential equation,itisnotobvious that alimiting form ofthissolution isasolution ofthecorresponding limiting form oftheequation. +See§3'1. Itisconventional towrite differentiations \Yith respecttozastotal differential coefficients while differentiations withrespecttovarewritten aspartialdifferential coefficients. Ofcourse, inmany parts ofthetheorj', variations invarenotcontemplated. XThesymbol r,j(2), which wasactually usedbyHankel,isused inthiswork todenote a function equal toI/tttimes Hankel's function(§3-54). §See, e.g.Hobson, FunctionsofaReal Variable(1921), §§312, 313. 351] BESSEL FUNCTIONS 30 sothat, AG OS) . Od, (2) ad, (2) +2r[d,(2)—(=)*I-g(2) Nowmakeyn,Alltheexpressions inthelastequationareZeontinuons functions ofv,and sowehave 9.[Op Om, where »istobemade equal toximmediately after thedifferentiations with respect to»have been performed, Wehave therefore proved that 8) Ta¥a(2)=0, 40that ¥4(z) is«solution ofBessel’s equation forsunctions oforder n Itistobenoticed that 4) inHOEP*Se) hin 2A, pe =RER whence followsaresultsubstantially duetoLommel*, @ ¥A) =(Ete) Again, ad(2 ‘@d_,(2) while,becauseJ;(2)isamonogenic functionof»atv=0,wehave Wane] _fee] __ foue(2)]BySame=lacy = ce and hence itfollows that Be (3) xonal Se Arestltequivalent tothis wasgiven byDuhamel as entlyas1840, BBL. Theexpansion of¥o(z) inanascending series Before considering theexpansion ofthegeneral fisnetion ¥, (2),itiscon vGhient toexamine thefunction oforder zero because theanalysis issimpler and theresulting expansion ismore compact. We nse the formata just obtained yaof? {ge (omaer™ 7] 1Coursmage, (Pasi, 184), pp12212 Digitized by Microsoft 60 THEORY OFBESSEL FUNCTIONS andtheresult ofterm-by-termdifferentiation is[chap,iti y(^)=2 =22 m=0V \)y%^)^log(1^)_flog r(l.+7/1+1) .-= {m !)2{logi^-l/rO»^ +l)}, wheret/tdenotes, asiscustomary,thelogarithmicderivate oftheGamma- function *. Since0<V' ('"+!)<™whenm=\, 2,3,...theconvergenceoftheseries forYo(2) maybeestabhshed bjusing D'Alembert's ratio-test fortheseries inwhich\//-(m-|-l)is replaced bym.Theconvergenceisalsoanimmediate consequenceofageneral theorem concerning analyticfunctions. SeeModernAnalysis, §5"3. Thefollowingforms oftheexpansionaretobenoticed : (1)Yo(^)=2 S^ \^]J {iog(i^)-^(,,,+ l)}, (2) Yo(^)=2log(i.)../o-(^)- S/\'\J ^{m+1)m=0 V''') (8)Yo(z)=2h+log(1.)} /o(^)-2J^^-y^T^ |i+2++1) mI- Thereader willobserve that 1Yo(z)+(log2-7)/o(^) isasolution ofBessel's equationforfunctions oforder zero.Theexpansionof thisfunction is m=l m=0 VW!)»(-)-(|^n1^ ^-"1+2'^ (m\yrii\ This function wasadoptedasthecanonical function ofthesecond kind oforder zeroby Neumann, Theorie clerBessel'sche?i Functionen(Leipzig, 1867), pp.42—44;see§3'57. Buttheseries wasobtained asasolution ofBessel's equation, long before, byEulerf. Euler's result inhisownnotation isthatthegeneral solution oftheequation xxddy+xdxdy+gx'^ycx^ =0 IS2Aq 6Ag'-„ ''n? 1.8?i5 9'22Ag^ 1.8. 27w^,.3«_100.4/ 1.8.27 .64?i9.t'4»+etc. nn 1 .4?r,,.2)1_ 9' 1.4. 9?i«a;3»+9^ +a—'^.3?"+--^^^ .r-"—«P'' 7l?l 1.4h^" 1.4. %i^.i;3"+1.4.9. 16?i8.r*"-etc. Ua' 1.4.9. 16?i8^«_etc., ModernAnalysis, Ch. xii. Itistoberemembered that,whenmisapositive integer, then ^(1): 7,^(m+l)=-+-+...+--Y, where 7denotes Euler's constant, 0-57721.57 .... tInst. Calc. Int. 11.(Petersburg, 1769), §977, pp.2.33—235. SeealsoActa Acad.Petrop.v. (1781) [published 1781], parsi.Mathematica, pp.186—190. 3-52] BESSEL FUNCTIONS 61 wliere Aandaarearbitraryconstants. Hegavethefollowing lawtodetermine successive numerators inthe first line : 6=3.2-1.0, 22=5.6-4.2, 100=7.22-9.6, 548=9 .100-16 .22, 3528=11 .548-25 .100etc. If2(1+^+...+!)= '^-^,\12 7nJml thislaw isevidently expressed bytheformula O-m+1=(2'«+1)a-m-rii-CTm-1• 3*52. Theexpansion ofYn(^)inanascendingseries and thedefinition of Weshallnowobtain Hankel's*expansionofthemoregeneralfunction Y,i(^), where nisanypositive integer. [Cf.equation (4)of§3'5.] Itisclear that dv „Zo^v[m!r(i'+m+1) =X(_)".( 1^).+. ^om.^.r{v+III+I)^^^^ '> when v-^n, where nisapositive integer.That istosay 'dJA2) (1)cv,^)r/^Vi-f'{hzY+-'''{\1 1 Theevaluation of\dJ_Az)ldv\=nisalittlemore tedious because ofthepole ofi/r(-I'+m+1)atV=nintheterms forwhich iii=0,l,% ...,n-\. We break theseries forJ_^{z)intotwoparts,thus low!r{-v+m+l) ;„=H7/i! r(-:/+m+l)' andintheformerpartwereplace 1 ,r(v—in)sin(v—m)TT 1)^^TT T(-v+m+l) Now,when ^m<n, ~d((|^)-''+-'» r(:^-7/0sin(v-m)ir ]' dv\TTJJf=' =[{^z)-''+'"'r(v-m) {tt-i yfr(v— III)sin(i/-m)7r +cos(v—m)tt-tt-^log(^z)sin(i/-?«)7r}]„=„ =(l^r)-«+27n r(w-m)cos(/I- III)TT. *3/ai/t. Anil. i.(1869), p.471. 62 THEORY OFBESSEL FUNCTIONS [chap.Ill Hence dv=2n-i(-)nr^n-m)(^z)—n+2m m +S\.//l: ,_.,{-loga^)+>/r(-7?+m+1)1, that istosaym=nw^!(—w+m)! (2), az; »(= x[log(-|-^)-A|r(»i+l)}, whenwereplacemby??+?7iinthesecond series. Oncombining (1)and(2)wehave Hankel's formula, namely 00(\m(\^\n+im, m'' ',«=oml{n+7n)\ m= X{2log{^z)-^lr{7n+l)-ylr (n+111+1)} =2{7+log(1^)1 /,,(^)-(1z)-f^^^~^^;r^^'(i^)"" - (-)'"(l^)»+^^ ^il1,111,1 ] Inthe firstterm{m=0)ofthelastsummation, theexpressionin {|is 11 1-+-+...+-. 12 n Itisfrequentlyconvenient(following Lommel*) towrite (4) sothat (5)%{^)-dv '^"^^^nZ^miriWm^l)-J^{z)\ogz, n {log2+'v/r(j;4-??l +l)} when Visanegative integer, ^^(z)isdefined bythelimit oftheexpression ontheright. Wethushave (6) Y„(z)=2J,(z)]ogz+%,{z)+{-r3-n {z). Thecomplete solution ofx -j-^^+ay=wasgivenintheform ofaseries(partofwhich contained alogarithmic factor) byEuler,Inst. Cede. Int. li.(Petersburg, 1769), §§935, 936;solutions ofthisequationare x^Ji(2ai^2)j ^4Yi(2aia;i). Euler alsogave {ibid. §§937,938)thecompletesolution oixi-j^-\-ay=Q);solutions of thisequation are x^Ji(4a2 ^i),.^iY2(4a* x^). *Studien ilber dieBesseVsehen Functionen (Leipzig, 1868), p.77. 3-53, 3-54] BESSEL FUNCTIONS 63 3-53. Thedefinition of^,(z). Hitherto thefunction ofthesecond kind hasbeen definedonlywhen its order isaninteger.The definition which wasadopted byHankel* forun- restricted values ofv(integralvalues of2vexcepted)is (1)' " sin2v7r This definition failsbothwhen visaninteger andwhen vishalf ofan oddinteger,because ofthevanishingofsin2p7r.The failure iscompletein thelatter case;but, intheformer case, thefunction isdefinedbythelimit oftheexpressionontherightand itiseasytoreconcile thisdefinition with thedefinition of§3*5. Toprovethisstatement, observe that Tre""^p-n J^{z)cosvtt—J_^(z)'limY^(3) lim ={-)"limcosVTTsni I'lr v—n J^,{z)cosVJT—J^y{£)' v—n (—)"cosvw—\ J^z)=Y,,[z)+lim =Y„(z), andsowehaveprovedthat (2) limY.(^)=Y,(4 Itisnowevident thatY^(^), defined eitherby(1)orbythelimitingform ofthatequation,isasolution ofBessel'sequationforfunctions oforder vboth when(i)vhasanyvalue forwhich 2visnotaninteger,andwhen(ii)visan integer:thelatter result follows fromequation (2)combined with§3*5(3). Thefunction Y,,(^),defined inthisway,iscalled aBessel functionofthe second kind(ofHankel'stype)oforder v;andthedefinition failsonlywhen v+^isaninteger. Note. Thereader should becareful toobserve that,inspiteofthechangeofform, the function Y^ (2),quafunction ofv,iscontinuous at i'=n,except when ziszero; and,in fact, Jt,{z) andY^(2) approachtheir limits /„(2)andY„ (2),asv^^n, uniformly with respectto2,exceptintheneighbourhood of2=0,where nisanyinteger, positiveornegative. 3*54. TheWeber-Schldfi function ofthesecond kind. The definition ofthefunction ofthesecond kind which wasgiven by Hankel(§3"5o) wasmodifiedslightly byWeber i'andSchlafii^'inorder to avoid theinconveniencesproduced bythefailure ofthedefinition when the order ofthefunction ishalfofanoddinteger. *Math. Ann. i.(1869), p.472. ^Journal furMath, lxxvi.(1873), p.9;Matli. Ann. vi.(1873), p.148. These papersare dated Sept. 1872and Oct.1872 respectively. Inapaperwritten afewmonths before these, JournalfilrMath. lxxv.(1873), pp.75—10-5, datedMay 1872, Weber hadusedNeumann's function ofthesecond kind (see §§3'57, 3-58). XAnn. (UMat.(2)vii.(1875), p.17;thispaperisdated Oct. 4,1872. 64 THEORY orBESSEL FUNCTIONS [CHAP.Ill Thefunction which wasadopted byWeber asthecanonical function ofthe second kind isexpressibleinterms offunctions ofthefirstkindbytheformula* J^{z)cosVTT—J_y(z) sinviT (orthelimit ofthis,when visaninteger). Schlafli, however, inserted afactor^tt;andhedenoted hisfunction by thesymbol K,sothat,with hisdefinition, ^^^^^J.^(^)cosz^^-J^)_smVTT Subsequent writers, however, haveusuallyomitted this factor^tt, e.g.Graf andGubler intheir treatisef,andalsoNielsen, sothatthese writers work with Weber's function. ThesymbolKis,however, usedlargelyinthiscountry, especially by Physicists,todenote acompletelydifferenttypeofBessel function(§3"7), andsoitisadvisable touseadifferent notation. Theprocedurewhich seems toproduceleast confusion istousethesymbol Y^,{z)todenote Weber sfunction, after themanner ofNielsenJ, and toadoptthis asthecanonical function of thesecond kind, save inrare instances when theuseofHankel's function of integialorder saves theinsertion ofthenumber irincertain formulae. Wethushave (1) n{z)=JA^)^o^v'--J-A^) ^cos^^^^ ^^smyTT Tre""^'^^ (2) F,.{z)=limJ^i^)^^^^'^-J-^i^) ^1Y„(.). [Note.Schlafli's function hasbeen used byBocher, Annals ofMath. vi.(1892), pp.85—90, andbyMcMabon, AnnalsofMath. viii.(1894), pp.57—61; ix.(1895), pp.23—30.Sehafheitlin andHeaviside useWeber's function with thesignchanged,so that thefunction which we(with Nielsen) denote byY^{z)iswritten as -.Yy(z)by Sehafheitlin§and(whenv=7i)as-Gn(z)byHeaviside||. Gray andMathews sometimesH useWeber'sfunction, andthey denote itbythe symbol Y„. *Weber's definition wasbyanintegral (see §6-1)which isequal tothisexpression ;the expression (with thefactor hwinserted) wasactually given bySchlafli. +EinleituiujindieTheorie derBeHsel'schen Funktionen,i.(Bern, 1898), p.Metseq. %Nielsen, asinthecase ofotherfunctions, writes thenumberindicating theorder asan index, thusY"[z),Handbuch derTheorie derCylinderfunktionen (Leipzig, 1904), p.11.There areobvious objections tosuch anotation, andwereserve itfortheobsolete function usedby Neumann(§3-58). §See, e.g.Journal furMath. cxiv.(1895), pp.31— 44,andother papers; alsoDieTheorie der Bessel'schen Funktionen(Leipzig, 1908). IIProc. RoyalHoc. liv.(1893), p.138,andElectroma<inetic. Theory,ii.(London, 1899), p.255; achange insignhasbeenmade from hisElectricalPapers,ii.(London, 1892), p.445. ^]ATreati'^e onBessel Functions (Loudon, 1895), pp.65—66. 3-55] BESSEL FUNCTIONS 65 Lomrael, inhislaterwork, usedNeumann's function ofthesecond kind(see §3'o7), but inhisStudien iiher dieBesseVsclien Functi<)7ien(Leipzig, 1868), pp.85—86,heused the function l-rrYn{z)+{^l^{n+^) +log2}Jn(z), where F„(z)isthefunction ofWeber. Onedisadvantageofthisfunction isthat the presenceoftheterm\|/-(«+i)makes therecurrence formulae forthefunction much more complicated; seeJulius, Archives iVeerlandaises, x.xviii.(1895), pp.221—225, inthis connexion.] 3"55. Heine's definition ofthefunction ofthesecond kind. Thedefinitiongiven byHeine* ofthefunction ofthesecond kindpossesses someadvantagesfrom theaspectofthetheoryofLegendre functions;it enables certaingeneralisationsofMehler's formula(§o'71), namely KmP„(cosdin)=/„{O), tobeexpressedinacompactform. The function, which Heine denotedby thesymbol Kn{z),isexpressibleinterms ofthecanonical functions, and itis equalto—\irYn{z) andto—|Y„(2);thefunctionconsequentlydiffersonly insignfrom thefunctionoriginallyusedbySchlafli. TheuseofHeine's function seems tohave diedoutontheContinent many years ago ; thefunction wasoccasionally usedbyGrayandMathews intheir treatiset,andtheyterm itGn{z).Inthisform thefunction hasbeenextensivelytabulated firstbyAldisJ and Airey§,andsubsequentlyinBritish AssociationBeports, 1913, 1914and1916. This revival oftheuseofHeine's function seemsdistinctly unfortunate, bothonaccount oftheexisting multiplicityoffunctions ofthesecond kindandalsoonaccount ofthefact (whichwillbecome more apparentinChaptersviandvii)that therelations between the functions ./„(z)andVu(z)present many pointsofresemblance totherelations between the cosine andsine; sothattheadoption |jofJn{z) and6-'„(2)ascanonical functions iscom- parabletotheuseofcoszand-^ttsinzascanoniciil functions. Itmust alsobepointed outthatthesymbol G^{z)hasbeenused insenses other than thatjustexplained byatleast twowriters, namely Heaviside, Proc. RoyalSoc. Liv.(1893), p.138(aswasstated in§3-54), andDougall,Proc. EdinhurykMath. Soc. xviii. (1900), p.36. Note. Anerror insignonp.245ofHeine's treatise hasbeen pointed outbyMorton, Nature,lxiii. (1901), p.29;theerror isequivalenttoachangeinthesignofyinformula §3"51(3)supra.Itwasalsostated byMorton that thiserrorhadapparently beencopied byvarious other writers, including (ashadbeenpreviouslynoticed byGrayH)J.J.Thomson, Rec&iit Researches inElectricity andMagnetism (Oxford, 1893), p.263.Afurther error *Haiidbuch derKugelfanctionen,i.(Berlin, 1878), pp.185—248. t.iTreatise onBessel Functions (London, 1895), pp.91,147,•242. JProc.RoijulSoc. lxvi. (1900), pp.32—43. §Phil.Mag. (6)xxii.(1911), pp.658—663. IIFrom thehistorical pointofview there issomething tobesaid forusing Hankel's function, andalso forusiug Neumann's function;butHeine's function, being moremodern than either, hasnoteven thisinitsfavour. 11Nature, xlix. (1894), p.359. W.B.F.5 66 THEORY OFBESSEL FUNCTIONS [CHAP.Ill noticed byMorton inThomson's workseems tobeduetoamost confusing notation employed hjHeine;foronp.245ofhistreatise Heine nsesthesymbol Kotodenote thefunction called -^TTJ'ointhiswork, while onp.248thesamesymbol Kqdenotes -W(J^o- ^'^'o)- 3-56. Recurrence formulae forV^(2)andY^{2). Therecurrence formulae which aresatisfied byY^(2)areofthesame form asthose which aresatisfied byJ^(z) ;theyareconsequentlyasfollows : (1)•Y,_,(2)+Y^^,(2)=^Y,{2), (2) F,_,(^)-F,+,(^)=2IV(^), (3) 2Y:{2)^vY,{2)=2\\_,{2), (4) zY:{2)-vY,{z)=-zY,.^,{z\ andinthese formulae thefunction Ymaybereplaced throughout bythe function Y. Toprove themwetake§3'2(3)and(4)intheforms ^[z^J",{z)]=z^/,_! (^),^[z^/_,{z)\=-z-/_,+! {z) ; ifwemultiplythesebycotvirandcosec vrr,andthen subtract, wehave d dz[z^F,{z)]=z"F,_, (2), whence(3)follows atonce.Equation (4)isderived inasimilar manner from theformulae ^{z-"J,(z)]=-z-"./,+! (z),j^[z-^/_, {z)]=?-"./_,_, {z). Byaddition andsubtraction of(3)and(4)weobtain(2)and(1). Theformulae are,sofar,provedonthehypothesisthat visnotaninteger; butsince F^{z)and itsderivatives arecontinuous functions ofv,theresult of proceedingtothelimitwhen vtends toanintegralvalue n,issimplyto replacevbyn. Again,theeffect ofmultiplyingthefourequations by7re'"'''sec vir,which isequalto7re'''='=^'"^ sec{v±l)7r,istoreplacethefunctions Fbythefunctions Ythroughout. Intliecase offunctions ofintegral order, these formulae weregiven byLommel, Studien iiher dieBesseVschen Fimctionen(Leipzig, 1868), p.87.Thereader will find it instructive toestablish them forsuch functionsdirectlyfrom theseries of§3"52. Neumann'sinvestigation connected with theformula(4)willbediscussed in§3-58. 3-56, 3-57] BESSEL FUNCTIONS 67 3"57.Neumann sfunction ofthesecond kind. Thefunction whichNeumann*adoptedasthecanonical function ofthe second kindpossessestheadvantagethat itisrepresented moresimply by integralsofPoisson'stypethan thefunctions ofthesecond kindwhich have been hitherto discussed ;butthis isitsonlymerit. We firstdefine thefunctionoforderzerof, which willbecalled F'"'(z). Thesecond solution ofBessel'sequationforfunctions oforder zerobeing known tocontainlogarithms, Neumann assumed asasolution theexpression Jo(Z)logZ+IU, wherewisafunction ofztobedetermined. IfthisexpressionistobeannihilatedbyV,,,wemust have VoW=-Vo{Jo(z)logz] =-2zJo'{z). But,by§2-12 (11), -2zJ,'(z)=2zJ,(^)=8i(-)»-'nJ,„{z); «=i and so,since Vq.72,^(2)=4/i-Jo,i(^), wehave V,w=2X(-r-'V,J,,,(z)/n -''11=1 =2V,S(-)--J",„(^)/r?.; M=l thechangeoftheorder oftheoperations SandVoiseasily justified. Hence apossiblevalue forivis 2ii-f-^ J,n{z)ln, u=l andtherefore Neumann's function F'"'(z),definedbytheequation (1)F(«'{z)=Jo{z)logz+2i(-)"-!^^^ , n=1 n isasolation ofBessel'sequationforfunctions oforder zero. Sincew-^as^^0,(theseries forwbeingananalyticfunction ofznear theorigin),itisevident that Jo(^)and F*"'{z)form afundamentalsystemof solutions, andhence Yo{z)isexpressibleasalinear combination ofJo{z)and V'^^iz); acomparisonofthebehaviours ofthethree functions near theorigin shews thattherelationconnectingthem is (2). F<»'{z)=1Yo{z)+(log2-7)Jo{z). *Theorie derBesseVsclien Functionen (Leipzig, 18G7), pp.42—44. Neumauu calls thisfunction Bessel's associatedfunction, andhedescribes another function, 0,^(z),asthefunction ofthesecond kind(§9-1). But,because 0„(z)isnotasolution ofBessel's equation, this descriptionisun- desirable and ithasnotsurvived. tNeumann's function isdistinguished from theWeber-Schladi function bythepositionofthe suffix which indicates theorder. 68 THEORY OFBESSEL FUNCTIONS [CHAP.Ill 3-571. Theintegral ofPoissonstypefor7'°'(z). Itwasshewn byPoisson* that gixcosa, ^Qg (a;sin-co)do) Jo isasolution ofBessel's equationforfunctions oforder zeroandargumentx; andsubsequent!}^Stokes obtained anexpressionoftheintegralintheform of anascendingseries (see §3'572). Theassociated integral - cos(zsin6).log(4^^cos^6)dO wasidentified byNeumannf with thefunction F<"'(z);andtheanalysis by Avhich heobtained thisresult isofsufficient interest tobegiven here, with someslightmodifications inmatters ofdetail. From§2'2(9)wehave n nirJo and so,ifAveassume that theorder ofsummation andintegrationcanbe changed,wededuce that »(-)»/,« (-2)4/•*- . .»cos2/?6' 22^~^-^=-cos{zcos0)z n=l n TTJ n=lde cos{zcos6).log(4sin^6)dO; TT./9rh from thisresult combined with Parseval'sintegral (§22)andthedefinition of F'"'{z),weatonce obtain theformula (1) F""{z)=-['"cos {zcos6).log{izsin-6)dd, fromwhich Neumann's result isobvious. Thechangeoftheorder ofsummation andintegration hasnow tobeexamined, because 2?i~i cos27i^ isnon-uniformly convergent near ^=0.Toovercome thisdifficulty weobserve that, since2(-)"Jin{^)lnisconvergent,itfollows from Abel's theorem%that 2(-)"^2«(2)/'i= lim 2(-)»a" J2«(2)/«= hm-2j'" cos{zcos6)'1=1 a-*-l-0 n=l a-».l-0 TTn=ly,a»cos2n^ ,. ctd. n=l *Journal de.VEcole B.Polytechnique,xii.(cahier 19), (1823), p.476.The solution ofan associatedpartial differential equation hadbeen given earlier{ihid. p.227). SeealsoDuhamel, Cours d'Analyse,ii.(Paris, 1840), pp.122—124, and Spitzer, Zeitschrift furMath, undFhys.ii. (1857), pp.16.5—170. tTheorie derBesseVsr.hen Functionen(Leipzig, 1867), pp.45—±9.SeealsoNiemoller, Zeit- schrift fiirMath, imdFhys. xxv.(1880), pp.65—71. :J:Cf.Bromwich, Theory ofInfinite Series, §-51. 3-571, 3-572] BESSEL FUNCTIONS 69 Now, since aislessthan1,2(a"cos2n6)/n does converge uniformly throughout the range ofintegration (bycomparison with2a"),andsotheinterchangeispermissible ;that istosay 2S /'*'" / /iN«"cos 2;i(9,-2fk" , ^«a»cos2?i(9,„ -2/cos(2cos5) d6—-Icos(scos^)2 doTn=iy ^ '^^ n=i n 1 /If=--Icos(scos^)log('l -2acos2^-|-a2)c/(9. Hence wehave 2^_^^ 22LW^_ijjj^_- ^.^,g (_^jQg^^log(1-2acos2(9+a'-i)C^5. Wenowproceedtoshewthat* lim I" cos(scos^){log(l -2aco.s2^ +a2)-log(4asin-^)}(;(9 =0. a-*'l-0.' Itisevident that 1-2acos 26l+a2-4asin2 ^=(1-af^O, andso log(1-2acos2^-j-a-)^log(4asin^6). Hence,ifAbetheupper bound tof jcos{zcos6)\when0^6^^tt,wehave Icos(sCOS^){log (1-2acos -2(9+a2)-log(4asin'^6)]d6 JQ ^.4 I{log (1-2acos26+a'-)-log(4asin^6)]d6 J /An-f»a»cos2/(^ "1 -1.22"-'^— -^—+log (1/a)-2log(2sind)\dd I"=l ^^) =*7rJlog(l/a), term-by-term integration being permissiblesincea< 1.Hence, whena< 1, \' cos{zCOS6){log (1-2acos26+a^)-log(4osin'-6)\d6 Jo asa-»-l-0;andthis istheresult tobeproved. Consequently i(-)"'^2''("^) =_lini1[-"cos(2COS^).log(4asin2^)c7^ ^' ''cos(2cos(9).log(4sin2^)rf^.<i7r/llog(l/a)^0, TT,,^ andtheinterchangeisfinally justified. Thereader willfind itinterestingtodeduce thisresult from Poisson's integralforJ^(z) combined with§3"5(5). 3-572. Stoles' seriesforthePoisson-Neumann integral. d^y \dy .,^ ,The differential equationconsidered byStokes+m1850was^+-^-m-i/=0,where misaconstant. This isBessel's equationforfunctions oforder zeroandargument-im:. Stokes stated (presumablywith reference toPoisson) that itwasknown thatthegeneral solution was f" i^c+Dlog{zsin26)}cosh(mzcos6)dO. J *Thevalue ofthislimitwasassumed byNeumann. tIfzisreal,^=1;ifnot,A$exp {]I{z)\\. +Trans. Camh. Phil. Soc. ix.(18-56), p.[38]. [Mathematical andPhijsicaJ Papers,iii.(1901), p.42.] 70 THEORY OFBESSEL FUNCTIONS [CHAP.Ill Itiseasytoseethat,withNeumann's notation, thevalue oftheexpressionontheright ^n{C-Dlog(4m)} Jo{imz)-^\tTDi^Wiimz). Theexpressionwasexpandedintoaseries byStokes;itisequalto WiC+D logz)J„(imz)+2Z)2-j^r-^cos^'^logsin dd, ,1=0 (,^'i;•J and,byintegrating byparts,Stokes obtained arecurrence forroula fi-omwhich itmaybe deduced that -j^%os2"^logsin^c^^=2#^|i7rlog2+i^(l+^+...+ ^)}. 3-58.Neumanns definition ofF<"'{z). TheBessel function ofthesecond kind, ofintegralorder n,wasdefined by Neumann* interms ofF<"'{z)byinduction from theformula (1) z^if^_nY^^'^(z)=-^F<'^+i) (z\ which isarecurrence formula ofthesametypeas§2'12(4).Itisevident from thisequationthat (2)F(«)(^)=(-^r(^)><«)(4 Now F'"'(z)satisfies theequation d and, ifweapplytheoperator -f-—r-tothisequationntimes, anduseLeibniz' theorem, weget andso Thisequationisatonce reducible to (4) V,F<»){z)=0, andsoF'"'(z)isasolution ofBessel'sequationforfunctions oforder ?i. Again, (3)maybewritten intheform d *Theorie derBesseVschen Functionen(Leipzig, 1867), p.51.Thefunction isundefined when itsorder isnotaninteger. tTheanalysisissimplified bytaking iz-=i,sothat d_d zdz~ d^' 3-58, 3-581] BESSEL FUNCTIONS 71 SOthat whence weobtain another recurrence formula (5) z\^+/iFe^){z)=^F"-^'(2). When wecombine(1)with(5)weatoncededuce theother recurrence ormulae (6)F'»-i'{z)+F'"+'){z)=^F'")(z),z r/T''"" iz\ (7) F<«-i'(^)-F'»+»(^)=2^ -* . ConsequentlyF*"'(^•)satisfies thesame recurrence formulae asJn{z\ F„(2) andY„{z).Itfollows from§3-57(2)that (8)F<")(^)=iTTF„(^)+(log2-7)/,{z) =iY„ (^)+(log2-7)./,(4 Asolution oftheequation Vh(y)=intheform ofadefinite integral, which reduce.s"to theintegralof J^3'571when n=0,hasbeen constructed bySpitzer, Zeitschrift furMath, undPhys.iii.(1858), pp.244-246;cf.§3-583. 3*581. Neumannsexpansion ofF'"'{z). Thegeneralisationoftheformula§8"57(1)hasbeengivenbyNeumann*; itis V(-)-^ {n+2m) 111 1 where s«=t+9+o+•••"•"' ^o=^- Toestablish this result, wefirst define thefunctions L^(z)andUn(2)by theequations (2) Z.,(.)=/.(.)log.-2^-~^,^, (3) U,,(z)=.,/,.{z)+S^ j^. ,;^^./™ (^), SOthat F»)(z)=Zo(^)-f/o(^). WeshallprovethatLn{z) andt^^C^) satisfytherecurrence formulae (4) X„^.: (z)=-L,:(z)+injz)L„(z), Un^, {z)=-U,:{z)+{niz) U,,(z\ andthen (1)willbeevident byinduction from§3-58(2). *Theorie derBcsseVschen Functioncn (Leipzig, 1867), p.o'i.SeealsoLommel, Stmiien iiher dieBesseVschen Functionen (Leipzig, 1868), pp.82—84;Otti,Bern Mitthellungen, 1898, pp.31— 3"); andHaentzschel, Zeitschrift fiirMath. uiidPIii/.^. xxxi. (1886), pp.25—33. z"72 THEORY OFBESSEL FUNCTIONS [CHAP,III Itisevident that dz\z''\~^^^dz\z'"] ^»+i ,,t^{n-m).m\dz\z'''^-'^ -Jn^. (z)log.+-^+^,in-m).ml f^^^"^^>^^^^^+"i^^^ -Jn^.{z)\ogz +^-+J^___|l+___^^_^^^ andthe firstpartof(4)isproved.Toprovethesecondpart,wehave dzlz''J''dz\z>'],^=1m{n^m) dz\z''\ \m J (z) 1*"(—Y=-Sn-"-^+-„J^„^(;^:^{^^Jn,.m-. (z)"(«+m)/„,.„,, (.)} ^n> andthesecondpartof(4)isproved.Itfollows from§3'58(2)that y"^+'>(Z)-Z„^, (Z)+Un^, (Z)^d {¥<->(Z)-Lnjz)+Un{z) z'^ dz\z^ andsince theexpressionontherightvanishes when n=0,itisevident by induction that itvanishes forallintegralvalues of ??.Hence andthetruth ofequation (1)istherefore established. 3'582. ThepowersefiesforUn(z). ThefunctionUn(z), which wasdefined in§3'581(3)asaseries ofBessel coefficients, hasbeenexpressed bySchlafli* asapowerseries withsimple coefficients, namely 1)1=0in:{n+m)i Toestablish this result, observe that itistruewhen 7i=by§3"51(3)and |3"57(1);andthat,bystraightforward differentiation, theexpressiononthe rightsatisfies thesame recurrence formula asthat of§3"581(4)forUn(z) ; equation (1)isthen evidentbyinduction. Note. Itwillbefoundinterestingtoestablish thisresult byevaluating thecoefficient of(|2)"+2wintheexpansion ontheright of§3-581(3). *Math. Ann. in.(1871), pp.146—147. 3-582-3-6] BESSEL FUNCTIONS 73 Thereader willnoweasily provethefollowing formulae : (2) %.(z)={7-log2}J,{z)-Un{z), (3) F""{z)=Ln{z)+%,(z)+{log2-y]J, (z), (4) 47rYn{z)=L„(z)+%,(z). 3'583. Theintagral ofPoisson^stypefory("){z). The Poisson- Neumann formula of§3-571for F(")(2)wasgeneralised byLommel, Studien fiber dieBesseVschen Functionen(Leipzig, 1868), p.86,with anotation ratlier different fromNeumann's; toobtain Lommel's result inNeumann'snotation, we first observe that,bydifierentiation ofPoisson'sintegral forJ^(2),wehave^-J,(z)log.= ^-^—^ly^^ jcos(.-sind)cos^" 6{log(^cos^d)-^{, +J)}dO, andso,from§3-582(3), ^'"'^")=r^fi^WM f'" ^'^^^''''''^)^'^«"' ^'1«S^^'^^^- >/'(n+1)-y}c/^+Ln(z), andhence, since\^(^)= \//-(1)—2log2=-y-2log2,wehave theformula (1)r(»){z)=^.^y ,1, /'" COS(^sin^)cos2»log(4cos^6)d6 -{^{n+h_)-y\r (i)}Jn(2)+/^«(2), inwhich itistoberemembered thatZ„(2)isexpressible asafinite combination ofBessel coefficients andpowersofz. 3*6.Functions ofthethird kind. InnumerousdevelopmentsofthetheoryofBessel functions, especially those which arebased onHankel's researches(ChaptersVIandVll)onintegral representationsandasymptotic expansionsofJ^{z)andY^{z),twocoiiibina- tions ofBessel functions, namely J^.{z) ±iY^,{z),areoffrequentoccurrence. Thecombinations alsopresentthemselves inthetheoryof"Bessel functions ofpurely imaginary argument" (§3"7). Ithasconsequentlyseemed desirable toNielsen* toregardthepairof functions Jv{z) ±iY^{z)asstandard solutions ofBessel'sequation,andhe describes them asfunctions ofthethird kirid; and, inhonour ofHankel, Nielsen denotes themb}^thesymbolH.Thetwofunctions ofthethird kind aredefined bytheequations f (1) Hl'^(z)=J.,(z)+iY,{z\Hf^ (z)=J,,(z)-iY^ (z). From these definitions, combined with§3-54(1),wehave \/^^„ \^jtsm^TTr / —ismi/TT When Visaninteger,theright-handsides aretobereplaced bytheir limits. Since J^{z) andY^{z) satisfythesame recurrence formulae(§§3-2, 3-56), inwhich thefunctions enterlinearly,andsince thefunctions ofthethird kind *Ofversigtoier detK.Vaiiskc Vide.nskahernes Selskabs Forhandlinger, 1902, p.125. Huiui- hucli derTheorie derCylindcrfunktionen (Leipzig, 1904), p.IG. tNielsen usesthesymbols Hi''{z), 11/[z). 74'THEORY OFBESSEL FUNCTIONS [gHAP.Ill arelinear functions (with constant coefficients)ofJ„(z)andY^(z),itfollows that these same recurrence formulae aresatisfied byfunctions ofthethird kind. Hence wecanatonce writedown thefollowing formulae : dH^^\z) „. ,,. dHf\z) ,,^ .,. rf^"^^'-1^^'dz ^a).„^ .;r,(2) (6).^^^-.jy^i)(.)=-.^(^>(.), z^^^^-rH''\z)=-zH%(z). dH^!;\z) a) dnf{z),,. Hm Note. Eayleighonseveral occasions, e.g.Phil.Mag. (5)xlhi. (1897), i").266;(6)xiv. (1907), pp.350—359[Scientific Papers,iv.(1904), p.290; v.(1912), pp.410—418],hasusedthe (2) symbol D^(z)todenote thefunction which Nielsen calls^iriH [z). 3'61. Relations connectingthethree kinds ofBesselfunctions. Itiseasytoobtain thefollowingsetofformulae, whichexpresseach function interms offunctions oftheother twokinds. Thereader willobserve thatsome oftheformulae aresimplythedefinitions ofthefunctions onthe left. ,-,, r, ^H^H^) +^f(^) Y.^z)-YA^)cosVTT ^^^ '^''^'^-2- ^STT^tT' e^^^H^\z) +e-^''^Hfiz) F-,(^)cos^tt-F,(^) ^^^ •^-A^)-2- sin;;7r' .oxY.,x_/.(^)cos.7r-J_.(^) _H^^\z)-Hf\z) ^"^^ ^^^^^^sirTW~ Yi' ...Y,,_/..(^)-/-.(^)cos^7r _e^''^Hl'^{z)-e-^-'Hf\z) W i^-.l^)-g-j^^^- ^^, (5)HW^s_J-A^)-e-'"''JA^) ^r-.(^)-e--"r,(^) "Isin T/TT sin i/tt' (6)jim,.^e^"'JAz)-J-A^) ^Y.,{z)-e^-iY,{z) "isinyTT sinvTT From(5)and(6)itisobvious that (7) H^]l (z)=e"-^i^^ (z),H?;(z)=e—' 7/f^ (z). 3-61-3-63]BESSEL FUNCTIONS 75 3"62. Besselfunctionswithargument—zand ze^"^"^. Since Bessel'sequationisunaltered ifzisreplaced by—z,wemustexpect thefunctions J±^(—2')tobesolutions oftheequationsatisfiedbyJj^^{z). Toavoid theslight difficulty produced bysupposingthat thephasesof both ofthecomplexvariables zand—zhave theirprincipal values*, we shall construct Bessel functions ofargument ^re""'', where inisanyinteger, arg2has itsprincipal value, and itissupposedthat arg (^^e""^')=mir+argz. SinceJj,(z)/2''isdefinable asaone-valued function, itisobviouslycon- venient toassume that,when thephaseofzisunrestricted, J^(z)istobe defined bythesame convention asthatbywhich z"isdefined; andaccordingly wehave theequations (1) J,{z&>'"')=e'"'-'J,(z), (2) J.y{ze""'')=e-'"""' J-„ (z). Thefunctions ofthesecond andthird kinds willnowbedefined forall values oftheargument bymeans oftheequations §3'o-i(1), §36(1);and then theconstruction ofthefollowingsetofformulae isaneasymatter: (3) Y^(ze"""')=e-"""' Y^(z)+2isinmvir C(jtvirJ^{z), (4) F_;, {ze"'"')=g-"""^'F_^{z)+2isinmvTtcosec virJy{z), (5) iT^'(^e'""') =e~""'"'i^' (^)-2e-''"'—^JAz) sini/TT"sini/TT (•>\ . ,-,f''! /N^•sininvirj- ,, (6) i/^^V2e'"'^0=e-'"''"^^^ (^)+2e*'"-- '^A^)\} V\ > V^'ginyj^ ^sin(1+m) v-JT^(2),.^^„„isinwu^^CD,. sinz/TT"^sini/TT" Ofthese results, (3)wasgiven byHankel, Math. Ann. viii. (1875), p.454,inthespecial casewhenm=\and visaninteger. Formulae equivalentto(5)and(6)wereobtained by Weber, Matk. Ann. xxxvii. (1890), pp.411,412,whenm=l;see§6-11.And amemoir byGraf, Zeitschrift filrMath, ttndPhys.xxxviii. (1893), pp.115—120, contains thegeneral formulae. 3-63. Fundamental systems ofsolutions ofBessel's equation. Ithasbeen seen(§3-12)thatJ,{z)and/_,{z)form afundamental systemof solutions ofBessel'sequation when, andonlywhen,i^isnotaninteger. Weshall nowexamine theWronskians ofotherpairsofsolutions withaview todeter- miningfundamental systemsinthecritical casewhen visaninteger. *ForArg(-z)-Arg2=ftt,accordiug as/{z)50. 76 THEORY OFBESSEL FUNCTIONS [CHAP.Ill Itisclear from§3-54(1)that m,[J,(z),ni^)]=-cosec vir .Wi[J.(z),J_. (^)l _2_ irz' This result isestablished onthehypothesisthati^ isnotaninteger;butcon- siderations ofcontinuitj^shew that (1) mi{JA^\ r,(^)l=2/(7r0), whether vbeanintegerornot.Hence Jv{z) andF^{z)always formafunda- mentalsystem ofsolutions. Itiseasytodeduce that (2) ra{/.W.Y.Wl=jjjj^, and, inparticular*, (3) im[Jn{z),^n{z)]=2lz. When weexpressthefunctions ofthethird kind interms of/„{z)and Y^(z),itisfound that (4)Wi[H? (z),hT C^)}=-^i(m{J. {z),F,{z)]=-ii/iTTz), sothat thefunctions ofthethird kind alsoform afundamental systemof solutions forallvalues ofv. Various formulae connected with(1)and(3)have beengiven byBasset, Proc.London Math. Soc.XXI. (1889), p.55;theyarereadily obtainablebyexpressingsuccessive differ- ential coefficients ofJ^iz) andYt,{z)interms ofJ^„(2),JJ{z),andr^(2)', JV(s)byre- peateddifferentiations ofBessel'sequation.Basset's results(ofwhich theearlier ones arefrequently requiredinphysical problems) areexpressedinthenotation used inthis workbythefollowing formulae : (5) J,(z)IV(z)- 1\{z)JJ' (2)=- J-2, (6) j;{z)}v (--)-yj{z)J," (z)=A (^1- ^), (7) J.(z)IV" (2)-n(2)J.'" (2)= ^.('^-1)' (8) j;(z)IV" iz)-r; (z)J.'" (z)=^2(I"- 1)' (9) J."{z) JV" (z)-Vu'iz) J.'" (2)=A(1- (10) J, {z)}v-)(-')-y.(z)J^^Kz)=^A,(1 2/v*+Uv'^ 2i/2+3 ,\ (11). j;{z)r,(iv) (z)-JV{z)^.(") (^)=-- (^^4 ^-+V Throughout these formulae 1\maybereplaced byJ_„iftheexpressions ontheright aremultiplied by-sini/7r; andJ^,Yymaybereplaced byS^^\ ZT^^^^throughoutifthe expressions ontherightaremultiplied by-2i. *Cf.Lommel, Math. Ann. iv.(1871), p.106,andHankel, Math. Ann. vni. (1875), p.4-57.2v2+l ^2 3-7] BESSEL FUNCTIONS 77 Anassociated formula, duetoLomtnel* Math. Ann. iv.(1871), [>.106,andHankel, Math. Ann. viii. (1875), p.458,is (12) J,{z)Y,,^{z)-J,,^{^z)l%{z)^-\.irz This isprovediuthesamewayas§32(7). 3'7. Besselfunctions ofpwely imaginary argument. The differentialequation which differs from Bessel'sequation onlyinthecoefficient ofy,isoffrequent occurrence inproblemsofMathematicalPhysics;insuchproblems,itisusually desirable topresentthesolution inarealform,andthefundamentalsystems J^(iz)and ./_^(iz)orJ^,(iz)andY^(is)areunsuited forthispurpose. However thefunction e~-'"'^'' J^(iz)isarealfunction ofzwhich isasolution oftheequation.Itiscustomarytodenote itbythesymbol /^(z)sothat When zisregardedasacomplex variable, itisusuallyconvenient todefine itsphase,notwith reference totheprincipalvalue ofarg iz,astheconsideration ofthefunction J^(iz)wouldsuggest,butwith reference totheprincipalvalue ofarg z,sothat (/„(z)=e-i"-'J„{zif\ (-vr<argz^^ir), \I,{z)=e^'"^*' J\,(ze--^'^'), (ITT<arg^<TT). Theintroduction ofthesymbol /;,(z)todenote "the function ofimaginary argument" isduetoBassetfand itisnowincommon use. Itshould bemen- tioned that fouryearsbefore thepublicationofBasset's work, NicolasJhad suggestedtheuseofthesymbol F^,{z), butthisnotation hasnotbeenusedby other writers. The relative positionsofPareandApplied Mathematics ontheContinent ascompared with thiscountryareremarkablyillustrated bythe fact that, inNielsen's standard treatise§,neither thefunctionIv{z)i northesecond solution K^iz), which willbedefined immediately,isevenmentioned, inspiteoftheir importanceinphysical applications. The function I-„{z)isalsoasolution of(1),and itiseasytoprove (c£ §3-12)that ^(3) m[L,{z\/_.(.)}=-^-^^. *Lommel gave thecorresponding formula forNeumann's function ofthesecond kind. fFvoc. Camb. Pliil. Soc. vi.(1889), p.11.[This paper was firstpublishediu1880.] Basset, inthispaper, defined thefunction ofintegralorder tobet+"J„((^),buthesubsequently changed it,inhisHydrodynamics,ii.(Cambridge, 1888), p.17,tothatgiveniuthe text. Themore recent definition isnowuniversally used, +Ann. Sci.deVEcole norm.S2ip. (2)xi.(1882), supplement, p.17. §Handbuch derTheorie derCylinderfunktionen (Leipzig, 1904). 78 THEORY OFBESSEL FUNCTIONS [CHAP.Ill Itfollows that,when visnotaninteger,thefunctions 7^(z)and/_^(z)form afundamentalsystemofsolutions ofequation (1). Inthecaseoffunctions ofintegral order, asecond solution hastobecon- structed bythemethods of§§3*5—3"54. Thefunction Kn(z),which willbeadopted throughoutthiswork asthe second solution,isdefined bytheequation (4) K^(z)=lim^-//_,(2)-/,« V—n Anequivalentdefinition (cf§3'o)is Itmaybeverified, bythemethods of§3'5,that /r„{z)isasolution of(1)when theorder visequalton. The function K^,{z) hasbeen defined, forunrestricted values of v,by Macdonald*, bytheequation (6) K,{z)=^7r^-''^''^~^'' ^.Sm I^TT and,with this definition, itmaybeverified that (7) Kn{z)=\miK,{z). Itiseasytodeduce from(6)that (8) K,(z)=^TTie-^-""'h! (iz)=iTTie-*'"'' H[,(iz). Thephysical importanceofthefunction Ky(z)liesinthefactthat itisa solution ofequation (1)which tendsexponentiallytozero asz-^^ x>through positivevalues. Thisfundamentalpropertyofthefunction willbeestablished in§7-23. The definition of^,,(2)isduetoBasset, Proc. Camh. Phil. Soc. vi.(1889), p.11,and hisdefinition isequivalenttothatgiven byequations (4)and(5) ;theinfiniteintegrals by which heactuallydefined thefunction willbediscussed in§i^6'14, 6:15. Basset subse- quentlymodified hisdefinition ofthefunction inhisHydrodynamics,il.(Cambridge, 1888), pp.18—19,andhisfinal definition isequivalent to-— r-,-^^— ^-^ Inorder toobtain afunction which satisfies thesame recurrence formulae as/^(z), Gray andMathews intheir work,ATreatise onBessel Functions (London, 1895), p.67, omit thefactor1/2",sothat their definition isequivalentto. ira/_^)_a7j^)-| Theonlysimple extension ofthisdefinition tofunctions ofunrestricted order isbythe formula K^ (2)=in-cot I/TT{/_^(2)-/^(3)}, *Proc.London Math. Soc.xxx. (1899), p.167. 3-71] BESSEL FUNCTIONS 79 (of.ModernAnalysis, §17'71) butthisfunction suffers from theseriousdisadvantage that itvanishes whenever 1visanoddinteger. Consequentlyinthiswork, ^lacdonald's function willbeused althoughithasthedisadvantageofnotsatisfyingthesame recur- rence formulae as/^(s). Aninspectionofformula(8)shews that itwould havebeenadvantageousifafactor^nhad beenomitted from thedefinition ofA"^{z) ;butinview oftheexistence ofextensive tables ofMacdonald's function itisnowinadvisable tomake thechange, andthepresence ofthe factor isnotsoundesirable asthepresenceofthecorrespondingfactor inSchliifli's function (§3'54) because linear combinations of/^(i)and K^,(z)arenotofcommon occurrence. 3'71. Formulae connected with /„(^)andKy{z). We shallnowgivevarious formulae forIv{z) andK^(z) analogousto those constructed in§§3*2—3*6fortheordinaryBessel functions. Theproofs oftheformulae arelefttothereader. (1) /.-: (Z)-L^.{Z)=^L(Z),/C-x (Z)-/C+i (^)=-V^^(')' (2) /.-, {z)+L^,{z)=21:{z\ /C-i {z)+/C+, {z)=-2K: (z), •(3) zlj(z)+vl,{z)=zl,_, {z),zK: {z)+vK, (z)=-zK,_, (z), (4) zi:(z)-vh(z)=zl,+, (z),zKJ {z)-vK, (z)--zK,^, {z), 5)(^)"{z^L (z)]=^"-'"/.-m {z),(^^)"[z^K. {z}]=(-)-^''-^._,„ {z\ ^^^\zdz) \s"Iz^-^'^^'\zdz) \Z^]^^Z^^'"' (7)I,'{z)=L(z), K:(z)=-K,(z), (8)I^n{z)=Iniz), K_,{Z)=K„(Z). Thefollowing integralformulae arevalidonlywhenR(v+h)>: id) lAz)=r7"XTVr7rxI'^osh(zcos6)sm'''dde (^zYe^^^o^^sin-" Odd =^ifll— -/'''cosh (zcosd)sin^'edd = liijJL[(1-t-y-'- cosh(^0dt. /, 80 THEORY OFBESSEL FUNCTIONS These results aredue toBasset. Wealsohave[chap.Ill (10) (11) (12) (18) (14) (15) (16) (17) (18) (19) (20)InH (^)=1 sj{'lirz) I-(n+h) {Z)=n (-)>-(n+r)! ,%r\{n-r)\{2zr n(^i4-7'")I ^^ ^,.ZQr\(n-r)\(2zyy :^(-Y(n+r)le^z ^(2-rrz) Ir=-orl{n-r)l(2zy ^^ ^,^orl(7i-ry.{2zyy ^„,,(^)=(^Ve-S(n+r)\ 2z) r=Qr\{n-ry.{2zy' K.(A=-log(iz).7.{z)+1^i^"f(m+1), 2,„=o m!(^^)«-«" {logi^z)- \y\r{m+1)- ^^/r(n+7n+1)},00 /'l9\n+2jn ,=0m!(M+Hi)! Z'o(^)=-- f" e^''"^{log(2^sin^^)+7]dO, K,(^e"-')=e-'"-'7C{z)-iri^^'^"^ /^(^),sin I/TT 5ie{7.,(2), A^(^)}=-1/^, /.(z)K,.^, (z)+/,+, (z)K^{z)=1/z. Theintegral involved in(16)hasbeen discussed byStokes(cf.§3-572). Theintegrals involved in(9)andtheseries in(14)were discussed byRiemann inhis memoir "Zur Theorie derNobili'schenFarbenringe," Ann. derPhysik undChemie, (2)xcv. (1855), pp.130—139,inthespecial case inwhich ^=0; healsodiscussed theascending powerseries for/q(2). Therecurrence formulae have beengiven byBasset, Proe. Camb. Phil. Soc. vi.(1889), pp.2—19;byMacdonald, Proc.London Math. Soc.xxix. (1899), pp.110—115;andby. Aichi, Proc.Phys. Math. Soc.ofJapan, (3)II.(1920), pp.8—19. Functions ofthistypewhose order ishalfanoddinteger,asinequations (10)and(12), were usedbyHertz inhisBerlinDissertation, 1880{Ges. Werl-e,1.(1895), pp.77—91J; andheaddedyetanother notation tothose described in§341. 3-8] BESSEL FUNCTIONS 81 3-8. TJiomsons functions ber{z)andbei{z)and theirgeneralisations. Aclass offunctions which occurs incertain electricalproblems consists of Bessel functions whose arguments have theirphases equalto{irorftt. Thefunctions oforder zerowere firstexaminedbyW.Thomson*; they maybedefinedbytheequation j- (1) ber{x)+ibei{x)=/o{^iVO=I^,(^vVO, where xisreal,andberandbeidenote real functions. Forcomplexaro-u- ments weadoptthedefinitionsexpressed bytheformulae (2) ber(z)±ibei(z)=./„(zi\I±i)=h{z^J± i). Hence wehave (3)bor(.)=l-y;^+g|-.... (2!)-^ (6!f' (10!)-^• Extensions ofthese definitions tofunctions ofanyorder ofthefirst,second and third kinds havebeen effected byRussell:J: andWhitehead§. Thefunctions ofthesecond kind oforder zerowere definedbyRussellby apairofequations resembling (2),thefunction I^being replaced bythe function K^,thus (5) ker{z)±ikei{z)=K^(zVi i). Functions ofunrestricted order i>were definedbyWhitehead with reference toBessel functions ofthe firstandthird kinds, thus (6) ber, (z)±ibei, (z)=J,(^e**'^'), (7) her, (z)±ihei,(z)-if,<" (ze^^'"'). Itwillbeobservedthat|| (8) ker(z)=—hirhei(z),kei(z)=^tther(z), inconsequenceof§3"7(8). Thefollowing series, due toRussell, areobtainable withoutdifficulty: (9) ker{z)= -\og(^z).her(z) +lTrhei{z) *Presidential Address totheInstitute ofElectrical Engineers,1889. [Math, andPIiijs. Papers, in.(1890), p.492.] fInthecase offLinetious ofzeroorder,itiscustomarytoomit thesuffix which indicates theorder. tPhil.Mag. (6)xvii.(1909), pp.524— 552. §Quarterly Journal, xlii.(1911), pp.316—342. IIIntegrals equal toker{z)andkei(z)occur inamemoir byHertz, Ann. derPhysik nndCheinie. (3)XXII.(1884), p.450[Ges. Werke,i.(1895), p.289]. \V.B.V. 6 82 THEORY OFBESSEL FUNCTIONS [CHAP.Ill (10) kei(z)=-log(1^).bei{z)- l-rrber(z) +2E.Wtw^<'"'^'^- Ithasalsobeen observed byEussell that the firstfewterms oftheexpansionof ber^(j)+bei^(2)havesimple coefficients, thus (11)- ber2(2)+bei2(.)=l+MV»+ii^+Mf,+..., butthis result hadpreviously been obtained, with adifferent notation, byNielsen(cf. §5-41) ;thecoefficient of(l^)^™intheexpansionontherightisl/[(m If.(2m) !]. Numerousexpansions involving squaresandproductsofthegeneral functions have been obtained byRussell ;forsuch formulae thereader is referred toRussell's memoir andalso toapaper bySavidge*. Formulaeanalogoustotheresults of§§8-61, 3"62havebeen discussed by Whitehead;itissufficient toquotethefollowinghere : (12) ber_„ (z)=cosvir .ber^ (z)—sinptt .[hei^ (z)—bei^ (z)], (13) bei_„ (z)=cosvtt .bei,,(z)+sinvtt .[her^ (z)—ber^ (z)], (14) her_^(2^)=cosvtt .her^ (z)—sinvir .he\^, {z), (15) hei_^ {z)=sinvir .her,, (z)+cosvtt .hei,, (z). Thereader willbeable toconstruct therecurrence formulae which have beenworked outatlength byWhitehead. Thefunctions oforderunityhaverecentlybeenexamined insome detail byB.A.Smithf. 3*9. Thedefinition ofcylinder functions. Various writers, especially SonineiJ: andNielsen§,have studied thegeneral theoryofanalyticfunctions oftwovariables "^^{z)whichsatisfythepairof recurrence formulae (1)• '^._, (5)+-^#.+1 (^)=-^.(^),z (2) "^.-1 (^)-'^Vi (^)=2*^;(^x inwhich zandvareunrestrictedcomplexvariables. These recurrence formulae aresatisfiedhyeach ofthethree kinds ofBessel functions. Functions whichsatisfy onlyoneofthetwoformulae arealsodiscussedby Sonine inhiselaborate memoir ;abrief account ofhisresearches willbegiven inChapterX. *-^Phil.Mag. (6)xix.(1910), pp.49—58. •fProc. American Soc.ofCivilEngineers,xlvi.(1920), pp.375—425. XMath. Ann. xvi.(1880), pp.1—80. §Handbttch derTiieorie derCylinderfunktionen (Leipzig, 1904), pp. 1,42etseq. 3-9] BESSEL FUNCTIONS 83 FollowingSonine weshall callanyfunction ^^{z),which satisfies hoihof theformulae, acylinder function.Itwillnowbeshewn thatcylinder functions areexpressibleinterms ofBessel functions. When wecombine theformulae(1)and(2),wefindthat (3) z%^' {z)+v^. (z)=z^tf..., (z), (4) zW; (z)-v'^, {z)=-z9^,^, (z), and so,if^bewritten forz(d/dz), wededuce that (5) {'^+v)'^?..(z)=z%\_,{z), (6) (^-")9^,.(z)=-z%^,^,(z). Itfollows that C^'-v')%,(z)=(^- i^){^'2?;_i (z)] that IStosay Hence<^,(z)=a,J,,(z)+b„F„(z), where a^andb,.areindependentofz,though theymaydependon v.When wesubstitute in(3)wefindthat a,./,_, (z)+6,r,_i (z)=«„_i,/,_i {z)+6,_i]"„_i (z), andso,sinceJ'^_i (z)/Y^_^ (z)isnotindependentofz,wemust have «„= a,,_i, b,=b,_,. Hence «„and b^must beperiodicfunctions of i>withperiod unity;and, conversely,iftheyaresuch functions ofv,itiseasytoseethatboth(1)and (2)aresatisfied. Hence thegeneralsolution of(1)and(2)is (8) r,(z)=n,,(v)J,,(z)+ST,(v)Y,(z\ where'!it^{v) and ^2(1/) arearbitrary periodicfunctions ofvwithperiod unity. Itmaybeobserved thatanequivalentsolution is (9) 'gf,(z)=^,(,.)i7,<i' (z)+^,(p)HJ^ {z). Adifferenceequation, which ismoregeneral than(1),hasbeenexamined byBarnes, Messenger, xxxiv.(19l>5), pp.52—71;incertain circumstances thesohition isexpressible ^JBessel functions, thoughitusuallyinvolves hypergeomctricfunctions. Note. Thenamecylinder functionisusedbyNielsen todenoteJ^,(z),Y^(z),H^W {z) and HiS^) {z)aswellasthemoregeneralfunctions discussed inthissection. Thisprocedui'e isinaccordance with theprinciplelaiddown byMittag-Lefflerthat itis,ingeneral, undesirable toassociate functions with thenames ofparticularmathematicians. Thenamecylinder function isderived from thefactthatnormal solutions ofLaplace's equation incylindrical coordinates are (cf.§4-8andModernAnalysis, §18-5). 6—2 84 THEORY OFBESSEL FUNCTIONS [CHAP.Ill Some writers*, following Heine twho called Jn{z) aFourier- Besselfunction,callJn{z) aFourierfunction. AlthoughBessel coefficients ofanyorder were used longbefore thetime ofBessel (cf.§§1-3, 1'4),itseems desirable toassociate Bessel's name withthem, notonlybecause ithasbecomegenerally customarytodoso,butalsobecause ofthegreat advance made by Bessel onthework ofhispredecessorsintheinvention ofasimple andcompact notation forthefunctions. Bessel's name wasassociated with thefunctions byJacobi, JournalfurMath. xv. (1836), p.13[Oes. Math. Werke,vi.(1891), p.101]. "Transcendentium/j.*naturam varios- queusus indeterminandis integralibusdefinitisexposuitill.Bessel incommentatione celeberrima." Amore recent controversy onthename tobeappliedtothefunctions istobefound in aseries ofletters inNature, LX.(1899), pp.101, 149,174; Lxxxi. (1909), p.68. *E.g.Nicolas, Ann. Sci.deVEcole norm.sup. (2)xr.(1882), supplement, fJonrnalfiirMath. lxix.(1868), p.128. Heine alsoseems toberesponsible fortheterm cylinder function. CHAPTER IV DIFFERENTIAL EQUATIONS 4'1.Daniel Bernoulli's solutionofRiccati'sequation. Thesolutiongiven byBernoulli* oftheequation (1) -^^az'' +bifdz-^ consisted inshewingthatwhen theindex nhasanyofthevalues 0'—A—l-_S_8- _12_12. Ifi Ifi. while aand hhaveanyconstant valuesf,then theequationissoluble by means ofalgebraic, exjxjnential andlogarithmic functions. Thevalues ofn justgivenarecomprisedintheformula Im±1 wheremiszero orapositive integer. Bernoulli's method ofsolution isasfollows :Ifnbecalled theindex ofthe equation,itisfirstprovedthatthegeneral equation Jofindex nistransformable intothegeneral equationofindex N,where (3) N=-'^ n+1' and itisalsoprovedthat thegeneral equationofindex nistransformable intothegeneral equationofindexv,where (4) v=-n- 4. The Riccatiequationofindex zero isobviously integrable,because the variables areseparable. Hence, by(4),theequationofindex—4isintegrable. Hence, by(3),theequationofindex—|isintegrable.Ifthisprocessbecon- tinued byusingthetransformations (3)and(4)alternately, wearrive atthe setofsoluble casesgiven above, and itiseasytoseethat these cases are comprisedinthegeneralformula(2). *Exercitationes quaedain iiuithematicae(Venice, 1724), pp.77—80;Acta Eruditorum, 1725, pp.473—475.Thenotation usedbyBernoulli hasbeen slightly modified;and inthisanalysis nisnotrestricted tobeaninteger. tItisassumed thatneither anor Jiiszero. Ifeither were zerothevariables would obviously beseparable. tThat is,theequation inwhich r;and hhave arbitraiy values. 86 THEORY OFBESSEL FUNCTIONS [CHAP. IV 4*11. Daniel Bernoulli stransformations ofRiccatisequation. Now that theoutlines ofBernoulli'sprocedurehavebeen indicated, we proceed togivetheanalysis bywhich therequisitetransformations areeffected. Take§41(1)asthestandard equationofindex nandmake thesubstitutions ^"+' „ 1 n+l'-^Y' [Note. Thesubstitutions arepossiblebecause —1isnotincluded among thevalues of n.The factor n+\inthedenominator wasnotinserted byBernoulli;the effect ofits presenceisthatthetransformed equationismore simple than ifitwereomitted.] Theequation becomes 1clY V-2,7*7^^' that is dY~=h{n+\YZ^+aY\ whereN=- n/(?i+1) ;andthis isthegeneral equationofindex N. Againin§4"1(1)make thesubstitutions• Theequation becomes where y=—M—4;andthis isthegeneral equationofindex v. Thetransformations described in§4"1aretherefore effected, and sothe equationissoluble inthecases stated. Butthisproceduredoesnotgivethe solution inacompactform. 4"12.Thelimiting form ofRiccatisequation,withindex—2. When theprocessesdescribed in§§4'1,411 arecontinually appliedto Riccati'sequation,thevalue towhich theindex tends, whenm-*xin §4'](2),is—2.Theequationwithindex—2isconsequentlynotsolubleby afinitenumber oftransformations ofthetypeshitherto under consideration. Tosolve theequationwithindex—2,namely writey=vjz,andtheequation becomes dv, ,zT^tt+V+bv- :dz andthis isanequation with thevariablesseparable. Hence, inthislimiting case, Riccati'sequationisstill soluble bytheuse ofelementaryfunctions. 4-11-4-13] DIFFERENTIAL EQUATIONS 87 This solution wasimplicitly given byEuler, Inst. Calc. Int. ii.(Petersburg, 1769), §933, [).185. Ifwewrite(cf.§4-14)y=~"r 'T^-'^^^equation which determinesr)is which ishomogeneous, andconsequentlyitisimmediately soluble. Euler does notseem tomention thelimiting case ofRiccati'sequation explicitly, although hegaveboth thesolution ofthehomogeneous linearequation andthetransforma- tionwhich connects anyequationofRiccati's typewithalinearequation. Itwillappear subsequently (§§4*7—4-75) that theonlycases inwhich Riccati'sequationissoluble infinite terms arethecaseswhich havenowbeen examined;that istosay,those inwhich theindex hasoneofthevalues 0-_4_4- _S_K. _9 andalsothetrivial cases inwhich aorh(orboth)iszero. This converse theorem, due toLiouville, is,ofC(jurse, muchmore recondite than Bernoulli's theorem thattheequationissoluble inthespecifiedcases. 4"13. Elder's solution ofRiccati'sequation. Apracticalmethod ofconstructingasolution ofRiccati'sequationinthe soluble caseswasdevised byEuler*, andthismethod (withsomeslight changes innotation),willnowbeexplained. First transform Riccati'sequation, §4"1(1),bytaking new variables and constants asfollows : (1) y=-v/h,ah=-c\ n=2q-2; thetransformedequationis (2) S+^'-'"'"'"=^ ' andthesoluble cases arethose inwhich1/qisanoddinteger. Define anewvariable wbytheequation 1dw ^ -^zuciz sothattheequationinwis p ^+2c.^-^+(?-1)^^^'-''^'^=0- Asolution inseries ofthelastequationis providedthat Ar+,_(2qr+q+l){2qr+q-1) ~A7~ 8^c(r+1)' -Nov. Comm. Acad. Petrop.viii.(1760—]761) [176:^], pp.:5—63 :and ix.(1762-1763) [1764], pp.154—169. 88 THEORY OFBESSEL FUNCTIONS [CHAP.IV andsotheseries terminates withtheterm^m^"*'" ifqhaseither ofthevalues +1/(27H+1);andthisprocedure givesthesolution* examined byBernoulli. Thegeneralsolution ofRiccati's equation,which isnotobvious bythismethod, was given explicitly byHargreave, Quarterly Journal, vn.(1866), pp.256—258, butHargreave's form ofthesolution wasunnecessarily complicated;twoyearslaterCayley,Phil.Mag. (4) xsxvi. (1868), pp.348—351[Collected Papers,vii.(1894), pp.9—12], gavethegeneralsolu- tioninaformwhichclosely resembles Euler's particular solution, thechief diiference between thetwosolutions beingthereversal oftheorder oftheterms oftheseries involved. Cayley usedaslightly simplerform oftheequation than(2),because hetookconstant multiplesofbothvariables inRiccati's equationinsuch awayastoreduce itto (5)|+,-2_^2,-2=0. 4'14.Cayley's generalsolittionofRiccati's equation. Wehavejustseen that Riccati'sequationisreducible totheform dz givenin§4'13(2);andweshallnowexplain Cayley'sf method ofsolving thisequation, which istoberegardedasacanonical form ofRiccati's equation. When wemake thesubstitution! r]=d(\ogv)/dz,theequation becomes (1) ^-c^z^-^-^v = ; and, iff/jandU^areafundamentalsystemofsolutions ofthisequation, the g'eneralsolution ofthecanonical form ofRiccati'sequationis Avhere CjandC^arearbitraryconstants andprimesdenote differentiations with respecttoz. Toexpress UiandU^inafinite form,wewrite V=wexp{cz^jq), sothat theequationsatisfiedbyzyis§4-13(4).Asolution ofthisequation inluproceedinginascending powersofz'iis 1-^-1.,^+ (iriKSi^l) ^,2,2.q{q-\) q{q-l)2q{2q-l) (q-l){Sq-l)(oq-l) q{q-l)2q(2q-l)Sq{Sq-l) andwetake Uj^tobeexp(czi/q) multiplied bythis series.cV9+..., *When theindex noftheEiccati equation is-2,equation (4)ishomogeneous, tPhil.Mag. (4)xxxvi.(1868), pp.348—351[Collected Papers,vir.(1894), pp.9—12]. Cf.also thememoirs byEuler which were cited in§4-13. JThisis,ofcourse, thesubstitution used in1702byJames Bernoulli; cf.§1-1. 4-14] DIFFERENTIAL EQUATIONS 89 Nowequation (1)isunaffected bychangingthesignofc,andsowetake (q-l){3q-l)U„U,=exip(±cz'i/(j) ^ qiq-1)'"^q{q- l)2q{2q^''''' (q-l){3q~l){5q-l)+ —(fz"i+... q(q-l)2q{2q-l)Hq{Sq-l) andbothofthese series terminate whenqisthereciprocalofanoddpositive integer.Since the ratio U^:U^istheexponential functionexp {2cz'^lq) multiplied byanalgebraicfunction ofz-J,itcannot beaconstant;and so Ui,U^form afundamental systemofsolutions of(1). Ifqwere thereciprocalofanoddnegative integer, weshould write equation (1)intheform <P(v/z) d{l/zy-c'{l/z)-"i-'{vlz)=0, dwhence itfollows that wherej^and72areconstants, and Fi ,V2=zexp (+cz^/q)-q{q+l) q{q+l)2q{2q +l)c-z-<i+ ... The series which havenowbeen obtained willbeexamined inmuchgreater detail in§§4-4—4-42. Thereader should haveuodifficultyinconstructingthefollowingsolutions ofRiccati's equation, when itissoluble infinite terms. 90 THEORY OFBESSEL FUNCTIONS [CHAP.IV Among thewriters whohave studied equation (1)areKummer, JournalfurMath. xii. (1834), pp.144—147, Lobatto, JournalfiirMath. xvii.(1837), pp-363—371, Glaisher(in thememoir towhich reference hasjustbeenmade), andSuchar, Bull, delaSac.Math, de France, xxxii.(1904), pp.103—116;forother references see§4-3. Thereader willobserve thatwhenq=0,theequation (1)ishomogeneous andimme- diatelysoluble;andthat thesecond order equationsolved byJames Bernoulli(§M)is obtainable bytaking j=2in(1),andsoitisnotincluded amongthesoluble cases. 4-15.SchldfliscoMonical form ofRiccati's equation. Theform ofRiccati'sequationwhich wasexamined bySchlafli*was (1) ^^t^-t-^-Ht\ This iseasilyreduced totheform of§4-13(2) bytaking —t-^'laasanew independentvariable. Tosolve theequation,Schlafli wrote ''"^ dt' andarrived attheequation 00pn Iff F(a,t)= X 1=0m!r(a+m+1)' thegeneralsolution oftheequationinyis 2/-c,F(a, t)+c.d-"F (-a,t). Thesolution of(1)isthen _cV^+^Fja +1,+c.,F{- a-l,t)"~ c^F{a, t)+cd-^'F (-a,t) Theconnexion between Riccati'sequation and Bessel's equationisthus rendered evident;butasomewhat tediousinvestigationisnecessary (§4*43) toexhibit theconnexion betweenCayley'ssolution and Schlafli's solution. Note. Thefunction ^:s,defined astheseries a 1 a^ 1 a^ '*'2• 2^2+1)•2T30^2+17(2 +2)"^*' which isevidently expressible interms ofSchlafli'sfunction, wasused byLegendre, Elements deGeometrie(Paris, 1802), note 4,inthecourse ofhisproofthat ttisirrational. Later thefunction wasstudied (with adifferent notation) byClifford; seeaposthumous fragmentinhisMath.Papers (London, 1882), pp.346—349. *Ann. diMat.(2)i.(1868), p.232.Thereader will seethatJames Bernoulli's solution in series(§1"1)istobeassociated with Schlafli's solution rather thanwith Cayley's solutiou. tThis notation should becompared withthenotation of§4-4. 4-15, 4-16] DIFFERENTIAL EQUATIONS 91 Itisobvious thatJ^{z)={\zyF{v,-j2-), and ithasrecently been suggested* that, because theSchliifli-ClifFord notationsimplifies theanalysisinthediscussion ofcertain problems onthestabilityofvertical wires under gravity,thestandard notation forBessel functions should beabandoned infavour ofa notation resembling thenotation usedbySchlafli-Cliftbrd :—aprocedure which seems com- parable toaproposaltoreplacetheordinarytables oftrigonometrical functionsbytables ofthefunctions 4'16. Miscellaneous researches onRiccati'sequation. Asolution ofRiccati'sequation, which involves definiteintegrals, wasgiven byMurphy, Trans. Camb. Phil. Sac. in.(1830), pp.440—443.Theequation which heconsidered is du.or, at and, ifabewritten forl/(?;i+2)andA~'^d{\ogy)\dtfor?f,hissolution (when ABd-=\)is y^-\t{/i-1[0(A)exp (^i/«//< )+(1//Oexp (/<<i.«)] dh, where ^('^O=^'''/<-"/'e"''h"-^dh=1-.~~,— '''^^. r. ^^ ^jo „=oa(«+l)(«+2)... («+/?.) IfXjhbewritten forhinthesecond partoftheintegral, then thelastexpression given foryreduces tonitmultiplied bytheresidue attheoriginof/i~^(p(A)exp {t^/"jh), andthe connexion betweenMurphy'ssolution and Schliifli's solution(§4'15)isevident. Aninvestigation waspublished byChallis, Quarterly Journal,vil.(1866), pp.51—53, which shewed howtoconnect twoequationsofthetypeof§4"13(2),namely inoneofwhichl/qisanoddpositive integer, and intheother itisanoddnegative integer. This investigationistobeassociated with thediscoveryofthetwotypesof solution givenin§4'14. Theequation ^*+^+bz" u'^-cz"'=0,dz z which iseasilytransformed intoanequationofRiccati'stypebytakingi»-«+iand z'^uas new variables, wasinvestigated byRawson, Messenger,vir.(1878), pp.69—72.Hetrans- formed itintotheequation y-— dz z bytaking bu=cz'^/y;twosuch equationsarecalledcognateRiccati equations. Asomewhat similarequation wasreduced toRiccati'stypebyBrassine, Journal deMath. xvi.(1851), pp.255—256. Theconnexions between thevariq^is typesofequationswhich different writers have adoptedascanonical forms ofRiccati'sequationhavebeen setoutinapaper byGreenhill, Quarterly Journal, xvi. (1879), p[>.294—298. *Greenhill, Engineering,cvii. (1919), p.334; Phil.Mag. (6)xxxviii. (1919), pp.501—528; seealsoEngineering,cix.(1920). p.851. 92 THEORY OFBESSEL FUNCTIONS [CHAP.IV Thereader should alsoconsult ashort paper bySiacci, Napoli Rendiconti, (3)vir. (1901), pp.139—143. Andamonograph onRiccati's equation,whichapparentlycontains themajorityoftheresults ofthischapter,hasbeen produced byFeldblum, Warschau Univ. Nach. 1898, nos.5,7,and1899,no.4. 4*2.ThegeneralisedRiccatiequation. Anobviousgeneralisationoftheequationdiscussed in§41is (1) %=P+Qy^Ry^ where P,Q,Rareanygivenfunctions ofz.Thisequationwasinvestigated byEuler*. Itissupposedthat neitherPnorRisidentically zero; for, if eitherPoxRiszero, theequationiseasily integrable byquadratures. ItwaspointedoutbyEnestrom, EncyclopMiedesSet.Math. ii.16,§10,p.75,thata special equationofthistypenamely nxxdx—nyydx+xxdy=xydx wasstudied byIManfredius, Deconstructioneaeqiiationmn differentialum pn'mi gradtts (Bologna, 1707), p.167."Sedtamen haeceademaequatio nonapparet quomodoconstrui- bilissit,neque enimvidemus quomod5illam integremus, neequomodo indeterminatas ab invicfemseparemus." Theequation (1)iseasilyreduced tothelinearequationofthesecond order, bytakinganewdependentvariable udefinedbytheequation f 1dlogu (2) y=-Rdz Theequation thenbecomes /r,\C^^^*(/^ 1dR]du Tin rv Conversely,ifinthegenerallinearequationofthesecond order, /,. dhi du (4)P^d?-^P^dz'-P''''=^' (where jJo,Pi,p^aregivenfunctions ofz),wewrite (5)it=e/.'/d2^ theequationtodetermineyis (6)^J=_ft_£!j,_y.dzpo /)o which isofthesametypeas(1).Thecomplete equivalenceofthegeneralised Riccatiequationwith thelinearequationofthesecond order isconsequently established. Theequationsofthissection havebeenexamined byAnisimov, Warschau Univ. Nach. 1896, pp.1—33. [Jahrhuchilber dieFortschritte der'Math. 1896, p.256.] *Nov.Comm. Acad. Petrop.viii.(1760—1761) [1763], p.32;seealsoashort paper byW.W. Johnson, Ann. ofMath. ni.(1887), pp.112—115. tThis isthegeneralisation ofJames Bernoulli's substitution (§1-1). Seealso Euler, Inst. Calc. Int. II.(Petersburg, 1769), §§831,852, pp.88,104. 4-2,4-21] DIFFERENTIAL EQUATIONS 93 4'21. Elder Htheorems concerningthegeneralised Riccatiequation. Ithasbeen shewn byEuler* that, ifaparticuhir solution ofthe generalisedRiccatiequationisknown, thegeneralsolution canbeobtained bytwoquadratures;iftwoparticularsolutions areknown thegeneral solution isobtainable byasingle quadrature f.And itfollows fromtheorems discovered byWeyrandPicard that, ifthreeparticular solutions arcknown, thegeneralsolution canbeeffected without aquadrature. Toprovethe first result, let i/,,heaparticularsolution of andwrite y=yo+^/v-Theequationinvis '^+{Q+2R>j,)v +R=0, ofwhich thesolution is Vexp{f{Q+2%o) dz]+JRexp{J{Q+2%,0 dz].dz=0, and, since v—l/(y— t/o)>thetruth ofthe firsttheorem ismanifest. Toprovethesecond, lety^andy^betwoparticular solutions, andwrite y-ihw=-—— . y-Vi The result ofsubstituting {yiW—y«)l{w—1)foryintheequationis 3/0-yidw _^wdy,_1^^p_^qIh'^v-Vq^^(y^'^^-y^ {w—\y- dzw—\dzw—1dz w-I \iv—1 and,whenwesubstitute for(dyjdz)and{dy^jdz)thevaluesF+Qy^+Ry^- andP+Qya+Ry^^,thelastequationisreduced to 1diu-r, r, tudz sothat tu=cexp|/(i?y„-Ryi) dz], where cistheconstant ofintegration. Hence, from theequation defining w, weseethatyisexpressedasafunctioninvolvingasingle quadrature. Toprovethethird result, lety^andy^bethesolutionsalready specified, le^2beathird solution, and let c'bethevalue tobeassignedtoctomake yreduce toy.,.Then y-ya^c_yo-yo y-yic' u-i-i/^' and this istheintegralinaform freefromquadratures. *Nov.Gomm. Acad.Fetvop.viii.(1760— 17G1) [1763], p.32. tIhid.p.59,and ix.(1762—1763) [1764], pp.163—164. SeealsoMinding, Journal fUr Math. XL.(1850), p.361. 94 THEORY OFBESSEL FUNCTIONS [CHAP.IV Itfollows that thegeneralsolution isexpressibleintheform Hence itisevident that, if^/i,y-u y-i,yibeanyfoursolutions, obtainedby givingCthevalues C\,Co,C^,0^respectively,then thecross-ratio {yi-y2)(y3-yA) (2/l-2/4)(2/3-2/2) isindependent ofz;foritisequalto (6\-a)((73-C,) (C.-QCCa-O,)' Inspiteoftheobvious character ofthistheorem, itdoes notseem tohave been noticed untilsomeforty years ago*. OtherpropertiesofthegeneralisedRiccatiequation maybederived from propertiesofthecorrespondinglinearequation (§4*2). ThusRafifyfhasgiven twomethods ofreducingtheRiccati equationtothecanonical form |+»'=^(f); thesecorrespondtothemethods ofreducingalinearequationtoitsnormal formbychangesofthedependent andindependentvariablesrespectively. Various propertie.softhesolution ofRiecati's equationinwhich P,§,Rarerational functions havebeen obtained byC.J.D.Hill,JournalfarMath. xxv. (1843), pp.23—37; Autonne, Comptes Reiidus,xcvi. (1883), pp.1354—1356; cxxviii.(1899), pp.410—412;and Jamet, Comptes Rendus deVAssoc. Francaise(Ajaccio), (1901), pp.207—228;Atm. dela Fac. desSci.deMarseille.,xii.(1902), pp.1—21. Thebehaviour ofthesolution nearsingularitiesofP,<^,Rhasbeen studied byFalken- hagen, NieuwArchiefroorWiskunde, (2)VI.(1905), pj).209—248. Theequationofthesecond order whose primitiveisofthetype Cir)i+C2r]2 +C3rj3^ C1C1+C2C2+CSC3' whereCi,c^, c-^areconstants ofintegration (whichisanobviousgeneralisation ofthe primitiveoftheRiccatiequation), hasbeen studied byVessiot, A7m. delaFac. desSci.de Toidouse,IX.(1895),no.6andbyWallenburg, JournalfitrMath. CXXi.(1900), pp.210 217; andComptes Rendus, cxxxvix.(1903), pp.1033—1035. *Weyr, Ahh.hdhm. Ges. Wiss.(6)viii.(1875—1876), Math. Mem. i.p.30;Picard, Ann. Sci. deVEcole norm. s^ip. (2)vi.(1877), pp.342—343. Picard'sthesis, inwhich theresult iscon- tained,isdevoted tothetheory ofsurfaces andtwisted curves— atheoryinwhich Kiccati's equation hasvarious applications. tNoiiv. Ann. deMath.(4)ii.(1902), pp.529—545. 4-3] DIFFERENTIAL EQUATIONS 95 4'3. Varioustransformations ofBessel'sequation. Theequationswhich wearenowabout toinvestigatearederived from Bessel'sequation byelementarytransformations ofthedependent andinde- pendentvariables. The firsttypewhich weshall consider is* where cisanunrestricted constant. Theequationisoffi-equent occurrence inphysical investigations, and, insuchproblems, pisusuallyaninteger. Theequationhasbeen encountered intheTlieoryofConduction ofHeat andthe TheoryofSound byPoisson, Journal deVEcolePolytechnique,xn.(cahier 19), (1823), pp.249—403; Stokes, Phil. Trans,oftheRoyalSoc.1868, pp.447—464[Phil. Mag. (4) XXXVI. (1868), pp.401—421, Math, andPhys. Papers,iv.(1904), pp.299—324]; Eayleigh, Proc. London Math. Soc. iv.(1873), pp.93—103, 253--283[Scientific Papers,i.(1899), pp.138, 139]. Thespecial equationinwhich p=2occurs intheTheoryoftheFigureof theEarth; seeEllis, Camb. Math. Journal,li.(1841), pp.169—177, 193—201. Sinceequation (1)maybewritten intheform ^d-{uz-^) d(uz-i) .„,,. ,1..,, _, . itsgeneral.solution is (2)ii=zi%+.{ciz). ConsequentlytheequationisequivalenttoBessel'sequation whenpis unrestricted, andnoadvantageistobegained bystudying equationsofthe form(1)rather than Bessel'sequation. But,whenjjisaninteger,thesolu- tions of(1)areexpressible"in finite termsf" (cf. §34),and itisthen frequentlydesirable toregard (1)asacanonical form. Therelations between varioustypesofsolutions of(1)willbeexamined indetail in§§4'41—4'4:). Tiiesecondtypeofequationisderived from(1)byatransformation of thedependentvariable which makes theindicialequationhave azero root. Theroots oftheindicialequationof(1)arep+l and—p,andsowewrite u=vz~P;wearethus ledtotheequation /Q\c?-y 2pdv2A ofwhich thegeneralsolution is (4) v=zP+^'&,,+,(ciz). •'SeePlana, Mevi. dellaB.Accad. delle Sci.diTorino, xxvi. (1821), pp.519—538, andPaoh, Mem. diMat. ediFis.della Soc.Italiana delle Sci.xx.(1828), pp.183—188. tThiswasknown toPlana, whostudied equations (1)and(5)inthepapertowhich reference Lasjustbeenmade. 96 THEORY OFBESSEL FUNCTIONS [CHAP.IV Equation (3),which hasbeen studied indetail byBach, Ann. Sci.deVEcole nm-m.sup. (2)III.(1874), pp.47—68, occurs incertain physical investigations;seeL.Lorenz, Ann. derPhysik undChemie, (2)xx. (188.3), pp.1—21[Oeuvres Scientifiques,i.(1898), pp.371— 396];andLamb, Hydrodynamics (Cambridge, 1906), §§287—291. Solutions ofequation (3) intheform ofcontinued fractions (cf.§§5-6,9-65)havebeenexamined byCatalan, Bulletin deVAcad. R.deBelgique, (2)xxxi. (1871), pp.68—73. SeealsoLePaige,ibid.(2)XLi. (1876), pp.1011—1016, 935—939. Next,wederive from(3),byachangeofindependent variable, anequation initsnormal form.Wewrite z=^^Iq,whereq=l/{2p +1),theequationthen becomes (5) .'^^,-cV^-'v=0, and itssolution is (6)v=a'i/qy'^'^^'^m,){ci^'^lq). When aconstant factor isabsorbed intothesymbol '^,thesolution maybe taken tobe r*"^l/(2,)(c^•W(Z)• Equation (5),which hasalready beenencountered in§4-14,hasbeen studied byPlana, Mem. della R.Accad. delle Sci.diTonno, xxvi. (1821), pp.519—538; Cayley,Phil.Mag. (4)XXXVI. (1868), pp.348—351[Collected Papers,vii.(1894), pp.9—12]; andLommel, Studien uher dieBesseVschen Functionen(Leipzig, 1868), pp.112—118. Thesystemofequationswhich hasnowbeen constructed hasbeen dis- cussedsystematically byGlaisher*, whoseimportant memoir contains an interestingaccount oftheresearches ofearlier writers. Theequationshavebeen studied fromadifferentaspect byHaentzschel f whoregardedthem asdegenerateforms ofLame'sequationsinwhich both of theinvariantsg.^andg-^arezero. Thefollowing papers byGlaisher should alsobeconsulted :Phil.Mag. (4)xliii.(1872), pp.433—438; Messenger,viii.(1879), pp.20—23; Proc.London Math. Soc. ix.(1878), pp.197—202. Itmaybenoted thattheforms ofequation (1)usedbyvarious writers areasfollows: ^J+y=^^^»(Plana), d^R n{n+\) /p• n -j^—a-u=" ",—-u.(Glaisher). Equation (5)hasbeenencountered byGreenhill|inhisresearches onthestabilityofa vertical poleofvariable cross-section, under theaction ofgravity. When thecross-section isconstant,thespecial equationinwhichj=fisobtained, andthesolution ofitleads to Bessel functions oforder+1. *Phil. Trans, oftheRoyal Soc. clxxii. (1881), pp.759—828;seealsoapaper byCurtis, Cam- bridge andDublin Math. Journal, ix.(1854), pp.272—290. tZeitschrtftfiir Math, undPhij<. xxxi.(1886), pp.25—33. XProc. Camb. Phil. Soc. iv.(1883), pp.65—73. 4-31] DIFFERENTIAL EQUATIONS 97 4'31. Lommel'stransformations ofBessel'sequation. Varioustypesoftransformations ofBessel'sequation wereexaminedby Lommel ontwooccasions; hisearlier researches* were ofasomewhatspecial type,thelaterfweremuch moregeneral. Intheearlierinvestigation,afterobservingthatthegeneralsolution of is (2) 3/=^'"g;(^), Lommelproceeded bydirect transformations toconstruct theequation whose generalsolution isz^''~''9$\{jz^), where a,/3,7areconstants. His result, which itwillbesufficient toquote,isthatthegeneralsolution of is (4)it=2^''-«'2?,(7^^). When /3=0,thegeneralsohition of(3)degeneratesinto andwhen7=0,itdegenei'atesinto unlessjiv\Hzero. Thesolution of(3)wasgiven expHcitly byLommel innvmierousspecial cases. Itwill besufficient toquote thefollowingforrefei'ence : (5)cfe^+.-rf^+n.^'-?)^*=^'^^-'^A^. (7),g*+(l_,)J+l„=0; «=.i''%^.(v/.).' ctz- az 4 (9)^2+/32/.-23--'u=0; u=zi-^i/c,^) (yz?). (11) P^±zu=0; u=zi^i(izi), 2H#j(§ul). Anaccount ofStokes' researches onthesolutions ofequation (11)willbegivenin §§6-4,10-2. *Studien liber dieBessel'scken Functionen (Leipzig, 1868), pp.98—120; Math. Ann. iii. (1871), pp.475—487. tMath. Ann. xiv.(1879), pp.510—536. w.B.F. 7 98 THEORY OFBESSEL FUNCTIONS [chap.IV Lommel's later researchesappearedatabout thesame time asamemoir byPearson*, andseveral results arecommon tothetwopapers.Lommel's procedure wastosimplifytheequation f d^jylxi^)] 2v-ld[ylx( z)] ,V^^ d[y^{z)Y yjriz) d^Jriz) %(^)' ofwhich thesolution is(§4'3) (12) y=x(^){^i^)V'^Mf(^)}- Onreduction theequationbecomes ^dz^[yjr(z) i^iz) x(^)Ja^ l[ylr (z) y}r{z) %(^)]%(^) %(^)^^^-'^ J-^ Now define thefunction(f)(z)bytheequation ItAvillbeadequatetotake (14) <^(^)=^'(^){%(^)Plt(^)}"-^- Ifweeliminate x(^)>itisapparentthatthegeneralsolution of ^ ^ dz"-^{z) dz^\_4^\<l,{z)\ 2cf>{z) 4<\^}r'(z)+ IS'Zyfr{z) (^(z) yfr(z)]+|^2(^)_^.+l yfr'(z)] ^'ir{z)\ j1J= Asaspecial case, ifwetake(p(z)=1,itisseenthatthegeneralsolution of y= <^^)2- IS (18)y=snir(z)/f'{z)}.<^^{.jr(z)}. Next, returningto(13),wetakex(z)=[y\r{z)Y-\ andwefindthatthegeneral solution of is (20)y=W{z)Y'i^^{y^{z)]. *Messenger, ix.(1880), pp.127—131.fThefunctions x(z)andi/-(z)arearbitrary. 4-32] DIFFERENTIAL EQUATIONS 99 Thefollowingarespecialcases of(17): (21) g+(e--^^),/=0;y=^g^{f), Theindependentresearches ofPearsonproceeded onverysimilar lines exceptthathestarted from Bessel'sequationinstead offrom themodified form ofit.Thereader willfindmany specialcases ofequation (17)worked outinhispaper. Apartialdiflferentialequation closely connected with(7)and(8),namely d-u ,, .vu du Z;^+(l+I/)--U. .-=0, hasbeeninvestigated byKepinski,Mat/>. Ann. Lxr.(1906), pp.397—405,andMyller- Lebedeff, Maih. Ann. Lxvi.(1909), pp.325—330. Thereader may verify thatKepinski's formula ''=— j""!• i—1\^^[-^^-r^)^^''^^"^ isasolution, when/(2/j) denotes anarbitrary function ofiv. Thespecial case oftheequation when i/=—1wasalsoinvestigated liyKepinski,Bull, int.de I'Acad, desSet.deCracuvie, 1905, pp.198—205. 4"32. Malmsten^sdifferential equation. Twenty yearsbefoi-e Lonnnelpu]:)lishedhisresearches ontransformations ofBessel's equation, Malmsten*investigated conditions fortheintegrabilityinfinite terms ofthe equation ,,, d-v rdvf , s\ which isobviouslyageneralisationofBessel'sequation; and itisaspecial caseof^4-31 (15). Toreduce theequation, Malmsten chosenewvariables defined bytheformulae wherepr,ndqareconstants tobesuitablychosen. Thetransformed equationis d^'^,^ n ,Idur, .,^,^„,„sq^-pq (pg+rq-q)~\ ^-^+(2M-g +l+ ry;-)^^-^= |^.4f/C"»"-''^--+^— ^-Y^^ ^'ju. ^^5?echoose pandqsothat thismayreduce totheequationof§4-3(1)consideredby Plana, andtherefore wetake 2pq-q+l+qr=0,{m+2)q=2, sothatp=-ir—|?n. Theequation then reduces to d^u^A^ y2{4g+(l-r)2}-l ]». *Camb. andDublin Math. Journal, v.(1850), pp.180—182.Thecase inwhich s=hadbeen previously considered byMiUmsten, JournalfilrMath, xxxix.(18.50), pp.108—11-5. 7—2 100 THEORY OFBESSEL FUNCTIONS [CHAP.IV By§4-3this isintegrableinfinite terms if where nisaninteger ;sothat J)4s+(l-r)^} Theequationisalsoobviously integrableinthetrivial cases.4=0andm=-2.(2),.+2=±^^^^ 4-4. Thenotation ofPochhamvier forseriesofhyper geometric type. Acompact notation, invented byPochhammer* andmodifiedb}^Barnesf, isconvenient forexpressingtheseries which aretobeinvestigated. Weshall writenowandsubsequently {0L)n=ci{a+ l)(a+2)...{a+n-1), (a)„=1. Thenotation which willbeused is,ingeneral, 00 ITT /. \_V («i)n {Cli)n• •(«p)n,. pJ^gK^i,0(2,...,ap, pi,p.,...,Pq,2)—wrr~Y~Y~\ 7T\^ n= ''\Pi)n \Pi/n \Pq)n Inparticular, ^{Ph 2;(p)2 ^.{ph -i-i°^ z- n=0^^-\p)n ,F,(p: z)=S n=0ni{p)n Thefunctions definedbythe first three series arecalledgeneralised hyper- geometricfunctions. Itmaybenoted here thatthefunction iF^(a;p;z)isasolution ofthe differentialequation and,whenpisnotaninteger,anindependentsolution ofthisequationis z^-^.,F,{a-p +l;-I-p; z). Itisevident that Variousintegral representations offunctions ofthetypes ji^j,0F2, 0-^3havebeen studied byPochhammer, 3Iath. Ann. xli.(1893), pp.174—178, 197—218. *Math. Ann. xxxvi.(1890), p.84;xxxviii.(1891), pp.227,586, 587. Cf.§4-15. tProc.London Math. Soc.(2)v.(1907), p.60.Theinodilieation duetoBarnes istheinsenion ofthesuffixes_pandqbefore andafter theFtorender evident thenumber ofsetsoffactors. 4-4,4-41] DIFFERENTIAL EQUATIONS 101 4'41. Various solutions inseries. Weshallnowexamine various solutions oftheequation ^-c^u=1-^-2u, andobtain relations between them, which will forthemostpartbeexpressed inPochhammer's notation. Itissupposedforthepresent thatpisnotapositive integerorzero, and, equally,since theequationisunalteredbyreplacing pby—^3— 1,itis supposedthatpisnotanegative integer. Itisalready known(§4-3)that thegeneralsolution* isz^^S'p+'^iciz),and thisgivesrisetothespecialsolutions zP+'.oF,{p +r,ic'z') ;3-P.oF,{l-p; ic'z'). Theequation maybewritten intheforms which aresuggested bythefactthat thefunctions e*"^^aresolutions ofthe original equationwiththeright-handsidesuppressed. When^iswritten forz{djdz),thelastpairofequations become (^- jt)-1)(^+jj).{ue^'') ±2c^^ (weT")=o. When wesolve these inseriesweareledtothefollowingfourexpressionsforu: zP+'e^'.^FAp +l;2p+2;-2cz); z'Pe'' .,F,(-p;-2p: -2cz); ^p+iQ-cz,^F,{p+l;2p+2;2cz) ;z-^'e''-.iF^{-p\ -2p; 2cz). Now,bydirectmultiplicationofseries, thetwoexpressionsonthe leftare expansibleinascendingseriesinvolvingzV'^'^, z^"^-, zP'^\....Andtheexpressions ontheright similarlyinvolve z'^, z^-p, z--'',....Since none ofthetwosetsof powersarethesamewhen2pisnotaninteger, wemust have (1) e'^'.,F,(p+l; 2p-v2--2cz)=e'" .,F,(p+I;2p^2; 2cz) ^oFAp +hic'^')^ J^) e''.,F,(-p;-2p;-2cz)=e-« .,F,(-p;-2p;2cz) =oF^{i-p; \<fn These formulae aredue toKummerf. When (1)hasbeenprovedforgeneral values ofp,thetruth of(2)isobvious onreplacing pby-p—lin(1). Wenowhave toconsider thecaseswhen 2pisaninteger. *Itfollows from §3-1thataspecial investigationisalsonecessary whenpishalf ofanodd integer. tJournal fUrMath. xv.(1836), pp.138—141. 102 THEORY OFBESSEL FUNCTIONS [CHAP.IV Whenphasanyofthevalues i,f,|,...,thesolutions which contain z'P asafactor have tobereplaced byseries involving logarithms (§§3-51, 3-52), andthere isonlyonesolution which involves onlypowersofz.Bythe previous reasoning, equation (1)still holds. Whenphasanyofthevalues 0,1,2,...acomparisonofthelowestpowers ofzinvolved inthesolutions shews that (1)stillholds;but itisnotobvious thatthere arenorelations oftheform z-P,F,{\-p- ic^z"-)=z-Pe'\FA-p- -'2p: -2cz)+kzP+\F,ip +^;\&z^ =z-Pe-'\F,{-p; -2p; 2cz)+hzP+\F, {p+^;\c'z% where kj,koareconstants which arenotzero. Weshallconsequentlyhave togiveanindependent investigationof(1) and(2)whichdependsondirectmultiplicationofseries. Note. Inaddition toKummer's researches, thereader should consult theinvestiga- tions oftheseries byCayley,Phil.Mag. (4)xxxvi.(1868), pp.348—3.51[Collected Papers, VII. (1894), pp.9—12] andGlaisher,Phil. Mag. (4)XLiii.(1872), pp.433—438; Phil. Trans, oftheRoyalSoc.CLXXii. (1881), pp.759—828. 4'42. Relations between thesolutions inseries. Theequation e'\F^{p +\\2p+2;-2cz)=e-'\F,{p +\;2p+2;2cz), which formspartofequation (1)of§4-41,isaparticularcase ofthemore generalformula due toKummer* (1) .FAa; p;=e^.FAp-a; p;-0, which holds forallvalues ofaandpsubjecttocertain conventions (whichwill bestatedpresently)which have tobemadewhen aandparenegative integers. We firstsupposethatpisnotanegative integerandthenthecoefficient of ^"intheexpansionoftheproductoftheseries fore^andiF^{p—a;p;-t)is „=o(w-m)! w!(p)^ w!(p)nm=o nl(p)n (a)«2nC„i.{p- «),„(1-p- n)r, 1=0 ,(1-a-n)n n !(p)n' ifwefirstuseVandermonde's theorem fandthen reverse theorder ofthefactors inthenumerator; andthelastexpressionisthecoefficient of^^iniFi(a, p',0- Theresultrequiredistherefore established when aandphavegeneral complex valuesJ. *Journal furMath. xv.(1836), pp.138—141; seealso|,Bach, Ann. Sci.deVEcole norm. sup. (2) III.(1874), p.55. tSee, e.g.Chrystal, Algebra,ii.(1900), p.9. XAnother proof depending onthetheoryofcontour integration hasbeen given byBarnes, Trans. Camb. Phil. Soc. xx.(1908), pp.254—257. 4-42] DIFFERENTIAL EQUATIONS 103 When pisanegative integer, equation (1)isobviously meaninglessunless alsoaisanegative integerand |a ]< |p|.Theinterpretationof(1)inthese circumstances willbederived byanappropriate limiting process. First letabeanegative integer (=—N)and letpnotbeaninteger,so that thepreceding analysisisvalid. The series iF-^{—N; p:^)isnowa terminating series, while ^F^ip+N;p;—^)isaninfinite series which con- sists ofiV+1terms followed byterms in\vhich theearlier factorsp+N, p+N+1,p+N+2,...inthesequencesinthenumerators canbecancelled with thelater factors ofthesequences p,p+I,p+2,...inthedenominators. When these factors havebeen cancelled, theseries fori^i(-iV;p;^)and li^'i(p+iV;p;—O^11'*^both continuous functions ofpnearp——M,where Misanyoftheintegers N,N -\-l,N +2,.... Hence we'may proceedtothelimitwhenp-*—M,andthelimitingform of(1)maythenbewritten* (2) ,i^,(-iV; -.¥;^)-]=e^,F,(N-M;-M; -0^, inw^hich thesymbol 1means thattheseries istostopattheterm in^^\i.e. the lastterm inwhich thenumerator does notcontain azero factor, w^hile thesymbol ']means thattheseries istoproceed normallyasfarastheterm in(^^'~^\ andthen itistocontinue wnthterms in^^^^"*"\ ^-^^+-,...,thevanishing factors innumerator anddenominator beingcancelled asthoughtheir ratio wereoneofequality. With thisconvention, itiseasytoseethat (3) ,F,(-N; -M:01=.F,(-.Y; -M;Ol When wereplaceNhyM—Nand^by- ^,wehave (4) ,F,iN-M; -M--^)^=,F,{N-M;-J/;-^1 N\{M-N)\.^^M+l^(,Y+1 :^v+2;-H- Asanordinarycaseof(1)wehave y^,F,(M-N-\-l- M+2; ^)=e^.FiN +1;M+2;-^), andfrom thisresult combined with(2),(3)and(4)wededuce that (.5) ,F,(-N; -M;0^=e^^F,{Iy-M,-^r:-n-^. Thiscould havebeen deriveddirectlyfi'om (1)bygivingp-a (insteadofa) anintegral value, andthenmaking ptend toitslimit. *Cf.Gayley, Messenger (old series),v.(1871), pp.77—82[Collected Papers,vm.(lB95),pp.4oS— 462],andGlaisher, Messenger,viii. (1879), pp.20— "23. 104 THEORY OFBESSEL FUNCTIONS [CHAP.IV Wenextexamine theequation (6) ^\F, (^+1 ;2p+2;-2cz)=o^^i(^+f;\c'z% which forms theremainder ofequation (1)in§4'41,andwhich isalsodueto Kummer*. Ifwesupposethat2pisnotanegative integer,thecoefficient of{czy'in theproductoftheseries onthe leftin(6)is «(-2r {p+1)„,^{-YI2,„(p+lX„(-n- 2;)-!)„_„, ,„=o{n-m)\m\ (2p+2),„ (2p+2)„,„^o ml{n~m)! 1"" Now— -S2'" .„C,rt (^J+l)m(—n-2p- l)n-misthecoefficient of«"inthe expansionof(1-2t)-P-' (1-t)'^+^'-^\ andsoitisequalto 1/•(0+)\no+) 27r*/•(0+) 1no+) (1-2t)-v-' (1-^)»+2i>+i r"-i dt=-—.\ (1-!/2)-p-h<-'»-irfM, where u=t/(l—t)andthecontours enclose theorigin butnoothersingularities oftheintegrands. B}^expandingtheintegrandinascending powersofu,we seethattheintegraliszero ifnisodd,but itisequal to~^i~~^ whennis even. Hence itfollows that _^jczT' andthis istheresult tobeproved. When wemakeptend tothevalue ofanegative integer,-N,wefindby thesamelimiting processasbefore that lim,F^{p+\:2p+2;-2cz)=,F,(1-N; 2-2iY;-2cz)1^'^ (-V-'(N-1)]N^^ i2N-2)l(2N) Y(-^"^>""'•^^^(^' '2^5-2cz). Itfollows that oF,{^-N; ic'z"-)=e'''.,F,{l-N; 2-2N; -2cz)n (-Y(]V-l)\Nl •^ (2i\r- 2)!(2^)!^^''^''''" "" •^^^^^" 2iV^' -2c^>- Ifwechange thesignsofcandzthroughout andaddtheresults soobtained,wefindthat (7)2.,F,{^-N; lc-^z')=e".,F,(l-N; 2-2N- -2cz)1 +e-'' .^F,(1-iY;2-2N; 2cz)~^ , *JournalfilrMath. xv.(1836), pp.138—141. Inconnexion with theproof given here, see Barnes, Trans. Camh. Phil. Soc. xx.(1908), p.272. 4-43] DIFFERENTIAL EQUATIONS 105 theother terms ontheright cancelling byauseofequation (1). Thisis,of course, theexpressionforJ-^r+}, (icz)infinite terms withadifferent notation. ForBarnes'proofofRummer's formulae, bythemethods ofcontour inte- gration,see§6"5. 4•43.Sharpes differential equation. Theequation (1) .g+|+(.+4),=0, which isageneralisationofBessel'sequationforfunctions oforder zero, occurs inthetheoryofthereflexion ofsoundbyaparaboloid.Ithasbeen investigated bySharpe*, whohasshewn that theintegralwhich reduces to unityattheoriginis (2) y=G\ (io^{zGo^6 +A\o^(tot\d)d6,Jo where •iff (3)1=C['"cos(Alogcothd)dd. Jo This istheappropriatemodification ofParseval'sintegral (§2'3). Toin- vestigateitsconvergencewrite cos6=tanh(f),and itbecomes (4)rcos(^0+. tanh<^) ^ '^ ./(, cosh(/)^ Itiseasytoseefrom thisform oftheintegralthat itconvergesfor(complex) values ofAforwhich j/(^)'<1,andf 2C=—cosh^ttA. IT TheintegTalhasbeeninvestigatedingreatdetail bySharpeandhehas givenelaborate rules forcalculatingsuccessive coefficients intheexpansionof yinpowersofz. Asimpleform ofthesolution (which wasnotgiven bySharpe)is 2/=e±'2,Fi(i +iU; 1;+2{z). Thereader should havenodifficultyinverifyingthis result. *Messenger,x.(1881), pp.174—185;xii.(1884), pp.66—79;Proc.Comb. Phil. Soc. x.(1900), pp.101—136. tSee, e.g.Watson, Complex IntegrationnndCanchtfs Theorem (1914), pp.64- -65. 106 THEORY OFBESSEL FUNCTIONS [CHAP.IV 4"5.Equations oforderhigherthan thesecond. .Theconstruction ofadifferentialequationofanyorder, which issoluble bymeans ofBessel functions, hasbeen effected byLommel*; itspossibility dependsonthe fact thatcylinderfunctions exist forwhich thequotient Wy(z)l^_y (2)isindependentof2. Each ofthefunctions Jn{z) andYn{z),ofintegral order, possessesthis property [§§2-31, 3-5]; andthefunctions ofthethird kindiT,''' {2), H^'^^ (z) possessit(§3"61), whether visanintegerornot. Nowwhen§3'9(5)iswritten intheform (1)^2^"^^ (7^2)=(Iy)-^*'"—)K-.n (7V^), thecylinderfunction ontherightisoforder—yifm=2v. This isthecase either(i)ifvisaninteger, n,andm=2n,or(ii)if V=n+landm=2?i+1. Hence if9^ndenotes either J,iorYn,wehave From thisequation weobtain Lommel's result thatthefunctions 2^^Jn(y V^)? 2i^Yn{'y ^/2)aresolutionsoff where 7hasanyvalue such that7^"=(—)"c^",sothat 7=icexp {r-rriln). (r=0,1,2,,..,n—1) Bygiving 7allpossiblevalues weobtain 2nsolutions of(2),andthese form afundamentalsystem. Next, if'^„+. denotesH^'\,+i, wehave'^_(n+A)=e'"+^"^'^n+i,sothat andhence 2'^'"''^^ H^^\+), (7^^z)isasolution of d^-^_(^cr^ where 7hasanyvalue such that7^''+!=c^"+i Q-^n+\)n%^ g^^j^g^^ y=-icexp {r7rij{n+^)}, (r=0,1,2,...,2?i) andthesolutions soobtained form afundamentalsystem. *Studien ilber dieBesseVschen Functionen(Leipzig, 18G8), p.120; Math. Ann. 11.(1870), pp.624—635. fThemore general equation hasbeen discussed byMolins, Mem. deVAcad. desSci.deToulouse, (7)viii.(1876), pp.167—189. 4-5] DIFFERENTIAL EQUATIONS 107 Forsomeapplicationsofthese results, seeForsyth, (Quarterly Journal, xix.(1883), pp.317—320. Inview of(1),which holdswhenmisaninteger, Lommel, Math. Ann. ii.(1870), p.635, hassuggested aninterpretationofa"fractional differential coefficient." Thus hewould /7\i interpret(-y-1" exp (+yv''^'^)tomean^r^^iyi Jz).The ideahasbeendevelopedatsome length byHeaviside invarious papers. Lommel's formulae maybegeneralised byconsidering equation (3)of §4"31, afterwritingitintheform (^+a)(^+a-2/3z')a=-^^'Z'^ ", thesolution oftheequation beingu—^^"""^^(7^^). For itiseasytoverity byinduction that, with thisvalue ofu, n-ln(^+a-2r/S) (^+a-2/3i/-2r/3)u=(-)"l3'-^c'"z'''^u, r= andsosolutions of 71-1 (4) n(^+a-2r/3)(^+a-2/3z/-2r/3)u=(-)"/3^V"^^'^ u areoftheform u=^^''~'"2f„ (72^), where 7=cexp (vTri/n). {r=0,1,...,n-1) Bygiving 7these values, weobtain 2nsolutions which form afundamental system. Inthespecialcase inwhich n=2,equation (4)reduces to (^+a)(^+ct-2^)(^+a-2/3v)(^+a-2/3z.-2/3)a=13'c'z'^a. Thisequation resembles anequationwhich hasbeenencountered byNicholson* inthe investigationoftheshapesofSponge Spicules, namely that istosay 5(5-1) (5+4,^-2) (S+4/i-3)ii=z^--i^u. Ifweidentifythiswith thespecialform of(4)weobtain thefollowingfourdistinct sets ofvalues fora,^,n,v : a 108 THEORY OFBESSEL FUNCTIONS [CHAP.IV These fourcases givethefollowing equationsandtheir solutions : (6)5=w;^^=^*{<^i(^) +#i(^2)}, (8)g|,¥g|=.l«; u=z-^{<^i{izi)+WU¥^h], These seem tobetheonlyequationsofNicholson's typewhich aresoluble with theaid ofBessel functions;inthecasen=2,theequation (5)ishomogeneous. Nicholson'sgeneral equationisassociated with thefunction ^3-2fx 2+2fil+2fiz^-'^\ "^^V4-2/x' 4-2/i' 4-2/x' (4-2^)V 4*6.Symbolicsolutions ofdifferential equations. Numerous mathematicians havegivensolutions oftheequation §4'3(1) namely (1) £--=^-^^«. insymbolic forms, whenj:)isapositive integer (zero included). These forms areintimatelyconnected with therecurrence formulae forBessel functions. Ithasbeen seen(|4'3)thatthegeneralsolution oftheequationis 2i%+^{ciz); andfrom therecurrence formula§39(6)wehave zi%^,(ciz)=(-ciyzP^^lj£j'[z-i'S^ {ciz)]. Sinceanycylinderfunction oftheform'W^{ciz)isexpressibleas where aand /3areconstants,itfollows that thegeneralsolution of(1)may bewritten (2) „=.P«(4Y?£!!±^£r.\zdzj z Amodification ofthis,duetoGlaisher*, is where a=ajc,/8'=— /3/c.Thismaybeseenbydifferentiating aV^+/3'e~'^^ once. *Phil. Trans, oftheRoyal Soc. clxxii. (1881), p.813. Itwasremarked byGlaisher that equation (3)issubstantially given byEarnshaw, Partial Differential Equations (London, 1871), p.92.SeealsoGlaisher, Quarterly Journal,xi.(1871), p.269,formula(9),and p.270. 4-6] DIFFERENTIAL EQUATIONS 109 Note.Aresult equivalentto(2)wassetbyGaskiti asaproblem*intheSenate House Examination, 1839;andaproofwaspublished byLeslieEllis, Camb. Math. Journal,ii. (1841), pp.193—195,and alsobyDonkin, Phil. Trans,oftheRoyal Soc.CXLVii.(1857), pp.43—57.InthequestionassetbyGaskin, thesignofc^wasclianged,sothatthesolu- tioninvolved circular functions instead ofexponentialfunctions. Nextweshallprovethesymbolic theorem, duetoGlaisherf, that z'-(*) ^""(:^J= ii^, dz) z-f•>/)—.' Inoperatingonafunction with theoperatorontheright,itissupposed that thefunction ismultiplied by1/^-^"-before theapplicationofthe operatorsz^(d/dz). Itisconvenient towrite .=.«,.|^=a. andthen tousethesymbolicformula (5) /(^).(e"'Z)=e"'./(^+a)Z, inwhich aisaconstant andZisanyfunction ofz. Theproofofthisformula presents nospecialdifficulties when/ (5)isapolynomial in S,asisthecase inthepresent investigation. See, e.g.Forsyth,Treatise onDifferential Equations (1914), §33. Itiseasytoseefrom(5)that =e^i-P)B (^_2^+2)(^- 2^)+4)(^- 2jj+6)...:^, whenwebringthesuccessive functions e~"^(beginningwith those ontheleft) pasttheoperatorsoneatatime,byrepeated applicationsof(5). Wenowreverse theorderjoftheoperatorsinthe last result, andbya reversal oftheprevious procedureweget zP+'(4-T=^*'"^" ^(^-2)(^--4)...(^-2p+2) =e<^-^'«[(^+2^-2) (^+2p-4)...(^+2)^ .e-t^i'-^t«] =e-<z'+^)e[^e^e^^ (e-o^)... (e-«^). e-'^^"-'"] ^p+i dzj z-i'^- *Theproblem wasthesecond partofquestion 8,Tuesday afternoon, Jan. 8,1839;seethe Cambridge University Calendar, 1839, p.319. tNouvelle Corr. Math. ii.(187(3), pp.240—243, 349—350;andPhil. Trans, oftheEoijalSoc. CLXXii.(1881), pp.803—80.5. XItwasremarked byCayley, Quarterly Journal, xn.(1872), p.132, inafootnote toapaper by Glaisher, that differential operatorsoftheform s«+'^-2-",i.e.S-«,obey thecommutative law. no THEORY OFBESSEL FUNCTIONS [chap.IV andthis istheresult tobeproved.Ifwereplace phyp+1,wefindthat (6)2P+1 ',dz dz) z-P When wetransform (2)and(3)with theaidof(4)and(6),weseethat thegeneralsolution of(1)isexpressibleinthefollowingforms : 1/„dyae'^'+^e- (7) (8)u= zP+^V dz) z If,dy+^(xe'^+/3'e—cz 1p—\' —cz "- ^i'+3\fdz) Z2P Thesolutions oftheequation d^v 2pdv dz- zdz [(3)of§4-3],whichcorrespondto(2), (3),(7)and(8)are—c^v=0, (9) (10) (11) (12)=!^"-P+idyae^^+^e-"' zdzj z d,\^+^ V=-\Z' V=dz) z^'' ,d_\P+'ae"^ 4-^'e' 'Tzj ^2p Adiflferent andmore direct method ofobtaining (7)isduetoBoole, Phil. Trans,ofthe RoyalSoc.1844, pp.251,252;Treatise onDifferential Equations (London, 1872),ch.xvii. pp.423—425; seealsoCurtis, Cambridge andDublin Math. Journal,IX.(1854), p.281. The solution(9)was firstgiven byLeslieEllis, Camb. Math. Joxirnal,11.(1841), -p\).169, 193,andLebesgue, Journal deMath. xi.(1846), p.338; developmentsinseries were obtained from itbyBach, A^m. Sci.deVEcole norm.sup. (2)in.(1874), p.61. dP'VSimilar symbolic solutions fortheequation -1-5-c-z^'i~^v=0 were discussed byFields, JohnHopkins University Circulars,vi.(1886—7),p.29. Atransformation ofthesolution(9),due toWilliamson,Phil. Mag. (4)xi.(1856), pp.364—371,is (13)v= c'">(^^.-\{ae''-\-Sie-"). -1p1s This isderived from theequivalence oftheoperators -—,-—, whenthey operate on functions ofcz. Wethus obtain theequivalenceofthefollowing operators ^2^+1 :ci)"j]=<«^>="*{C:^*)"a =(p.^y-p+1-{cz) _\cz^ccj czj \_\cdcj cj' itbeing supposedthattheoperators operate onafunction ofcz;andWilliamson's formula isthen manifest. 4'7] DIFFERENTIAL EQUATIONS HI 4*7. Liouvillesclassification ofelementary transcendentalfunctions. Before wegiveaproofofLiouville'sgeneral theorem (which wasmentioned in§4-12) concerningtheimpossibilityofsolvingRiccati'sequation "infinite terms" exceptintheclassical cases discoveredbyDaniel Bernoulli (and the limitingform ofindex —2),weshallgiveanaccount ofLiouville's*theory ofaclass offunctions known aselementary transcendentalfunctions; andwe shall introduce aconvenient notation forhandlingsuch functions. Forbrevity wewritef k(z)=l{z)^ log z, I,{z)=I(I(z)), I,(z)=I(I,(Z)), ..., e,{z)=e(z)=e', e.(z)=e(e(z)), e-Az)=e(e,(z)), ,,f(z)=,f{z)=Jf{z)dz, ,,f(z)=,[,f{z)], ,,/(,)=,l,,/(^)},.... Afimction ofzisthen said tobeanelementary transcendentalfunction\ ifitisexpressibleasanalgebraicfunction ofzandoffunctions ofthetypes lr(j)(z), Cr^iz), '?rX(^).where theauxiliaryfunctions^{z), -^(z), xi^)^^^ expressibleinterms of ^^andofasecond setofauxiliary functions, andsoon; providedthatthere exists afinitenumbern,such thatthenthsetofauxiliary functions are allalgebraicfunctions ofz. Theorder ofanelementarytranscendental function ofzisthen defined inductivelyasfollows: (I)Any algebraicfunction ofzisoforderzero|. (II)If/,.[z)denotesanyfunction oforder r,thenanyalgebraicfunction offunctions ofthetypes lfr{z), efr{z\ ,f,(z), f.(z), fr_,(z),...f(z) (intowhich atleastoneofthe firstthree enters)issaid tobeoforder ?'+1. (III)Anyfunction issupposedtobeexpressedasafunction ofthelowest possibleorder. Thuselfr (z)istobereplaced byfr{z),and itisafunction of order rnotoforder ?•+2. Inconnexion with thisandthefollowing sections, thereader should study Hardy, Orders ofInfinity (Camb. Math. Tracts, no.12,1910). Thefunctions discussed byHardy were ofaslightly more restricted character than thosenow inider consideration, since, for hisBtirposes, thesymbolsisnotrequired, andalso,forhispurposes,itisconvenient to postulate therealityofthefunctions which heinvestigates. Itmaybenoted that Liouville didnotstudy propertiesofthesj'mbolyindetail, but merely remarked that ithadmany propertiesakin tothose ofthesymbolI. *Journal deMath. ii.(1837), pp.56—10.5;in.(18.38), pp.523—547 :iv.(1839), pp.423—45(5. tItissupposed thattheintegrals are alliudeiinite. X"Une fonction fiuieexplicite." §Forthepurposes ofthisinvestigation, irratioiKtl powersof:,sucli asz^,ofcourse must notberegarded asalgebraic functions. 112 THEORY OFBESSEL FUNCTIONS [CHAP.IV 4-71. Liouvillesfirsttheorem* concerninglineardifferential equations. Theinvestigationofthecharacter ofthesolution oftheequation d}u , , (1)^^=^^%(^>' inwhichx{z)isatranscendant oforderf n,hasbeenmade byLiouville, who hasestablished thefollowingtheorem : Ifequation (1)hasasolution luhich isatranscendant oforder ni+\,where m^n,theneither there exists asolutionoftheequationwhich isoforderf n, orelsethere exists asolution, u^,oftheequation expressibleintheform (2)u,=(f)^{2).ef^{z), whereft,.{z)isoforderfu,,and theorderof4>^{2) does notexceedfj.,and/mis such thatn^fi<:m. Iftheequation (1)hasasolution oforderm+1,letitbefm+i{^);then /m+i (^)isanalgebraicfunction ofoneormore functions ofthetypes If^ (z)^ ?/m(^X^fm(^)aswell as(possibly)offunctions whose order does notexceed m.Letusconcentrate ourattention onaparticularfunction ofoneofthe three types,and letitbecalled ^,^or©accordingtoitstype. (I)Weshall firstshewhow toprove that, if(1)hasasolution oforder m+1,then asolution canbeconstructed which doesnotinvolve functions of thetypes6and^. For, ifpossible,let/,„+i (z)=F(z, 6),whereFisanalgebraicfunction of6; andanyfunction ofz(other than 6itself) oforder ni+1which occurs inF isalgebraically independentof6. Then itiseasytoshew that ^^'dz''^^^~ dz^^f^Xz)dzdOdz \1df,,{z)Yd-^F \d\\df,,{z)\-\dF ^ (/,„{z) dz f8^"^ ^dz (/,„{z) dz\\de"^ •^^^'' itbeing supposedthat zand 6aretheindependentvariables inperforming thepartialdifferentiations. Theexpressionontherightin(3)isanalgebraicfunction of6which vanishesidentically when 6isreplaced bylf„^{z).Hence itmust vanish identicallyforallvalues of6;forifitdidnot,theresult ofequatingitto zerowouldexpress lfm{z)asanalgebraicfunction oftranscendants whose orders donotexceed nitogether with transcendants oforder wi+1which are, exhyputhesi, algebraically independentof6. *Journal deMath. iv.(1839), pp.435—442. tThisphraseisused asanabbreviation of"elementary transcendental function oforder «.-' tNull solutions aredisregarded ;ifuwere oforder lessthan n,then,would beoforder udz'' lessthann,which iscontrarytohypothesis. 4-71] DIFFERENTIAL EQUATIONS 113 Inparticular,theexpression ontherightof(3)vanishes when 6isreplaced hy6+c,where cisanarbitraryconstant;andwhen thischangeismade the expressionontheleftof(3)changesinto which istherefore zero. That istosay (4)^^-^^-F{z,0 +c).x(2)=0. When wedifferentiate(4)partiallywithregardtoc,wefindthat dF(z, e+c) d'Fjz, +c) ac' dc''••• aresolutions of(1)forallvalues ofcindependentofz.Ifweputc=after performingtheditferentiations, theseexpressions become dF{z, d) d-F{z,e) which areconsequentlysolutions of(1).Forbrevity theywill becalled ^0'Ke,• Now eitherFandFgform afundamentalsystemofsolutions ol'(1)or theydonot. Iftheydonot,wemusthave* Fe=AF, whereAisindependentboth ofzand 0.Onintegrationwefindthat F:=(^e^', where$involves transcendants (oforder notexceeding m+1)which are algebraically independentof6.But this isimpossiblebecause e^^ isnotan algebraic function of9;andtherefore FandFqformafundamental system ofsolutionsof(1). Hence Fg^ isexpressibleinterms ofFandFgbyanequationoftheform Fee=AFe+BF, whereAandBareconstants. Now thismayberegardedasalinearequation in6(with constant coefficients) and itssolution is F^^.e'^^ -^^,e^^ orF^e'^^[<i>,+<i>.d\, where<l>iand<I>2arefunctions ofthesame nature ascj)^while aandjBare theroots oftheequation X-—Ax—B=0. Theonlyvalue ofFwhich isanalgebraicfunction of6isobtained when a=/3= ;andthenFisalinearfunction <f0. Similarly,iff^,^^^ {z)involves afunction ofthetype ^,wccanprovethat itmust bealinear function of'^. *SinceFmust involve0,b'ecannot beidenticallyzero, w.B.¥. 8 114 THEORY OFBESSEL FUNCTIONS [CHAP.IV Itfollows that, insofarasfm^^ {z)involves functions ofthetypes6and^, itinvolves themlinearly,sothatwemaywrite /„^, {z)=te,{z)d,{z)... dp{z).^,{z)%{z)...\(z).ir,^ ,(z), where thefunctions^p,q(z)areoforderm+1atmost,andtheonlyfunctions oforderm+1involved inthem areofthetype0. Takeanyoneoftheterms inf,n+i (z)which isofthehighest degree, qua function ofO^, 6.,,...^i,^2, •••,and letitbe e,(z)6,{z)... dp{z).^,{z)... ^Q{z).tp,Q{z). Then, byarguments resemblingthosepreviously used, itfollows that dd d dd ^ -f (\ isasolution of(1);i.e.i/rp^(z)isasolution of(1). But'\jrpQ(z)iseither afunction oforder notexceeding m,orelse itisa function oforderm+1which involves functions ofthetypeandnotof thetypes6and^. Intheformer case,werepeattheprocessofreduction tofunctions oflower order, andinthelatter caseweseethatsome solution oftheequationisan algebraicfunction offunctions ofthetype@. Wehave thereforeproved that, if(1)hasasolution which isatranscendant ofordergreaterthan n,then either ithasasolution oforder norelse ithasa solution which isanalgebraicfunction offunctions ofthetype ef^{z)and <^^{z),wheref^{z)isoforderfiand0^{z)isofanorder which doesnotexceed/x. (II)Weshall nextprove that,whenever(1)hasasolution which isa transcendant ofordergreaterthan n,then ithasasolution which involves thetranscendantef^{z)onlyinhavingapowerofitasafactor. Weconcentrate ourattention onaparticulartranscendant oftheform ef^{z),andthen thepostulatedsolution maybewritten intheform G{z, 0), whereGisanalgebraicfunction of;andanyfunction(other thanitself) oforderyu.+1which occurs inGisalgebraically independentof0. Then itiseasytoshew that Ovr+^[/m"(^)+{/m'W]^^3-G^.x(^). Theexpressionontherightisanalgebraicfunction of which vanishes when isreplaced byef^{z), andsoitvanishesidentically, bythearguments used in(I).Inparticularitvanishes when isreplaced byc0,where cis independentofz.But itsvalue isthen ^^-'-(?(5,c0).X{z). 4-71] DIFFERENTIAL EQUATIONS 115 SOthat (6) ^'-G{z,c%).x{^)=^- When wedifferentiate thiswithregardtoc,wefindthat dG{z,c%)d-Hl(z^&) dc' d^'' ••• aresolutions of(1)forallvalues ofcindependentofz.Ifweputc=1,these expressions become ds'd^''•• Hence, bythereasoningused in(I),wehave^)Gq=AGorelse ®-'Ge@=AeGe +BG, whereAandBareconstants. Intheformer caseG=^%'^, andinthelatterGhasoneofthevalues <J»jBy+(J)„e« or0y[(l>i+<!>,log0|=6^{Oj+$,/; (z)], where<$>, <t>i,<J>oarefunctions ofzoforder/x+1atmost, anyfunctions of order/u,+lwhich areinvolvedbeing algebraically independentof©;while 7andSaretheroots oftheequation a;(x-1)-Ax—B=0. Inanycase,Geither contains @onl}^byafactor which isapowerof orelse Gisthesum oftwoexpressionswhich contain Honlyinthatmanner. Inthe latter case*, G(z,ce)-c^G{z,f)) isasolution of(1)which contains Bonlybyafactor which isapowerofB. Byrepetitionsofthisprocedure, weseethat, ifBj,Bo,...B,.are allthe transcendants oforder/a+1which occur inthepostulated solution, wccan derive from that solution asequenceofsolutions ofwhich thesthcontains Bj,Bg,...B^onlybyfactors which arepowersofBj,Bo, ...B^;andtherth meniber ofthesequence consequentlyconsists ofaproductofpowersof Bi,Bo, ...B^multiplied byatranscendant which isoforder/j.atmost; this solution isoftheform </>M(^)exp\S^7.log <^).|, which isoftheform^^(z).e/^(z). *If<i>iisnotidentically zero;ifitis,then<!>._,0^isasolution ofthespecified type. S—2 116 THEORY OFBESSEL FUNCTIONS[CHAP. IV 4*72. Liouvilles second theorem concerninglineardifferential equations. Wehavejustseen that, iftheequation (1)'J="%W [inwhich x(^)i^oforder7i]hasasolution which isanelementarytran- scendant ofordergreaterthan n,then itmust have asolution oftheform 0M(^)e/;{z), where/u.^/;.Iftheequationhasmore thanonesolution ofthistype,leta solution forwhich/u.hasthesmallest value bechosen, and letitbecalled u^. Liouville's theorem, which weshallnowprove,isthat,forthis solution, the orderofd(logu^jdzisequalton. Let dlog u,^^ dz~ andthen tisoforderyu.atmost;lettheorder oftbeiV,whereN^/x. IfN=n, thetheorem requiredisproved.IfN>n, then theequation satisfied byt,namely (2)^1+^'=^(^)' hasasolution whose orderNisgi'eaterthan n. Now tisanalgebraicfunction ofatleast onetranscendant ofthetypes IfN-ii^), <;fN-i{z), ef^-iiz)and(possibly)oftranscendants whose order does notexceed iV—1.We callthe ijrst tliree transcendants 6,^,@respectively. Iftcontains more than onetranscendant ofthetype d,weconcentrate ourattention onaparticularfunction ofthistype,andwewrite t=F(z,e). Byarguments resemblingthose used in§4"7l,wefind that, ifiV^> n,then F(z,d+c) isalsoasolution of(2).Thecorrespondingsolution of(1)is expjF(z,6+c)dz, and this isasolution for allvalues ofcindependentofz.Hence, by differentiation withrespecttoc,wefindthatthefunction u.defined as l{exil>jF{z,e+c)dz] _dc isalsoasolution of(1) ;andwehave U2= itiJFgdz, sothat du.y dui ,„c=o 4-72, 4-73] DIFFERENTIAL EQUATIONS 117 ButtheWronskian ofanytwosolutions of(1)isaconstant*; andso ih'Fe=C, where Gisaconstant. If=0,J"isindependentof^,which iscontrarytohypothesis ;soC:^0,and u,=sJiCIF,). Hence Mjisanalgebraic function of6 ;andsimilarlyitisanalgebraicfunction ofallthefunctions ofthetypes6and^which occur in t. Next consider anyfunction ofthetype©which occurs int;wewrite t=Cr{z, ©), and,byarguments resemblingthose used in§4-71andthose used earlier in this section, wefindthatthefunction it.,defined as 7)V expJG^(^, cQ-y)dzdc isasolution of(1) ;andwehave0=1 sothat du; da, ^^ ThisWronskian isaconstant, C'l,andso Consequently u-^isanalgebraic function, notonlyofallthetranscendants of thetypes6and ?r,butalsoofthose oftypewhich occur int;andtherefore ^/lisoforder N.This iscontrarytothehypothesisthat u,isoforder/a+1, where ^^N,ifN>n. Thecontradiction shews thatNcannot begreaterthann;hence theorder ofd([ogu,)ldzisn.And this isthetheorem tobeestablished. 4"73. Liouville'stheorem-]-thatBesseVsequationhasnoalgebraic integral. Weshallnowshew thattheequation hasnointegral (other than anull-function) which isanalgebraicfunction ofz. We firstreduce theequationtoitsnormal form bywriting y=uz~,i^==±v—\- *Seee.g.Forsyth, Treatise onDifferential Equations (1914), §65. tJournal deMath. iv.(1839), pp.429—435;vi.(1841), pp.4—7. Liouville's firstinvestigation wasconcerned with thegeneral case inwhich xi^)isanypolynomial;theapplication (with "various modifications)toBessel's equation wasgiveninhislater paper.Journal deMath. vi. <1841), pp.1—13, 36. 118 THEORY OFBESSEL FUNCTIONS [CHAP.IV This isoftheform dht where (2) xW=«^^-i^^-i. Ifpossible,letBessel'sequationhaveanalgebraic integral;then(1)also hasanalgebraic integral.Lettheequationwhichexpressesthisintegral, w, asanalgebraicfunction ofzbe (3) 64{u,z)=% where^isapolynomialboth in iiandinz;and itissupposedthatS€is irreducible*. Since uisasolution of(1)wehave (4) MuuMi-'l£4uz^uUz-\-r9lzz-S€r^^-6€^^ux{z)=^. Theequations (3)and(4)have acommon root,andhence alltheroots of(3)satisfy (4). For, ifnot,theleft-hand sides of(3)and(4){(luafunctions ofii)would have ahighestccrmmon factor other thanS4 itself, and thiswould bea polynomialinuandinz.Hence s€would bereducible, which iscontraryto hypothesis. Let alltheroots of(3)beMj,lu,...w,^.Then,ifsisanypositive integer, isarational function ofz;andthere isatleast onevalue ofsnotexceedingMforwhich thissum isnotzerof. Letanysuch value ofsbetaken, and let M—— tfmm=\ M'du.Also let Tf,=5(5-1)...(s-r+1)^iz/^*-'"r'^X ,m=\ \dzJ wherer= 1,2,...s.Sinceu^,lu,... u^jare allsolutions :Jof(1),itiseasyto provethatdW (5)'^•=,r., dW (6)~^=^Vr^^+r{s-r+l)x{z)W,-u (r=1,2,....-1) *That istosay,.^4hasnofactors which arepolynomials innorinzorinboth «and z. tIfnot, alltheroots of(3)would bezero. +Because(4)issatisfied byalltheroots of(3),quaequation in«. 4-73] DIFFERENTIAL EQUATIONS 119 Since W^isarational function ofz,itisexpressibleinpartial fractions, sothat ,.._VJ„ ,V__?«.? whereA^andBn,qareconstants, kand A.areintegers,nassumespositive integralvaluesonlyinthelastsummation andciq^0. Letthehighest powerofl/(^— a.q)which occurs inTFobe\j{z—a^y. Itfollowsbyaneasyinduction from(5)and(6)thatthehighest powerof 1/(2— a,y)inWris1/(2— UqY'^'',where ?=1,2,... .9. Hence there isahigher poweronthe leftof(7)thanontheright.This contradiction shews thatthere arenoterms ofthetype Bn,q (z—aq)~^inW^ n=-K Wemaynowassume that J.a=^0, because thisexpressionforW^must have alastterm ifitdoesnotvanishidentically. From(5)and(6)itiseasytoseethattheterms ofhighest degreeinz which occur inIfo, l^i>^2, 1^3.•••are* A^z\ \A>,z^-\ A^sz\ \A>,{'^s-^2)z^-\.... Byasimpleinduction itispossibletoshew thattheterm ofhighest degree inTfo.is ^;,^M.3...(2r-l).s(s-2)...(s-2r+2). Aninduction ofamorecomplicatednature isthennecessarytoshew thatthe term ofhighest degreeinW^r+\is \A,z>^-' 2.4...(2r). (s-1)(s-3).. .(,§-2r+\)..^, (i,-!«;i-hj] l),-+a, where thesuffix r+1indicates thatthe first ?•+1termsonlyofthehyper- geometricseries aretobetaken. Ifsisodd,theterms ofhighest degreeonthe leftandrightof(7)are ofdegrees X—2and A.respectively,which isimpossible.Hence TFovanishes whenever sisodd. When 5iseven, theresult ofequatingcoefficients ofz^~'^in(7)is \Ak.s\ =-\A,. s\,F,(1,-is;i-hs]l)y. That istosay XA^.sl oFi{h,-^s; ^-hj;1)=0, and so,byVandermonde's theorem, ^^1.3.0 ...(.9—1) Theexpressiononthe leftvanishesonlywhen X,iszero^f*. *Itistoberemembered thattheterm ofhighest degreeinxi^)^^-1- tThe analysis given byLiouville, Journal deMath. vi.(1841), p.7,seems tofail atthis point, because heapparentlyoverlooked thepossibilityof\vanishing. The failure seems in- evitable inview ofthefactthatjf^^j(z)+J'tn-h('>^*^^^^algebraicfunction ofz,by§3-4.The subsequent partoftheproof givenhere isbased onasuggestion made byLiouville, Journal de Math. IV.(1839), p.435; seealsoGenocchi, Mem. Accad. delle Sci. diTorino, xxiii.(18(51)), pp.299—362;Coniptes Eendus, lxxxv.(1877), pp.391—394. 120 THEORY OFBESSEL FUNCTIONS [CHAP, IV Wehave thereforeproved that,when sisodd,Wovanishes, andthat,when siseven,Wnisexpressibleintheform V4~—n H=() whereJ.o,sdoesnotvanish. From Newton's theorem whichexpressesthecoefficients inanequation interms ofthesums ofpowersoftheroots, itappearsthatMmust beeven, andthattheequationrS4{u,z)= isexpressibleintheform (8) u^'+ '1ii''-'^'-}%.(l/z)=0, r=l where thefunctions ^^arepolynomialsinl/z. When wesolve(8)inaseries ofascending powersofl/z,wefind that each ofthebranches ofuisexpressibleintheform w= where nisapositive integer and, inthecase ofonebranch atleast, Cqdoes notvanish because theconstant terms inthefunctions^,.arenot allzero. Andtheseries which areoftheform 00 ?n= areconvergent*forallsufficiently largevalues ofz. When wesubstitute theseries intotheleft-hand sideof(1),wefindthat thecoefficient oftheconstant term intheresult isCo,and so,forevery branch, Comust bezero, contrarytowhat hasjustbeenproved. The contradiction thusobtained shews that Bessel'sequationhasnoalgebraic integral. 4"74. Ontheimpossibility ofintegratingBessel'sequationinfiniteterms. Wearenow inapositiontoproveLiouville's theorem fthat Bessel's equationforfunctions oforder vhasnosolution(exceptanull-function) which isexpressibleinfinite terrasbymeans ofelementarytranscendental functions, if2visnotanoddinteger. Asin§4"73,wereduce Bessel'sequationtoitsnormal form (1)£=«^<^)- where-^{z)=-I+p(p+l)/2''andp=±v—^. Now writed(logu)/dz=t,andwehave (2)^^^.,i_£0-l).o. *Goursat, Cours d'Analyse,ii.(Paris, 1911), pp.273—281. Manytreatisestacitly assume the convergence ofaseries derived inthismanner fromanalgebraic equation, tJournal deMath. vi.(1841), pp.1—13, 3G. 4-74] DIFFERENTIAL EQUATIONS 121 Since%(z)isoforder zero, itfollows from§4'72 that, ifBessel'sequation hasanintegral expressibleinfinite terms, then(2)must have asolution which isoforder zero, i.e.itmust haveanalgebraic integral. If(2)hasanalgebraic integral,lettheequationwhichexpressesthis integral,t,asanalgebraicfunction ofz,be (3) S4(t,z)=0, where S^-isanirreducible polynomialintand z. Since tisasolution of(2),wehave (4) az+[x(^)- 1']^*=0. Asinthecorresponding analysisof§4*73, allthebranches oftsatisfy (4). Firstsupposethatthere aremore than twobranches oft,and letthree ofthem becalled ti,to,U,thecorrespondingvalues ofu(definedasexp^tdz) being «i,Uo,u^.These functions areall.solutions of(1)andsotheWronskians du-i du2 dui dii-^ duo dui '"''di~ ""'~dJ''''^~'''~cU''''Jz" '''~d^ areconstants, which willbecalled C\,C^,0^. Now itiseasytoverifythat ri dtt-i dvo ,,'(JI=Mo—JK'S-J-—U-iUs{is— 1.2}'. and ^3— 1.2isnotzero, because, ifitwere zero, theequation (3)would have a pairofequal roots, andwould therefore bereducible. Hence Cj^0,andso ii^Us=Gi/{ts- to). Therefore U0U3(and similarly UsUiand MjMo)isanalgebraicfunction oiz. But Ui=./-" , andtherefore u^isanalgebraicfunction ofz.This, aswehave seen in§4-73, cannot bethecase,andsothasnotmore thantwobranches. Nextsupposethat thastwobranches, sothat -SW {t,2)isquadraticin t. LetTthe branches beU±\/V,whereUandVarerational functions ofz.By substitutingin(2)wefindthat ^^^[F'+4t7F=0. LetVbefixctorised sothat V=Az^U{z-a,;)'''t, whereAisconstant,k,^andXareintegers,andk,,anda,,arenotzero. 122 THEORY OFBESSEL FUNCTIONS [CHAP. IV From thesecond member of(5)itfollows that rr__^_<? ^9 4^^4<{z- Clq)' andthenbysubstitutingintothe firstmember of(5)wehave Now consider theprincipal partoftheexpressionontheleftneara^.It isevident thatnone ofthenumbersk^canbelessthan—2,and, ifanyone ofthem isgreaterthan—2itmustsatisfytheequation Kg+^fCq^U, SOthatKgisor—4,which arebothexcluded from consideration. Hence all thenumbersKgareequalto—2. Again,ifweconsider theprincipal partnear oo,weseethatthehighest powerinVmust cancel with the—1in^(z),sothatX=—^Kg. Itfollows that\JVisrational, andconsequently /-4{t, z)isreducible, which iscontrarytohypothesis. Hence tcannot have asmanyastwobranches andsoitmust berational. Accordingly,lettheexpressionfor tinpartialfractions be ^ B Zg{Z~ag)-' t=1Anz''+ ^ n=-K n. where AnandBn^gareconstants, kand X,areintegers,iiassumespositive valuesonlyinthelastsummation andUg^0. Ifwesubstitute thisvalue oftin(2)wefindthat 2nAnZ^^-^-t'"^''' l^^+ ISAnz-+t^^J'+1-PSp±1^ =0. n=-K n,q\^ (^q) (h=-k \Z— Ojg) JZ Ifweconsider theprincipal partoftheleft-hand sideneara^weseethat \l{z—ag)cannot occur in^toahigher powerthan the firstandthat 5,,,-B\ g=0, SOthat^1^5=1. Similarly,ifweconsider theprincipal partsnear and oo,wefindthat K=l,{A_y-A_,==p(p+1); \=0,^o'=-l. Sincep=±v-^, wemaytake^_i=-pwithout lossofgenerality. Itthen follows that u=z-Pe^''U(z-ag). (J Accordingly,ifwereplace ubyz'Pe^'^win(1),weseethattheequation must have asolution which isapolynomialinz,andtheconstant term in thispolynomialdoesnotvanish. 4-75] DIFFERENTIAL EQUATIONS 123 When wesubstitute Sc^nZ^ forxoin(7)wefindthattherelationconnecting successive coefficients is m{m-2p-l) c,a±"2iCm-i (ni-p -1)=0, andsotheseries forlucannot terminate unlessm—j)—1canvanish, i.e.unless piszero orapositive integer. Hence thehypothesisthat Bessel'sequationissoluble infinite terms leads ofnecessitytotheconsequencethatoneofthenumbers +y—^iszero ora positive integer; andthis isthecaseif,andonl}^if,2visanoddinteger. Conversely wehave seen(§3'4) that,when 2visanoddinteger, Bessel's equation actually possessesafundamentalsystemofsolutionsexpressiblein finite terms. Theinvestigationofthesolubilityoftheequationistherefore complete. Someapplicationsofthistheorem toequationsofthetypes discussed in§4'3have been recorded hyLebesgue, Jotmial deMath. xi.(1846), pp.338—340. 4*75. Ontheimpossibility ofintegratingRiccati'sequation infiniteterms. Bymeans oftheresultjustobtained, wecandiscuss Riccati'sequation ClZ with aview toprovingthat itis,ingeneral,notintegrableinfinite terms. Ithasbeen seen(§4"21) that theequationisreducible to wheren=2q—2;and,by§•i"3,the lastequationisreducible toBessel's equationforfunctions oforderl/(2q)unlessq=0. Hence theonlypossiblecases inwhich Riccati'sequation,isintegrablein finite terms arethose inwhichqiszero or1/qisanoddinteger ;and these arepreciselythecases inwhich nisequalto—2orto ^"' (..=0,1,2,...)2m+1 Consequentlytheonlycases inwhich Riccati'sequationisintegrableinfinite terms aretheclassical cases discovered byDaniel Bernoulli (cf§-i'l1)andthe limitingcasediscussed after themanner ofEuler in§4'r2. This theorem wasproved byLiouville, Journal deMath. vi.(1841), pp.1—b3. It seems impossibletoestablish itbyanymethod whicli isapi)reciablymore brief than the analysisused intheprecedingsections. 124 THEORY OFBESSEL FUNCTIONS [CHAP. IV 4"8. SolutionsofLaplace's equation. The firstappearanceinanalysisofthegeneralBessel coefficient hasbeen seen(§I'S)tobeinconnexion with anequation, equivalenttoLaplace's equation,which occurs intheproblemofthevibrations ofacircular membrane. WeshallnowshewhowBessel coefficients arise inanatural manner from Whittaker's* solution ofLaplace's equation d'V d'V d-'V _ Thesolution inquestionis (2)^^^ I/(^+*^^^^ ^*+*V^i^^*'^)^"' J—TT inwhich /denotesanarbitraryfunction ofthetwovariables involved. Inparticular,asolution is ek(z+ixcosu+iysinu)QQ^.^^^^^,^^ '-TT inwhich kisanyconstant andmisanyinteger. Ifwetakecylindrical-polar coordinates, defined bytheequations x=pcos (f),y=psin^, thissolution becomes gfcz gikpcoi{u-4,) (jQgy^n^du=e^^ Iei*p'=°s« cosm(v+0)dv, .—It J—IT _2e^^IeifcpCOS» QQg j^^yQQgj,j0^y^ J =27ri'" e*^cosm(j>./,„ (A;/3), by§2'2.Inlikemanner asolution is J—IT and this isequalto27rt"* e^"^sinm^.J",^ (A:/?). Both ofthese solutions are analyticnear theorigin. Again,ifLaplace's equationbetransformed!^^cylindrical-polarcoordi- nates, itisfound tobecome dp"-'^ pdp'^ p-d(f>^-'^dz^~' *Monthly Notices oftheR.A.S.lxii.(1902), pp.617—620; 3Iath. Ann. lvii. (1902), pp.333—341. tThesimplest method ofeffecting thetransformation isbyusing Green's theorem. See W.Thomson, Camb. Math. Journal, iv.(1845), pp.33—42. 4-8,4-81] DIFFERENTIAL EQUATIONS 125 andanormal solution ofthisequationofwhich e*^isafactor mustbesuch that isindependentof(f),and, ifthesolution istobeone-valued, itmust beequal to—m-where niisaninteger. Consequentlythefunction ofpwhich isa factor ofVmust beannihilated by dp^ pdp\' p' andtherefore itmust beamultipleof./,„(kp)ifitistobeanalytic alongthe linep=0. Wethus obtain anew thesolutions e^^ .md) .J,n(kp).sm These solutions have been derived byHobson* from thesohition ^^J„(kp) byClerk Maxwell's method ofdifferentiating harmonics with respecttoaxes. Another solution ofLaplace's equation involving Bessel functions hasbeenobtainedby Hobson{ibid. p.447)from theequationincylindrical-polarcoordinates byregarding cjdz asasymbolic operator. Thesolution soobtained is sin^- "'"V'^rf^ where/(2)isanarbitraryfunction ;buttheinterpretationofthissolution when'^'„jinvolves afunction ofthesecond kind isopentoquestion. Other solutions involving aBcs.sel function ofanoperator acting onanarbitraryfunction havebeen given byHobson, P)-oc. London Math. Soc.xxiv.(1893), pp.55—67; xxvi. (1895), pp.492—494. 4*81. Solutions oftheequations ofivave motions. Weshallnowexamine theequationofwave motions ^^dx'"^ dy'"^ dz'~ crdt^' inwhich trepresentsthetimeand cthevelocityofpropagationofthewaves, from thesameaspect. Whit taker's fsolution ofthisequationis I'TT i'TV (2)y= \/(^sinucos y+ysinvsinv+scosu+ct,u,v)dudv, wtere/denotes anarbitraryfunction ofthethree variables involved. Inparticular,asolution is „ikixsmucasi> +1/sinusill i}+Z<:osu+ct)^ /f^f ^i\dudi* TT. (I whereFdenotes anarbitraryfunction ofuand v. *Froc. Loudon Matli. Soc. xxii. (1892), pp.431—449. tMath. Ann. lvii. (1902), pp.342— H4o. SeealsoHavelock, Proc. London Math. f^oc. (2)ii. (1904), pp.122—137, andWatson, SIr.ssrnijcr, xxxvi, (1907), pp.98—106. 126 THEORY OFBESSEL FUNCTIONS [CHAP.IV Thephysical importanceofthisparticularsolution liesinthefactthat it isthegeneralsolution inwhich thewaves allhave thesamefrequencykc. Now letthepolarcoordinates of{x,y,z)be(r,6,<^),and let(&>, y\r)bethe angularcoordinates ofthedirection (m,v)referred tonewaxes forwhich the polaraxis isthedirection {6,4>)andtheplane -^=passes throughthe ^-axis. Thewell-known formulae ofspherical trigonometrythenshew that cos ft)=cos^cosu+sin6sinucos{v- (f)), sinusin(v— (f))=sin cosin^fr. Now take thearbitraryfunction F(n,v)tobeSn(u,v)sin u,where S,,de- notes asurface harmonic in{u,v)ofdegreen;wemaythen write Sn{U,V)=Sn{d, (/);Oi,ir), where Snisasurface harmonic* in(w,^)ofdegreen. Wethusgetthesolution Vn-e^^"* gikrcosmSn(d, (});ft),i/r)sincodwdyjr. J-TT.' Since Snisasurface harmonic ofdegreenin(&>, \jr),wemaywrite Sn{6, <f>;(0,ir)=An(0, (f>).Pn(cOS ft)) 11 -\-S[AJ"'\6, (f))cosm^fr+BJ>"^6,(i))sinmylr\Fn"' (cos (o), where An{6, 4>),An^"'^{6, </>)and £„<'"> (^,<^)areindependentoftoand-v/r. Performingtheintegrationwithrespecttoyjr,weget Vn=lire'^'^^An (0, (t>) Ie^fc'-cos.op^^ ^qq^^)sin(odo) =(2'7r)U-e''^<'^'I^^^An(e,<i>) by§8-32. Now theequationofwave motions isunaffected ifwemultiplya;,y,zand tbythesame constant factor, i.e.ifwemultiplyrand tbythesame constant factor, leaving6and<^unaltered ;sothatAn{0, (f))maybetaken tobein- dependent!oftheconstant kwhichmultipliesrand t. Hence lim(^•~''Vn) isasolution oftheequation ofwave motions, that is tosay,r''An{d, <f>)isasolution(independentoft)oftheequationofwave motions, and isconsequentlyasolution ofLaplace's equation. Hence An(^, </>) This follows from thefactthatLaplace's operatorisaninvariant forchanges ofrectangular axes. tThis isotherwise obvious, becauseS,^maybetaken independent of A-. 4-82] DIFFERENTIAL EQUATIONS 127 isasurface harmonic ofdegreen.Ifweassume ittobepermissible totake Anid,</))tobeanysuch harmonic, weobtain theresult that eikcty-hj ^kr)F,r(cos6)^^^ vid>sm isasolutionoftheequation ofwave motions*; andthemotionrepresented by thissolution hasfrequencykc. Tojustifytheassumption thatA,^{6, cf))maybeanysurface harmonic ofdegree n,we construct thenormal solution oftheequationofwave motions ,bV\ 1 / .^dV\ 1 a-'I'i^d^V /,cV\1c/ .^dV\1 di-Vdrj sin6d0\ ddJ^sin^6dcfi^c"-dfi' ,coswhich hasfactors oftheform t;'*"' .mcb.The factor which involves 6must then heofsm theform P„"*(cos 6) ;andthefactor which involves risannihilated bytheoperator dr\dr sothat ifthisfactor istobeanalyticattheoriginitmust beamultipleofe/^+^ {kr)j^h:-(r'^§yn(n+l)U-^r^ 4'82. Theorems derived fromsolutionsoftheequations ofMathematical Physics. Itispossibletoprove (or,atany rate, torenderprobable) theorems con- cerningBessel functions byacomparisonofvarious solutions ofLaplace's equationoroftheequationofwave motions. Thus, ifwetake thefunction e^^J^i [k^J{p'+a"—lapcos^)], bymakingachangeoforigintothepoint (a,0,0),weseethat itisasolution ofLaplace's equationincylindrical-polarcoordinates. This solution has e*-'^as afactor and itisanalyticatallpointsofspace.Itistherefore natural to expectittobeexpansibleintheform 00 e^^ J.ot/o {kp)-1-22{A,ncosmj)+B„isinm^) J,„(kp) _ ?«=1 Assumingthepossibilityofthisexpansion,weobserve thatthefunction under consideration isaneven function ofcf),andsoB,n= ;and,from thesymmetry inpanda,A,n isoftheform Cm.Jm{ka), where c,„isindependentofpand a. /Wethusget Jo\k's/{p"+a---lapcos^)}=Se,„c,„Jm 0>'p) 'fn{ka)cosmcf). 111=0 Ifweexpandboth sides inpowersofp,aand cos(f),andcomparethe coefficients of(k-pacos<^)™,weget C,)i=i, *Cf.Bryan, Nature, lxxx.(1909), p.309. 128 THEORY OFBESSEL FUNCTIONS [CHAP.IV andsoweareledtotheexpansion* 00 Jo{k\/(p-+a--2apcos</))}=Sf,„,/m (kp)Jm(ka)cos7n(f,,m= ofwhich amore formalproofwillbegivenin§11-2. Again,ifwetake e'*<'^*+^*, which isasolution oftheequationofwave motions, andwhichrepresentsawavemovinginthedirection oftheaxis ofzfrom +00to-00withfrequencykcandwave-length 27r/Ar,weexpectthisexpression tobeexpansible fintheform (^^^kctICni^^J^^^^(kr)Pn{cOS 6), where Cnisaconstant;sothat gikrcose=r^yXc,,i'^/„^. (kr)Pn(cOS 6).\KrJ n=0 Ifwecomparethecoefficients of(krcos6)^oneach side,wefindthat nl^''2'»+*r(n +|)'2«.(w!/' andsoc„=n+^;wearethus ledtotheexpansion:}: \icr/ ,1=0 ofwhich amore formalproofwillbegivenin§H'o. 4*83. Solutions ofthewaveequationinspace ofpdimensions. Theanalysis justexplainedhasbeenextendedbyHobson§tothecaseof theequation •d^-V d-V d^V 1d'^V 1 1-...H= dx{' dx.rdxp^c-dt- Anormal solution ofthisequationoffrequencykcwhich isexpressibleasa function ofrand tonly,where r=vW-+«2-+ •••+VX must beannihilatedbytheoperator d^p-1d j^ r (jr andsosuch asolution, containingatime-factore^*^^must beoftheform e""'''^^^p.,,(kr)/{kr)i^p-^K *This isduetoNeumann, Theorie derBessel'sclten Functionen(Leipzig, 1867), pp.59—65. tThe tesseral harmonics donotoccur because thefunction issymmetrical about theaxisofz. XThisexpansionisduetoBauer, Journal furMath. lyi.(1859), pp.104, 106. §Proc. London Matli. Soc. xxv.(1894), pp.49—75. 4-83] DIFFERENTIAL EQUATIONS 129 Hobson describes thequotient '(^jip-a) ikr)/{kr)^^P~''^asacylinderfunction ofrankp;such afunction maybewritten intheform Byusingthisnotation combined with theconceptof^j-dimensional space, Hobson succeeded inprovinganumber oftheorems forcylinderfunctions of integralorder andoforderequaltohalfanoddinteger simultaneously. Asanexampleofsuchtheorems weshall consider anexpansionfor J\ks/'if+a"-%ircos</>)' p], where itisconvenient toregard (^asbeing connected withXpbytheequation Xp=rcos0.Thisfunction multiplied bye'^'^* isasolution ofthewaveequation, andwhenwewritep=rsin<p,itisexpressibleasafunction ofp,<^,tandof noother coordinates. Hence e''*«*./{kV(r-+a--2arcos</>)jp] isannihilatedbytheoperator dp'^ pdp dec./' that istosay,bytheoperator dr^ rdr r-sm cf) d(f)r-d(f)'^ Nownormal functions which areannihilatedbythisoperatorareoftheform oa where Fn(cos^!^)isthecoefficient* ofaV'intheexpansionof (l-2a cos(f)+a'y-^p. Bythereasoningused in§4*82,weinfer that J{k\l{r-+a^—2arcos<^)\p] 1={kap=Hh^^^^^n-/«+i^-i i]^r) Jn+,,,., {ka)Pn (cos <t>!p). /N^owexpandalltheBessel functions andequatethe coefficients oi' {k'^arcos<^)'*oneach side;wefindthat 2^ A^2»r(?t+i/J-1) ^M+hp-, ,^11^^^^+ip^- )2»+iP-i r{n+lp)f" n\r(1^-1)^ ' sothatAn=2flP-^ (n+h^p-l)V {hp-1). *Sothat,inGegenbauer's notation, I'u(cos<p\p)^ C^/'-^cos./.). W. 15.K. !) 130 THEORY OFBESSEL FUNCTIONS [CHAP. IV Wethus obtain theexpansion /jp_, {kV(r2+a^- 2arcos<f))] (r^+a^—2arcos(f))^^'* 2iP- '/\;f_,^^i(*^+*p-l)e/,^i^-i(^^r)/^,p_,(A:a)Cf-^(cos(/>). Ananalytical proofofthisexpansion,which holds forBessel functions of allorders(thoughtheproof givenhere isvalidonlywhen})isaninteger),will begivenin§11'4. 4"84. Batemari's solutionsofthegeneralised equation ofwave motions. Twosystemsofnormal solutions oftheequation 8^a^rr d-V_1B^F havebeeninvestigated byBateman*, whoalsoestablished aconnexion between thetwosystems. Ifwetakenewvariablesp,a,^y"^definedbytheequations Xi=pcos^,Wx=acos-yfr, ^2=psinXr ^i=(^sin-\/r, theequationtransforms into Anormal solution ofthisequationwithfrequencykcis J^(kpcos<t>)J^{kasin<f>)e'('*x+''>^+^fO^ where <I>isanyconstant. Further, ifwewrite p=?•cos(f>,(T=r sin<^, sothat(r, %,i/r,^)form asystemofpolar coordinates, equation (2)transforms into ,g.^ir.??7,l?!r cot</>-tan</ )aF ^'ar^ rdr r-d(l>^"^ r^ df 1 a'^F 1a^Fia^F "*" r^cos-<^a^'^ r-sin-</>a-»/r-~ c^1)¥' Nownormal solutions ofthisequation which have e«"('^x-l-'"/'-t-^cO asafactor areannihilatedbytheoperator K^"+7,5^+ '^+^loT^+(cot6—tan <i)-^r~-. ),or-rdr v[a<^2vr r/^^^^^^^ sm'<j>\' *Messenger, xxxm.(1904), pp.182—188; Proc. Lo7idon Math. Soc.(2)in.(1905), pp.111—123. 4-84] DIFFERENTIAL EQUATIONS 131 andsince such solutions areexpressibleastheproductofafunction ofrand afunction ofcf)theymust beannihilatedbyeach oftheoperators a^ 3a 4\(x+i) +(cot(^-tan0)-+4X(\+1)-^^,;---/'t , d0- d^cos-(f)sin^(^ where A,isaconstant whose valuedependsontheparticularsolution under consideration. Thenormal solutions soobtained arenoweasilyverified tobe oftheform (At)-i JoK+i (kr)cos'"^sin"^ XJ\(^—^-\,^^+\+1 ;I.+1;sin'^(f^e»(Mx+.'*+/^<'0. Itisthereforesuggestedthat Jfj, (A;?"cos^cos^)Jy(krsin^sin<t>) isexpressibleintheform Sa;,(kr)-' /^A+i (kr) cos'"<^sin"(ft.-.F,(^^^-\^i^+A.+1;i;+ 1 ;sin'<j)\, where thesummation extends over various values ofX,andthecoefficients a^ dependon A,and<P,butnotonror (j).Bysymmetryitisclear that ttA=6acos'"(I> sill"O .oi^i(^-^-^>^^^-^+A+1;i^+1;sin-a> J, where b^isindependentof^. Itisnotdifficult toseethat \^h(fi +v)+n, .,- (?i=0,1,2,...) andBateman hasprovedthat 6^=2(-)"(/i+i'+2n+ 1) „ir(fM+n+i){V{v +iyr Weshall notgiveBateman'sproof,which isbased onthetheor}'oflinear differentialequations,but later(§11'6)weshall establish theexpansionof ./^(^cos cos't>)Jy{krsin(/>sin<P)byadirect transformation. 9—2 CHAPTEK V MISCELLANEOUS PROPERTIES OFBESSEL FUNCTIONS 5*1.Indefinite integrals containingasingleBesselfunction. Inthischapter weshall discuss somepropertiesofBessel functions which have notfound aplaceinthetwopreceding chapters,andwhich havebut onefeature incommon, namelythattheyareallobtainablebyprocessesofa definitely elementarycharacter. Weshall firstevaluate some indefiniteintegrals. Therecurrence formulae§3'.9(5)and(6)atonce lead totheresults (1)1%"+^'^.(^)dz=z^+^^C+i {z), (2)l~z--^^ <^,{z)dz=-z-"^''e,^ (z). Togeneralisethese formulae, consider j%''^^f(zy^^{z)dz; letthisintegralbeequalto z''+^{A(z)<^^(z) +B(z)K^,(z)l whereA(z)andB{z)aretobedetermined. The result ofdifferentiation isthat ^"^Vl^)K(z)^z"^'\a'(z)<^^{z)+A{z)^''^<^^{z)-A{z)^^^,(^)| +z"-^^[B'{z)-^^.^j {z)+B{z)<W.{z)\. Inorder thatA{z)andB{z)maynotdepend onthecylinder function, we takeA(^)=B'(z),andthen f{z)^A'{z)^^^^A{z)^B{z). Hence itfollows that (3)j' z^^^ \^"{z)+^^B'(z)-fB(z)].^^{z)dz =z^^^[E{z)^^{z)+B{z)%\^,{z%. This result wasobtainedbySonine, Math. Ami. xvi.(1880), p.30,thoughanequivalent formula (withadiflPerentnotation) hadbeen obtainedpreviously byLommel, Studien iiher dieBesseVschen Functionen(Leipzig, 1868), p.70.Somedevelopments offormula(3)are duetoNielsen, Nyt Tidsskrift, ix.(1898), pp.73—83 andAnn. diMat.(3)vi.(1901), pp.43—46. Forsome associatedintegrals which involve thefunctions berandbei,seeWhitehead, Quarterly Journal,XLii.(1911), pp.338—340. 5-1,5-11] MISCELLANEOUS THEOREMS 133 Thefollowingreduction formula, which isanobviousconsequenceof(3), should benoted: (4) I~ z'^^''€,{z)dz=- ifji'-V-)rz>--'K{z)dz +[^'^^''C+i {z)+{^J^-v) zi^^, {z)'\. 5"11.Lommel'sintegrals containingtwocylinder functions. Thesimplest integralswhich contain two Bessel functions arethose derived from theWronskian formula of§312(2),namely /.iz)J'_. (.)-/_.{z)j; (z)=-^^^^ , TTZ whichgives (I)Pdz^ITJ-.{z ) JzJ^^{z)2sinVTTJ^{z) ^^jZ./.,(Z)J_,(Z)2sinVTT^^/,(z)' andsimilarly,from§3"6o(1), dz 77V^{z) (3).;zJ^H^) 2J^{z)' ^^'zJ,{z)YAz)-2'''^ JA^)' dz TTJ^(z) (5) zY:-{z)2F,(2r)' Thereader should havenodifficultyinevaluatingthesimilar integralswhich contain anytwocylinderfunctions ofthesame order inthedenominator. Thefornmlaeactually givenareduetoLommel, Math. Ann. rv.(1871), pp.103—116. Thereader should compare (3)with theresult duetoEuler which wasquotedin§1'2. Some moreinteresting results, alsodue toLommel*, areobtained from genei'alisationsofBessel'sequation. Itisatonce verified bydifferentiation that, ifyand?;satisfytheequations thenf(P-Qyy'}'^^=y£'''£• *Math. Ann. xiv.(1879), pp.520—536. 134 THEORY OFBESSEL FUNCTIONS [chap. V Nowapplythisresult toanytwoequationsofthetypeof§4"31(17).If I,^vdenote anytwocylinderfunctions oforders\xandvrespectively,wehave (6) .,^^^^.,.^^Ap^dz dz cj^'iz) 4>(z) -2;p(7)^ 4|^P(7)I"'^^'^"'+^^l^(-)n where (f)(z)andy{r(z)arearbitraryfunctions of ^^. Thisformula istoogeneraltobeofpracticaluse.Asaspecial case, take (f){z)and^{^(z)tobemultiplesof2^,saykzand Iz.Itisthenfound that (7) {k^-l^)z-IX'—V %(kz)'<^Mlz)dz =z\k%^, {kz)f,{Iz)-m^(kz)^?,+i (Iz)]-ifji-v) 9^^{kz)%\{Iz). Theexpression ontheleftsimplifiesstillfurther intwospecialcases(i)/u,= i/, (ii)k=l. Ifwetakeix=v,\i isfound that (8) {kz)6^{tz)dz= k''-V Thisformula maybeverifiedbydifferentiatingtheexpressionontheright. Itbecomesnugatory when k=l, forthedenominator isthen zero, while the numerator isaconstant. Ifthisconstant isomitted, anapplicationofTHospital'sruleshews that, when l-^k, (9) z"^^{kz)%\{kz)dz=-^[kz^^^, (kz)'6;{kz) -kz90^ (kz)f^V+i {kz)-'^^(kz)#^+1 (kz)}. The result ofusingrecurrence formulae toremove thederivates onthe rightof(9)is (10) z^^(kz)9^^(kz)dz=\z-'[W^ (kz)%%(kz)-9^^_, (kz)%%+, (kz) -^,^,{kz)¥f,-,{kz)]. 5-11] MISCELLANEOUS THEOREMS 135 Specialcases ofthese formulae are : (11)f~^K'{kz)dz=^z'{'e;~{kz)-'-^^_. {kz)'^,^, {kz)] =Iz^- |(l-^\9B,'{kz)+'^;^ (/.^)J, a2)^~z'W^{kz)f_^{kz)dz=\z"-{2'^^^ {kz)'f_^{kz)+W^_, {kz)Wf_^_, {kz) +9?,^,{kz)^_^^,{kz)], thelatterequation beingobtainedbyregardinge~'^'''-Yi} ^{kz)asacylinder function oforder— fi. Toobtain adifferent class ofelementary integralstakek=l in(7)and it isfound that (13) rfe-^(fe)-g, iic.)^=-^"^^'*' ^'"^^'^^"^" ^'^'<^"^'^'- <^'^^' ^S^{kz)^,{kz) Theresult ofmakingz/^-/ainthisformula is (-14) I?^^(^^)6.^(A;^ ^T"" 2~ 1'"^' '— a _^^ (yr,^)^M+i(^)l_^^g'M(A:^)^^.(A^-^) The lastequationisalsoreadilyobtainablebymultiplyingtheequations ^.'^,(^)=0,V,^^^^=2/.t^(^) y'~a'~^>^\^) respectively, subtractmg andmtegrating,andthen re-ZfJi Z placingzbykz. Asaspecialcasewehave (15)j./m-H^^)^=|^{/..,(A-,^)!i|M(^-^)-/.(^-^)3M+.(/^-^)}+2^^.M^-4 Arfalternative method ofobtainingthisresult willbegiven immediately. Resultsequivalentto(11)areasoldasFourier'streatise, LaTheorieAnalytiquedela Chaleur(Paris, 1822), §5^318—319, inthecaseoffunctions oforder zero;butnone ofthe other formulae ofthis section seem tohave been discovered before thepublicationof Lommel's memoir. Variousspecial cases oftheformulae have beenworked outindetail byMarcolongo, Napoli Rendiconti, (2)ni.(1889), pp.91—99 andbyChessin, Trans. Aaid. Sci.ofStLouis, XII.(1902), pp.99—108. 136 THEORY OFBESSEL FUNCTIONS [CHAP. V 5'12. Indefimte integrals containingtwocylinder functions; LommeVs second method. Analternative method hasbeengiven byLommel* forevaluating some oftheintegrals justdiscussed. Bythismethod their values areobtained ina formmore suitable fornumerical computation. Themethod consists inaddingthetworesults ^{z"^^(z)#.(z)]=-z<'{'^,(z)W,^, (z)+%^, (z)W'.(z)] +(p+^+v)z'>-'1^^{z)W,{z), ^{p-^-v-2)z<>-^^^+, {z)%\^, {z),, sothat (p+/x+v){~z^-' %\(z)^,{z)dz +(p-fM-v- 2)rZP-'^^+, (z)W,+1(z)dz =z^re(z)f.{z)+^,+, (z)^,+1 (z)l andthengiving specialvalues top. Thuswehave (1)fz->^-''-^%^Az)'^^^,{z)dz 2(/7^+l)^"^^^'^^^^'^+'^'^-^^ ^'^'^"^^ ^^>^' (2) J^'^<-''+i '^e(^)%%(^)f/^= g^^^^_^^^{^,(z)^^(z)+-e+^ (z)'^^^, (zy^. Asspecialcases ofthese (3)f^-'"-'"^V+i (^)c^^=-4^l^,' (z)+<^W, (z)}, (4)' ['z-^-+^'^C^{z)dz=^^^(<^,^ (z)+<^^,+, (^)}. Again,ifpbemade zero,itisfound that =K(z)%\{z)+^^^, {z)%\^, (z), sothat,bysummingformulae ofthistype,weget (5) if.+v)f%\(z)W,(z)^-(p,+,+2n)j'W,^,, (z)^?.+, (z)^ Math. Ann. xiv.(1879), pp.530—536. 5'12-5-14] MISCELLANEOUS THEOREMS 137 Inparticular,if/u.=z^=0, dz (6)j%\{z)'&n{z)~ 1r, r- ^"-ll '^0(^)^0(^)+22'&,n{z)'^\n{z) +'^niz)'^n{z)m=I where n=1,2,3,....Butthere seems tobenosimple formula for j%{z)%\{z)^. Foraspecial caseof(1)seeRayleigh, Phil.Mag. (.5)xi.(1881), p.217.{Scientific Papers, I.(1899), p.516.] 5"IS. So7mie'sintegrals containingtioocylinder functions. Theanalysisof§.5-1hasbeenextended bySonine, Math. Ann. xvi.(1880), pp.30—33, tothediscussion ofconditions that maybeexpressibleintheform ^(^)-^M i'^ (^")J;5.{^(^)H^ (s)-2?^^, {0(^)}-f,{>/^(^)} +C(z)<^^{0(.-)}%-^^^ {V.(3)}+i> (.-)-gf^^j {0(z)}^^^, {^{z% buttheresults aretoocomplicated andnotsufficiently importanttojustifytheir insertion here. 5'14.Scliafheitlinsreduction formula. Areduction formula for I'z>^9^%''{z)dz, which isanatural extension oftheformula§5"1(4),hasbeen discoveredby Schafheitlin* andapplied byhim todiscuss therateofchangeofthezeros of ^v{z) asVvaries(§15"6). Toobtain theformula weobserve that ' 2>^{z'-v')'i^J'{z)dz --fz^W.i^)\z^^^z^m^)dz -^[_^^+2^>^_ (^)r^;(^)]+ I' {^^+-2^;2 (^)+(^+1,^^+1 -i^^(^)'^-;(^)|dz. Now,byapartial integration, (/j.+S)\'z'^+"(^J-'(z)dz=[z>^+''^\''{z)] +21%'^+'K'(z)[z"^: {z)+{z"--v')K(^)ldz, *Berliner Sitzunf/nberichte,v.(1906), p.88, 138 THEORY OFBESSEL FUNCTIONS [CHAP. V andso {fjL+1) I%'^+^ '^/^ {z)dz=r^'^+^K" {z)]+2 ['2'^+^(^'^- v'^)'^,(^)-g^;(^)dz. Hence, onsubstitution, {lji+\)\%>^{z'--v')9^-'{z)dz =[^'^+« -^/^(2)-(/x+1)z^^' '^v(z)'^J(z)] +21%^^+-' 9^.(z)%%'(z)dz+{{fi+If- -22.^11%'^+'^,(z)W;{z)dz =[z>'+' "W:^ (z)-(fi+1)z>'+'"^.{z) "^J{z)+z>^+-''g?,2{z) +{hAjJi+lf-v^z>^+'^ii,'{z)] -{fi+3)[V+^W^{z)-(/x+1){1{^l+1)^- v'\J^*^'i'/{z)dz. Byreari'anging wefindthat {fji+2)[' z>^+^^i^fj'{z)dz={ti+l)[v'-\{fi +^f^^'z^9Sj'{z)dz +1 [^'^+1 {z9S: (^)-i(m+1)'^.(^)}^+^'^+^[z'-7.^+H/^+^)1'^^.^(^)]. andthis isthereduction formula inquestion. 5'2.ExpansionsinseriesofBesselfunctions. Weshallnow discuss some ofthesimplest expansionsofthetypeob- tained for(1^)'"^in§2'7.Thegeneral theoryofsuchexpansionsisreserved forChapterXVI. The result of§27atoncesuggeststhepossibilityoftheexpansion 00 (1) (i.)^=s(^i±MIV+ .>)^^^^(,)_ which isduetoGegenbauer* and isvalidwhenfiisnotanegative integer. Toestablish theexpansion, observe that isaseries ofanalytic functions whichconverges uniformly throughout any bounded domain ofthe^;-plane (cf§3'13); andsince d (^z)~'^^{(i^)"^ J^+on (Z)]=- ^_^^^{nJ^+^n-l (z)-(fl+ n)J^+2n+l (z)], itisevident thatthederivate oftheseriesnowunder consideration is i¥n 1= n- n=o ^• *Wiener Sitzungsberichte, lxxiv.(2),(1877), pp.124—130.=0, 5-2,5-21] MISCELLANEOUS THEOREMS 139 andsothesum isaconstant. When wemake z-*0, weseethattheconstant isunity;that istosay andtherequiredresult isestablished. Thereader will findthat itisnot difficult toverify thatwhen theexpansion onthe rightin(1)isrearrangedinpowers ofz,allthecoefficientsexcept that ofzf^vanish;but this isacrude method ofprovingtheresult. 5'21. Theexpansion ofaBesselfunction asaseries ofBesselfunctions. Theexpansion (1) {\z)-^J^{z)={U)->^T{v +\-,jL) iix+2n)r(/x+n)XSJon\T{v+l-^,-n)T{v +n+1)^^+'^" ^^^ isageneralisationofaformulaproved bySonine* when thedifference v— ijl isapositive integer;itisvalid whenfi,vand y— //arenotnegative integers. Itismosteasilyobtainedbyexpandingeachpowerofzintheexpansion of{^zY'^J^iz)with theaidof§o"2,andrearrangingtheresultingdouble series, which iseasilyseen tobeabsolutely convergent. Itisthusfound that 00 /\m(^^\ii+-2m 00 (-V Hi^Q-mW {v+7n+ (_)m _V ,,,'^0^1! T{v+m+ _VJV 1")""r(/A+m+n)_^(yx+2/M+2jj)r{iJL+2ni +])) ^(/i+2w)r(^+»i+w) 1)«=,» (w-m)! (j,i^o'}nl{n—)n)ll {v+m+ ])\ /i=o(/»=owi!(n-)n)lr(v+m+ 1)1 /— „=()i?!1(;^+1—/Lt—?0i(v+H+1) byVandermonde's theorem;andtheresult isestablished. IfweputV=yu-\-w.,wefindthat which isSonine's form oftheresult, and isreadily pru\ed byinduction. *Math. Ann. xvi.(1880), p.22. 140 THEORY OFBESSEL FUNCTIONS [CHAP. V Byaslightmodification oftheanalysis, wemayprove that, ifkisany constant, Xoi^i{fi+n,-n\ V+1\ A--) (yu,+2n)J^,+^n {z). Thisformula willberequiredinestablishingsome moregeneral expansions in§11-6. 5'22.LommeVsexpansions of{z+A)**''./^ \\/{z+h)]. Itisevident that{z+A)~^" J"^[s/{z+h)],quafunction ofz+h,isanalytic forallvalues ofthevariable, andconsequently, byTaylor'stheorem combined with§.3"21(6),w^ehave 00Am(Jm (1) {z+hr^^j.yiz +h)]=s--^[z-i^j. (v^)l w=mf+Wl(V^). Again, (^+A)"'/,. {\/('2'+^)}isanalytic except when z-\-h=0;and so, providedthat |A |< j^^|,wehave (2) (.+/o^'' J.{V(^+h)]=1-.^{.i^/. (V^)] These formulae aredtietoLommel*. Ifwetake y=-^in(1)and v=^in (2)wededuce from§3*4,aftermakingsomeslight changesinnotation, (3) l^\cos^(z^-2zt)=^--^J.n-i (z), yirzj ,n=om. (4) —sin^/{z^-+2zt)=S—./i_^ (z), equation (4)beingtrueonlywhen|i|<^|2^j.These formulae aredue to Glaisherf, whoregardedtheleft-hand sides asthegeneratingfunctions associated with thefunctions whose order ishalf ofanoddinteger, justas exp f^^' (^—1/0}isthegeneratingfunction associated with theBessel co- efficients. Proofs of(3)and(4)bydirectexpansionoftheright-handsides have beengiven byGlaisher;thealgebrainvolved ininvestigationsofthisnature issomewhat formidable. *Studien ilber dieBesseVschen Functionen(Leipzig, 1868), pp.11—16.Formula (1)wasgiven byBessel, Berliner Abh. 1824[1826], p.35,fortheBessel coefficients. tQuarterly Journal, xii.(1873), p.136;British Association Report, 1878, pp.469—470. Phil. Trans, oftheRoyalSoc. clxxii.(1881), pp.774—781, 813. 5-22] MISCELLANEOUS THEOREMS 141 Weshallnowenumerate various modifications of(1)and(2). In(1)replacezand //byz-and^2^andthen (5) J,{W(l+h)\=(1+hrS^^ ;;,-^ /.^„, (^), and, inparticular, (6) /.(^V2)^2^^ S- «7fJ^^-Az). Ifwedivide(5)by(1+A:)^"andthenmake ^-^—1,wefindthat Inlikemanner, from(2), (8) J.{W(l+A-)H (1+AO-i"i^-^^^- /.-,„ (^), providedthat |.^^ |<1. Ifwemake A'^-1+0,wefind,byAbel's theorem, A--*-l+0 m=0 W'i providedthattheseries ontherightisconvergent. Theconvergenceisobvious when Visaninteger.Ifvisnotaninteger, then, forlargevalues of?», ??l! TT .in]^ ^ In Hence thecondition forconvergenceisR{v)>0,and ifthecondition is satisfied, theconvergenceisabsolute.Consequently, whenR(v)>0,andalso when Visanyinteger, (9) S^-^ ^f^J.-,„{z)=0. ,Inlikemanner, ifR(v)>—1,andalsowhen visaiii/integer, wehave (10) /.(W2)=2-^"t^^J,_„, {z). III-- Itshould beobserved that functions ofthesecond kindmaybesubstituted forfunctions ofthe firstkind in(1), (2),(5)and(8)providedthat |A j< |^ | and j^• I<1;sothat 111= "*-! (12) (z+h)i''r,y{2+h)]=5^i^^]:"'^!-"-) Y,_,,wz), (13) F.[zV(l+AOJ= (1+Z^)^"S^ ^ ^7"^^ iW-« (--), /«= (14) Y^{zv(i+^01=(1+k)-^-"t^ I}-;_,„(4 142 THEORY OFBESSEL FUNCTIONS [CHAP. V These maybeproved byexpressingthefunctions ofthesecond kind asa linear combination offunctions ofthe firstkind;byproceedingtothelimit when Vtends toanintegral value, weseethattheyhold forfunctions of integralorder. Bycombining (11)—(14)with thecorrespondingresults forfunctions of the first kind,weseethatwemaysubstitute thesymbolWforthesymbol Ythroughout. These lastformulae werenoted byLommel, Stadien, p.87.Numerous generalisations ofthem willbegiveninChapterxi.Ithasbeenobserved byAirey,Phil.Mag. (6)xxxvi. (1918), pp.234—242,that theyareofsome useincalculations connected with zeros of Besael functions. When wecombine (5)and(13),andthenreplace \/(l -\-k)byX,wefind that,when |X^—1 1<1, 00(_\m (\-2_-\\mil^\m (15) ^.(M=x^SU__^^_Ji-Ai^<g',^„, {z), and, inparticular,whenXisunrestricted, 00(—\m (\2_-\\m.(lp'\m (16) /.(X^)=X-'S^^^^ ,^^''^ J.^,n {z). These two results arefi*equentlydescribed* asmultiplicationtheorems for Bessel functions. Itmaybeobserved that theresult oftreating (14)inthesamewayas(8)isthat (whenVistaken equaltoaninteger n) (17) -{a-l)\ (2/zr=n2i-i^ F„_,„(4 ?n=0'''' • Analternative proofofthemultiplication formula hasbeen given byBohmer, Berliner Sitzungsberichte,xiii. (1913), p.35,with theaidofthemethods ofcomplex integration ; seealso Nielsen, Math. Ann. lix.(1904), p.108,and(fornumerous extensions ofthe formulae) Wagner, BernMittheilungen, 1895, pp.115—119; 1896, pp.53—60. [Note. Aspecialcase offormula(1),namely that inwhich ^=1,wasdiscovered by Lommel seven yearsbefore thepublication ofhistreatise;seeArchiv derMath, xxxvii. (1861), p.356.• Hismethod consisted, intakingtheintegral ^IIc,os,{^r cos6+r]rsinQ)d^d-q overthearea ofthecircle^~+rf'—l., andevaluatingitbytwo difif'erent methods. Theresult ofintegrating withresi:)ecttorjis 27r /_isin(^rcos6+r^rsin6)^i^-^") di _-v'(i-{=)'''sin 6 =- jcos{^cose)iimU{l-P).riim6}~^^ 7r„i=o (2?n+l)! 7_i^^ /\b/ <= -(-)'»(lrsing)2"^J^^i(?-cosg) TO=o ml(/-cos^)'"*!' See, e.g.Schafheitlin, DieTheorie derBesselschen Funktionen (Leipzig, 1908), p.83. 5-23, 5-3] MISCELLANEOUS THEOREMS 143 andtheresult ofchangingtopolarcoordinates{p,(p)is 1 /"jrri 1/""^ r^-—I Icos{ri}cos{(f)—d)}pdpd(f)=^—I cos(rpcoh(f))pdpdcf)'2nI—TTJ^n'./—nJI) Ifwecompare these equations weobtain(1)inthecase i/=lwith zandhreplaced by r'^cos''^^ and r^sin-^.] 5*23. Theexpansion ofaBessel function asaseriesofBesselfunctions. From formula§5'22(7),Lommel hasdeduced aninterestingseries of Bessel functions whichrepresents anygivenBessel function. Iffjband Vareunequal,andjxisnotanegative integer, wehave Therepeatedseries isabsolutely convergent; consequently wemayre- arrangeitbyreplacing pbytn—n,andthenwehave andhence, byVandermonde's theorem, Thisformula wasgiven byLommel, Studien ilher dieBessel'schen Functionen(Leipzig, 1868), pp.22—23,inthesjjecialcase/i=0;bydifferentiating withrespecttovandthen puttingv=0,itisfound that (2)knY,{z)= J,{z)\0g{hz)-''^f^-}^i^-^~^{¥r-''J,..,n{z) M~MJm=0 l«l} and,when/i=0,wehaveLommel's formula (3) 1^Y,(z)=./o(^){y+log(1-0}+2^^'ff "f;/'^. J/l^\lit •ilv • Thisshould becompared withNeumann's expansion givenin§3'571. 53.Anaddition formula forBesselfunctions. Anextension oftheformula of§2'4toBe.ssel functions ofanyorder is (1) J,{z+t)=iJ.-„^{t)J,n{z), 7/1=—00 where |2; |< j^],vbeingunrestricted. This formula isduetoSchlafli*;and thesimilar butmoregeneralformula (2) %\{z+t)=X%\-,n{t)J,.{z)m=—X isduetoSoninef. *Math. Ann. in.(1871), pp.13-')— 137. t^'^'''- x^'i-(1^80), pp.7—8. 144 THEORY OFBESSEL FUNCTIONS [CHAP. V Itwill firstbeshewn that theseries ontherightof(1)isauniformly convergentseries ofanalyticfunctions ofboth zand twhen I2^ I^r, 7?^ I^ I^A, where r,R,Aareunequal positivenumbers inascendingorder ofmagnitude. Whenmislargeandpositive, Ju-m(i)Jm(^)iscomparablewith r(m-v)sinvir.i^Ry.ir/Ry m andtheconvergenceoftheseries iscomparablewith that ofthebinomial series for{l—rjR)". Whenmislargeandnegative {=—n),thegeneral term iscomparablewith V{v+n+\).n\ andtheuniformityoftheconvergencefollows forboth setsofvalues of iiiby thetestofWeierstrass. Term-by-termdifferentiation isconsequently permissible*,sothat il-P)iJ.-.n{t)Jrr,{z)= l\J\-n,{t) J^{z)-J,_,n{t) J'm{z)] \<jt CZj^=-tx^ 1,1=-ao 1* ^m=—<x> 1°°- 9SJ^-m(0[Jm-l {2)-Jm+i (^)}, and itisseen, onrearrangement,that alltheterms ontheright cancel, so that X Hence, when 12^ ]<ji|,theseries 2J^,-m (t)Jm{z)isananalyticfunction m=-X ofzand twhich isexpressible asafunction ofz-\-t only,since itsderivates withrespecttozand tareidentically equal.Ifthis function becalled F{z+1),then m=—oc Ifweputz=0,weseethatF{t)^J^{t), andthetruth of(1)becomes evident. Again,ifthesignsofvandwin(1)bechanged,wehave in=-» andwhen thisresult iscombined with(1),weseethat (3) Y,{z+t)=IY^_rr,(t)J,n{z).m=- CO *Cf.ModernAnalysis, §5-8. 5'4r] MISCELLANEOUS THEOREMS 145 When this iscombined with(1),equation (2)becomes evident. Thereader willreadily prove bythesamemethod that,when |^ |<i<|, (4) J,{t-Z)= iJ,^,n{t)J,,{z\ (5)- 'i^,{t-z)= I%%^,, (t)J,n{Z), (6) Y,{t-z)= 2F,+„,(0</„.(4 )«=—«; Ofthese results, (3)wasgiven byLommel, Studien iiber dieBesseVschen Functionen (Leijjzig, 1868),when visaninteger;while(4),(5)and (6)weregiven* explicitly byGraf, Math. Ann. XLiii. (1893), pp.141—142. Various generahsationsofthese formulae willbe giveninChapterxi. 5'4. Products ofBesselfunctions. Theascendingseries fortheproduct J^(z)J^(z)hasbeengiven byvarious writers; theexpansionissometimes stated tobedue toSchonholzerf, who publisheditin1877, but ithad, infact,beenpreviously published (in1870) bySchlafli^. Morerecentlytheproducthasbeenexamined byOrr§,while Nicholsonjlhasgiven expansions (c£§5-42)forproductsoftheforms J^(z)Yn(z)and l^^(z)Yn(z). Inthepresentsection weshall construct thedifferentialequationsatisfied bytheproductoftwoBessel functions, andsolve itinseries.Weshallthen (§541) obtain theexpansionanewbydirectmultiplicationofseries. Given twodifferentialequationsintheirnormal forms az- dz- ifydenotes theproduct vw,wehave y"=v"iu+2v'w'+vtu" =-{I+J)y+2v'tu', whereprimesindicate differentiations withrespecttoz. *SeealsoEpstein, DievierReclinungsoperationenmitBesseVschen Ftniciionen (Bern, 1894), [Jahrbuchiiber dieFortschritte derMath. 1893—1894, pp.845—846]. yXJeber dieAuswertlmng bestimmterlntegrale mitHiilfe vonl^mndertmgendesIntegratioimceges (Bern, 1877), p.13.Theauthorities who attribute theexpansiontoSchonholzer include Grafand Gubler, Einleitung indieTheorie derBessrVschen Funktionen, ii.(Bern, 1900), pp.85—87, and Nielsen, Ann. Sci.deVEcole norm. sup. (3)xviii. (1901), p.50;Handbuch derTheorie derCylin- derfiinktionen (Leipzig, 1904), p.20.AccordingtoNielsen, None. Ann. deMath.(4)ii.(1902), p.396,Meissel obtained some series forproductsintheIserhhn Frogramm,1862. $Math. Ann. iii.(1871), pp.141-142. Atrivial defect inSchlaHi's proofisthatheusesa contour integral which(ashepoints out)converges onlywhen R(fi+i>+l)>0. §Proc. Camb. Phil. Soc. x.(1900), pp.93—100. IIQuarterly Journal,xliii. (1912), pp.78—100. 10 146 THEORY OFBESSEL FUNCTIOjNS [CHAP. V Itfollows that^Jy"+(^+^) 2/1=2^"^'+2v'rv" =-2Ivw'-2Jv'w andhencey'"+2(1+J)y+(/'+J')y={I-J)(v'w- viv'). Hence, inthespecialcasewhenI=J,ysatisfiestheequation (1) 2/'"+4//+2/'2/=0; but, ifI^J,it-iseasytoshewbydifferentiation that This istheform ofthedifferential equation usedbyGit; inconnexion with(1),see Appell, Comptes Rendus, xci.(1880), pp.211—214. Toapplythese results toBessel'sequation,theequationhastobereduced toanormal form;both OrrandNicholson effect thereductionbytaking z^^^{z)asanewdependent variable, but, forpurposesofsolution inseries, it issimplertotakeanewindependentvariable bywriting _ g d^_d^_ sothat^-^^+^^''-^'^^^^^^=^• Hence theequationsatisfied byJ^{z)Jy(z),when/x-^v',is 5?S+2(2^=--.'•-.^^+ 4e-,}+(^=- .=)•=y=0, that istosay (3) [^^-2(fi'+v')^2+(fM'-vj]y+4e-<'(^+1)(^+2)3/=0, andtheequationsatisfiedbyJ^{z)J±„(z)is (4) ^(^^-4z;2)2/ +4e2«(^+l)?/=0. Solutions inseries of(8)are m=0 wherea=±/j^±vand 4(a+2m-l)(a +2m)c,„_iCm— {a+fx+u+2ni){cc+/J,—V+2m)(a— yu,+z/+2m) {a— fi—v+2m)' Ifwetake a=/m+pand* 1 ^'~2''+''r{fi +i)r(v+i)' weobtain theseries ^(-)'" (1^)'^+'^+"" T(fi+v+2m +l) ,„^omlr{fi+V+m+l)r {/jl+7n+l)r(v+m+1)' andtheother series which aresolutions of(3)areobtained bychangingthe signsofeitherfxorvorboth/xand v. 5-41] MISCELLANEOUS THEOREMS 147 Byconsideringthepowersofzwhich occur intheproduct J^{z) J^{z)it iseasytoinfer that, if2/li, )>vand2(/i+z/)arenotnegative integers and if /u,^^V",then (5)/(^)J(A=^^~^'" iUY^"^"-^'^ r(/.4-^+2m+1) Inlikemanner, bysolving (4)inseries, wefindthat,when 2visnota negative integer,then and,when z^isnotanegative integer,then (-)"H^zY"' (2m)l (7) J,{z)J_,{z)= S o(Hil)-r(z^ +m+l)V{-v +m-\- !)• Byreasoningwhich resembles thatgivenin§4'42, itmaybeshewn that (6)holdswhen vishalfofanoddnegative integer, providedthatthequotient r{-Iv+2iii -\-l)/r (2i/+m+1)isreplaced bytheproduct (2v+m+!),„. * 5"41. Products ofseries representingBesselfunctions. Itiseasytoobtain theresults of§5"4bydirectmultiplicationofseries. Thismethod hastheadvantagethatspecial investigations,forthecases in which /x-=V'andthose inwhich/a+t-isanegative integer,aresuperfluous. The coefficient of(-)'« (^^)'^+''+-'»intheproductofthetwoabsolutely convergentseries « (-)'" {UY^'''' ^{-)"(^zy+^" ,Zqm !r(^+m+1)„ro7* !T(y+ ?(+1) isSm -\ „ton':T(v+n-\-l). (m-n)]T(/u,+nt-n-\-\) (\rii m mli(/A+rii+1)1 \i'+ni+l)„=o ' mlr(fj,+m+l )r(v+rii+l) (/z+ z'+m+lU m!r(/A+m+1)r(z^4-m+1)' when Vandermonde's theorem isused tosumthefinite series. Hence, forallvalues of/xand v, (1) J^{z)J,{z)= X ,=0^'^! r{fM+m+l)r(z'+z»+1)' andthisformulacomprisestheformulae(5),(6)and(7)of§5*4. 10—2 148. THEORY OFBESSEL FUNCTIONS [CHAP. V This obvious mode ofproceduredoes notseem tohave been noticedbyanyofthe earlier writers;itwasgiven byNielsen, Math. Ann. lii.(1899), p.228. The series forJq{z)cosz andJ(i{z)ii\m were obtained byBessel, Berliner Abh. 1824, [1826], pp.38—39,andthecorrespondingresults forJ^{z)cos,z andJ^(2)sin2were deduced from Eoisson's integral byLommel, Studien iiher dieBesseV schen Functionen (Leipzig, 1868), pp.16—18.Some deductions concerningthefunctions berand beihave beenmade byWhitehead, Quarterly Journal, XLii. (1911), p.342. Moregenerally,ifwemultiplytheseries forJ^(az) andJ^ibz), weobtain anexpansioninwhich thecoefficient of(-)"'a'^h"(|2r)^+''+-"»is fow!T{v+n+l).{m -n)\ r{fi+m-n+l) '_tt'-»'oF^(-m,- //--m;1/+1;6-/a^) m!r(yLt+m+l)r(i/ +l) andso {hazY (Ibz)" (2)J^(az)J,{bz)= r(^+l) ^(-)'» (|a^)^'» 2^1(-m,-fi-m; v+1;¥/a') Jlo mWiji +m+l) and this result canbesimplified whenever thehypergeometricseries is expressibleinacompactform. One case ofreduction isthecase b=a,which hasalreadybeen discussed; another isthecase b=ia,providedthat/i-=v". Inthiscaseweusetheformula* Fia^-a-^4-1- -1)=r(«-^-fl)r(i) andthen Aveseethat /ox I-/\A/XV i-y"(^azy''+^coshiiTT m- =2(-)'" (^a^)2''+^'« m=omlT{v+m-^l)r{i' +2m+ 1)' (4)J_.(az)L(az)= X(-)-(i«^r cos(I.- lm)7r Ifwetakea=e^'^«in(3)wefindthat (6) ber,2 (^)+bei.H^)=S~-^, ^^i2i/+4W anexpansionofwhich theleading terms Averegivenin§SS. *Cf,Kummer, Journal furMath. xv.(1830), p.78,formula(53). 5-42] MISCELLANEOUS THEOREMS 149 Theformulae(3), (4),(5)were discovered byNielsen, Attidella R.Aecad. deiLincei, (5) XV.(1906), pp.490—497andMonatshefte farMath,undPhys.xix.(1908), pp.1G4— 170, from aconsideration ofthedifferential equationsatisfiedbyJ^,{a.z)J^v{bz). Some series have been given, Quarterly Journal, XLi.(1910), p.55,forproductsofthe types J^^{z)andJ^^(z)J^„(/:),buttheyaretoocumbrous tobeofanyimportance. Bygiving /*thespecialvalues +|in(2),itiseasytoprovethat (7) e--« /._, {zsin^)=^(2sinBy-^SrT^^TZT^^'"''^^°' ^>-iy\) M=o1-\^v+n) Thespecialcase ofthisformula inwhich 2i/isanintegerhasbeengiven byHobson*. 5'42. Products involvingBesselfunctions ofthesecond kind. The series fortheproducts J^{2)F„(z),J,n{z)Yn(z),and Y,n(z)F„(z) havebeen thesubjectofdetailedstudy byNicholsonf ;thefollowingisan outline ofhisanalysiswithsome modifications. Wehave TtJ, (z)Yn{Z)= I;{J^(Z)/.(Z)]-(-rI[J,(^)/-. (^)l, where vistobemadeequaltonafter thedifferentiations havebeenperformed. Now |;{/.(^)J.(^)}=l0g(|^)./.(^)^.(^) +2 r=0rlT{fi +v+r+l)r{fM+r+l)r{v+ri-l) x{f(fjL +v+2r+l)-ylr{/u.+v+r+1)- i/r(z.+?•+1)} and 1^[J,{z)J_.(z)}=-logiiz). J,{z}/_,(z) 50V(_)'•(i^)^--+^'T(/^-i^ +2r+l) .=0irlr(fM-v+r+1)r(fi+r+1)r(-V+r+1) {^|r{f^-v +2r-\-l)-^lr{fl-v +r+l)-^jr{-v +r+l)] Wedivide thelastseries intotwoparts,Sand% .Intheformer partwe r=o r=n have ^^„1(-i'+r+l) *Proc. Lotidon Math. Soc.xxv.(1894), p.0(3;seealso Cailler, Mem. de l<iSoc. dePhijg.de Geneve, xxxiv. (1902—1905), p.316. tQuarterly Journal,xliii. (1912), pp.78—100. TheexpansionofJo(z)Yo(z)hadbeen giveu previously byNielsen, Handbuch derTheorie derCylinderfunktionen (Leipzig, 1904), p.21. 150 THEORY OFBESSEL FUNCTIONS [CHAP. V while inthelatterpartthere isnoundetermined form tobeevaluated. When risreplacedinthispartbyn+r,itisseen that --^(l^r-»+^'- (;.-n+r+1),(n-r-1)! (1) 7r/,(^)J.(^)=- 2^ r!r(;a+r+l) \.to r\{n+r)\T{fi +r+\) X[21og(i^) +2>/r(yLt+«+2r+l) - -v/r(yu,+?i+r+1)- >/r(yLt+r+1) -^/rO^+r+l)-^/r(r+l)}. The expressionontherightisacontinuous function of/aatyu-=mwhere m=0,1,2,...,andsotheseries for7rJ'm{z) Yn{z)isobtained byreplacing fx, bymontherightin(1). The series forYj,i{z)F„{z)canbecalculated byconstructingseries for ' d-J±y.{z)J±^{z)' dfxdv inasimilar manner. The details oftheanalysis,which isextremely laborious, havebeengiven byNicholson, and willnotberepeatedhere.fj.•=m,c=M 5*43. Theintegral forJ^{z)J^{z). AgeneralisationofNeumann'sintegral (§2*6)forJn^(z)isobtainableby applyingtheformula* totheresult of§5*41;theintegralhasthisvaluewhenm=0,l,2,...,provided thatR(fi+v)>-l. Itisthen evident that J^{z)J^{z)=-X ,T^, , ;--—COS(fi-v)6dO, sothat,whenR{/jb+v)>—l, (1) J^(z)J,(z)=-('^J^+, {2zcos6)cos(fi-v)dde;TJo thechangeoftheorder ofsummation andintegration presentsnoserious difficulty. *This formula isduetoCauchy ;foraproof bycontour integration, seeModern Analysis, p.263. 5-43-5-51] MISCELLANEOUS THEOREMS 151 Ifwbeapositive integerandR(/ji—n)>—1,then (2) J^(z)J,,(z)=^-t)!['V^_, {2zcosd)cos(ix+n)\ede, andthisformula isalsotrue if/aandnarebothintegers, butareotherwise unrestricted. Formula(1)wasgiven bySchlafli, Math. Ann. iii.(1871), p.142,when yi±v areboth integers;thegeneral formula isdue toGegenbauer, WienerSitzungsherichte, cxi.(2a), (1902), p.567. 5*5.Theexpansion o/QzY'^''asaseriesofproducts. Anaturalgeneralisationoftheformulae ofNeumann(§"2'7)andGegenbauer (§5-2)isthat a^(i,w..-r(,.+i)r(.+i) (i)(2^r- r(/x+.+i) X Theformula istrue ifp,and varenotnegative integers,butthefollowing proof applies onlyifi2(^+i'+1)>—1. From§5"2wehave (zcosey+''= s^^—-—^—T^ ^/^+,+,„,(22cos^). Ifwemultiply bycos(p—v)6andintegrate,itisclear from§5"43that cos'*+'' 6cos(/u-p)Odd= S—^/^—^ 77"Jo . m= «i' X'-/^+„i,(2^) t/,,+m(2), andtheresult follows byevaluatingtheintegralonthe left;forother values offxand Vtheresultmaybeestablishedbyanalyticcontinuation. Theformula isatonce deducible fromformulae given byGegenbauer, WienerSitzungs- herichte, Lxxv.(2),(1877), p.220. 5'51.LommeVs seriesofsquares ofBesselfunctions. Anexpansionderived byLommel* from theformula 2vdJ^^{z) ^-j"irz~'='^^-'^'^-'^^^'^^^ r, /,xV2(v+2n)dJU,,,(z)IS J-^_i {z)=Z ~r- , 71=0 ^ a^ sothat 2JzJ\_i(z)dz=i(v+2n)J\^,,,(z) a= *Theresults ofthissection willbefouud inMath. Ann. 11.(1870), pp.632—633;xiv.(1878), p.532;Milnchener Ahh. xv.(1886), pp.548—549. .152 THEORY OFBESSEL FUNCTIONS [CHAP. V Hence, by§5'11 (11),wehave ..- (1) Iz^{./^_i (z)-J,-, (z)J,(z)}=i(i/+2n)J%+,n{z), n= ontakingzero asthelower limitwhenR{v)>0\ byaddingonterms atthe beginningoftheseries, itmaybeseen thattherestriction R{v)>0issuper- fluous. Ifwetake inturn v=h;,v=%,andaddandsubtract theresults soobtained, wehave(§3*4) z (2) -=2(w+i)/Wi(4 (3)sm2.^» (_)n(„+i)/.^^^^(^)^ while, bytaking7-=1.weseethat (4) Iz^[J:-(z)+J,'(z)]=i(2n+1)J%„^, (z). Another formula ofthesametypeisderivedbydifferentiatingtheseries ^fn"'v+n (3'); M=0 for itisevident that d"^00 J-2^end'v+n\2)=2Z<e,iJv+n (^)^v+ni^) 00=S€nJ„+n (2)[Ju+n-i (^)- '/^-f-n+i i^)} =J,(z)lJ,_,(z) +J,^,(z)] =2vJj^{z)/z, and so.AvhenR(v)>0,weobtain amodification ofHansen's formula(§2-5), namely (5) 2enJ-\^,,{z)=2v\J.'{t)~. «=o .'0 t Animportant consequenceofthisformula, namelythevalue ofanupper bound for |J^{x)\,willbegivenin§1342. Bytakingy=i,itisfound that 2en/-n+i {z)=-\sin^ t i=" TTJo 2sin^^ 2r^.^.cZ^+-sni2t- TT; ^ andso (6) IJ\+,{z)=-Si{2z),. n=^0 TT where, asusual, thesymbol Sidenotes the"sineintegi'al."This result isgiven byLommel inthethird ofthememoirs towhich reference hasbeenmade. 5-6] MISCELLANEOUS THEOREMS 153 5"6.Continued fraction formulae. ExpressionsforquotientsofBessel functions ascontinued fractions are deducible immediatelyfrom therecurrence formulagiven by^3'2(1); thus, iftheformula bewritten itisatonceapparentthat (uAM=W^ i^V{^ (1^+1)1 i^V{(^+i)(^-f 2)1 ^ ^ J^_,iz)1-1 -1 \z-/{(v+ rn~l)(v+m)} ^zJ^+rn^i(z)l{v +ni)—i^ —J„+,„ (2) Thisformula iseasilytransformed into ^^^ /,_, (z) 2vlz-2(v+l)/z-...-2(v +m)!z-./,+„, {z)' These results aretrue forgeneralvalues ofi^;(1)wasdiscovered by Bessel* forintegralvalues ofv.Anequivalent result, duetoSchlomilchf,is that, ifQ„{z)=/,+!{z)l[^z J„{z)],then Other formulae, given byLommel:|:,are J^+^{z) _Z Z"' Z" Z"-Zj,+,n+l{2) (4) J,{z) 2{v+\)--l{v +2)-2{v\-^)-...-2{v +m)- J.^m{z)' ^^JJz)' ^2{v+l)-2(v +2)-...-2(,^ +m)- J.+.ni^)' TheBessel functions inallthese formulae mayobviouslybereplaced byany cylinderfunctions. Itwasassumedb}^Bessel that,when /m-^00 ,the lastquotient maybe neglected,sothat JAz) _hzlv \z"-l\v{v+l)] \z^'l[{v^l){v+2)]^ J,_,{z)1-1 _1 _.... *Berliner Abh. (1824), [1820], p.31.Formula (2)seems nottohavebeen given bytheearlier writers;seeKncydopediedesSci.Math. 11.28,§08,p.217.Aslightlyditt'erent form isusedby Graf, Ami. diMat. (2)xxiii.(1895), p.47. tZeitschrift fiirMath, uiidPlnjs.11.(1857), p.142;Schlomilch considered integral values of ponly. IStiidien iiher dieBessePsclien Fiinctiunen (Leipzig, 1868), p.5;seealso Spitzer, Archir dcr Math, undPhys.xxx.(1858), p.332,and Giinther, Archiv derMath, uiidPhys-.lvi.(1874), pp.292—297. 154 THEORY OFBESSEL FUNCTIONS [CHAP. V Itisnotobvious that thisassumptionisjustifiable, thoughithappensto beso,andarigorous proofoftheexpansionofaquotientofBessel functions intoaninfinite continued fraction willbegivenin§9'65with thehelpofthe theoryof"Lommel'spolynomials." [Note. Thereason whytheassumptionisnotobviouslycorrect isthat, eventhough thefraction pjom tends toalimit as»i-*-co,itisnotnecessarilythecasethat^'"^"'^"'+^ tends tothat limit;thismaybeseenbytaking jt?„i=m+sinw, q,a=m,a,„=-l.]• Thereader willfindanelaborate discussion ontherepresentationoiJ^{z)/J^_i (s)asa continued fraction inamemoir* byPerron, MUnchenei'Sitzttngsberichie, xxxvii.(1907), pp.483—504; solutions ofRiccati's equation, depending onsuch arepresentation, have been considered byWilton, Quarterly Journal,XLVI.(1915), pji.320—323.Theconnexion between continued fractions ofthetypesconsidered inthis section andtherelations con- necting contiguous hypergeometricfunctions hasbeen noticed byHeine, Journalfilr Math. LVii.(1860), pp.231—247 and Christoffel, JournalfiirMath. Lvur.(1861), pp.90—92. «•^'«=j™,i>Tl)-'^-(^'''^^+i-4fJ5*7.Hansen'sexpression forJ^(z) asalimitofahypergeometric function. Itwasstated byHansenj-that Weshallprovethisresult forgeneral (complex)values ofvandzwhen\and fjLtend toinfinity through complexvalues. IfX=1/8, fi=I/tj,the(m+ l)thterm oftheexpansion ontherightis _i )\2^) n[(l+rS)(l+r7,)].m!V{v+m+1)^=1' This isacontinuous function ofhand77 ;and, if80 ,770arearbitrary positive numbers(lessthan 2 |z|~i),theseries ofwhich itisthe{m+l)thterm con- verges uniformlywithrespecttohand?;whenever both |S |<Soand 177 1^770. Fortheterm inquestionisnumericallylessthanthemodulus ofthe{m+1)th term ofthe(absolutely convergent) expansionof andtheuniformityoftheconvergencefollows from thetestofWeierstrass. Since theconvergenceisuniform, thesum oftheterms isacontinuous *Thismemoir isthesubject ofapaper byNielsen, Milnchener Sitzungsberichte, xxxviii. (1908), pp.85—88. tLeipziger Abh. 11.(1855), p.252;seealsoaHalberstadt dissertation byF.Neumann, 1909. [Jahrbuch ilber dieFortschritte dcrMath. 1909, p.575.] 5-7,5-71] MISCELLANEOUS THEOREMS 155 function ofboth thevariables(8,77)at(0,0),andsothelimit oftheseries is thesum ofthelimits oftheindividual terms;that istosay andthis istheresult stated. 5*71. BesselfunctionsaslimitsofLegendre functions. Itiswellknown that solutions ofLaplace's equation, which areanalytic near theoriginandwhich areappropriateforthediscussion ofphysical problemsconnected with asphere, maybeconveniently expressedaslinear combinations offunctions ofthetype cos r"P,, (cos 0), 7->'P,r (cos0)md>;sin these arenormal solutions ofLaplace's equation when referred topolar coordinates(r,6,cf)). Now consider thenature ofthestructure ofspheres,cones andplanes associated withpolarcoordinates inaregionofspaceatagreatdistance from theoriginneartheaxisofhaniionics. Thespheres approximatetoplanerand theconesapproximatetocylinders, andthestructure resembles thestructure associated withcylindrical-polarcoordinates;andnormal solutions ofLaplace's equationreferred tosuch coordinates areoftheform(§4'8) '^sm Itistherefore tobeexpected that,when rand narelarge*while 6issmall insuch awaythat rsin(i.e.p)remains bounded, theLegendrefunction shouldapproximatetoaBessel function;inother words, wemustexpect Bessel functions tobeexpressibleaslimits ofLegendrefunctions. The actual formulaebywhich Bessel functions aresoexpressed are, in effect, specialcases ofHansen's limit. Themostimportantformula ofthistypeis (1) limPjcos-)=./o(4 Thisresult, which seems tohave beenknown toNeumannt in1862, hasbeen investi- gated byMehler, JournalfiirMath. Lxviii. (1868), p.140; Math. Ann. v.(1872), pp. 1.36, 141—144; Heine, JourimlfilrMath. lxix.(1869), p.130; Rayleigh,Proc.London Math. Socy^. (1878), pp.61—64\Proc.RoyalSoc. xcii.A,(1916), pp.433—437[Scientific Papers, I.(1899), pp.338—341;vi.(1920), pp.393—397]; andGiuliani,Giorn. diMat.xxn. (1884), pp.236—239. The result hasbeen extended togeneralised Legendrefunctions byHeine andRayleigh. Ithasusuallybeenassumed that ntends toinfinity through integral values inproving (1);but itiseasier toproveitwhen ntends toinfinityas acontinuous realvariable. *Ifnwere notlarge, theapproximate formula forP,J" (cos Q)would be(sm"'tf)/Hi!. tCf.JournalfiirMath. Lxn.(1863), pp.36—49. 156 THEORY OFBESSEL FUNCTIONS [CHAP. V WetakeMurphy'sformula P„(coszjn)=o/^i(-n,n+1 ;1;sin-^z/n); andthereasoningoftheprecedingsection isapplicablewith theslight modification thatweusetheinequality when\z\^2\n\,andthenwecancomparethetwoseries ,F,(-n,w+1;1 ;sin^'^z/n), ,F,(1/So, 1|^+1;1;^B^|^^), whereBqisanarbitrary positivenumber lessthanf j^|~^andthecomparison ismadewhen\n\> l/8o-Thedetails oftheproofmaynowbelefttothereader. When nisrestricted tobeapositive integer,theseries forP„(cosz/n) terminates, and itisconvenient toappealtoTannery'stheorem*tocomplete theproofThis factwas firstnoticedbyGiuliani;theearlier writers took for grantedthepermissibilityofthepassagetothelimit. Inthecase ofgeneralised Legendrefunctions(ofunrestricted order'7>i), thedefinitiondependsonwhether theargumentofthefunctions isbetween +1and—1ornot;forrealvalues ofx(between andir)wehave P„-(cos^)=^^^^^^.F,(-.,n+l;m+l;sm^|.r/.), sothat (2) lim«-P,r"'(cos-)=j;«(^), butotherwise, wehave Pn'"^ (cosh ~]=p^^^\^ j^^^ oPi(-n,n+1;m -\-1;-smh^|-^/»), sothat (3) limn^-P,r-^(cosh -)=/,,{z). Thecorrespondingformula forfunctions ofthesecond kindmaybededuced from theequation whichexpressesf Q„"*interms ofP„"*andP„~"' ;itis ?i~'"sinnIT (4) limQn"'[cosh =Km (2). Thisformula hasbeengiven (with adifferentnotation) byHeine ;]:;itismost easily proved bysubstitutingtheintegralofLaplace's typefortheLegendre function, proceedingtothelimitandusingformula(5)of§6-22. *Cf.Bromwich, Theory ofInfinite Series, §49. +Cf.Barnes, Quarterly Journal, xxxxx.(1908), p.109; theequationis _„ ,,sinmirsinnir P~"^ P'"2r(-??^-n) 5 g/»=" T-^"r(l-m +n)T{l+m+n) inBarnes' notation, which isadopted inthiswork. tJournalfiirMath. lxix.(1868), p.131. 5-72] MISCELLANEOUS THEOREMS 157. Another formula, slightlydifferent from thosejust discussed, is (5)limP,,g^)=/o(2.) ; this isduetoLaurent*, and itmaybeproved byusingthesecond ofMurphy's formulae, namely Pn(cose)=cos"^e.2F1(-n,-n;1 ;-tan-^^). [Note. Theexistence oftheformulae ofthissection must beemphasized because it used tobegenerallybelieved thatthere wasnoconnexion between Legendre functions and Bessel functions. Thus itwasstated byTodhunter inhisEleuientan/ Treatise onLaplace's Functions^ Lamfs Functions and BesseVs Functions (London, 1875), p.vi,that"these [i.e.Besselfunctions]arenotconnected with themainsubjectofthisbook."] 5"72.Integralsassociated luithMehlers formula. Acompletelydifferent method ofestablishingtheformulae ofthe last section wasgiven byMehler and also, later, byRayleigh;thismethoddepends onauseofLaplace's integral, thus: P,j(cos6)=^ \(cos6+isin6cos</))"^cZd) 1T"__g»log(cos0+i sinecos</))^^Jj Since nlog{cos{zjn)+isin{zjn)cos0}-^izcos uniformlyasn-^gowhen ^^^tt,wehave atonce 1f" limP„(cos2/n)=-e'''"''"^ d(f)=J^(z). Heine fanddeBall.Ihavemade similarpassagestothelimit withintegrals ofLaplace's typeforLegendrefunctions. InthiswayHeine hasdefined Bessel functions ofthesecond andthird kinds;reference willbemade tohis results in§6'22whenwedealwithintegral representationsofY^,{z). Mehler hasalsogivenaproofofhisformulabyusing theMehler- Dirichletintegral 2/'^cos{n+^)(f)d(f) ^JV{2(^os (p-cos(9)} Ifn<p= y\r^itmaybeshewn that 2/"^cos^|rd^P„(coss/«)^- --r:y,—-.u, '^J\'V-Y) bvitthepassagetothelimit])resents some littledifficultybecause theintegralisanim- proper integral. Various formulae havebeengiven recentlywhich exhibit thewayinwhich *Juuriml deMath.(3)i.(1875), pp.384—385; theformula actually given byLaurent is enoneous onaccount ofanarithmetical error. tJournalJ'iirMath. lxix. (1808), p.131. SeealsoSharpe, (juarterli/ Journal, xxiv. (1890), pp.383—386. tAi<tr. Nacli. cxxviii.(1891),col.1—4. 158 THEORY OFBESSEL FUNCTIONS [CHAP. V theLegendrefunctionapproachesitslimit asitsdegreetends toinfinity. Thus, aformalexpansiondue toMacdonald* is (1) P^-'-lcos^) =(n+1)-™ (cosi^)-"» [Jm (^') where a;=(2?i+1)sin^6. Other formulae, which exhibit anupperlimit fortheerrorduetoreplacing aLegendrefunction oflarge degree byaBessel function, aref (2)Pn(cos 7])±ITT-lQn(cOS 7]) =V(sec 7;).e±("+*)»(" -tan')) [/,,|(^+1)tan77}±iY^{(n+1)tan77}] 4<9i\/(sec 11) (3) Pn(cosh ^j=(^)/o(nf )+^-^-^, (4)Qn(cosh f)=e-(«+4)(f-tanhj)^(gech I).7^0{(w+i)tanh^| §6>3V(sech|).e-<^^+^)^^ i?(n)+i, where, in(2),^?;<^ir,and, in(3)and(4),f^;thenumbers 6^,62,6sare lessthanunityinabsolutemagnitude, andnmaybecomplex providedthat itsrealpartispositive. Buttheproofofthese results istoolengthytobe givenhere. 5•73.Theformulae ofOlbricJd. The factthat aBessel function isexpressible byHansen's formula asa limit ofahypergeometricfunction hasledOlbricht;):toinvestigate methods bywhich Bessel'sequationisexpressibleasaconfluent form ofequations associated withRiemann's P-functions. Ifwetaketheequation S-^|-(-'^>=o. ofwhich afundamentalsystemofsolutions isthepairoffunctions andcomparetheequation with theequation definedbythescheme 'a, b, c, pU ^, 1, A, /3', 7', *Proc. London Math. Soc.(2)xiii.(1U14), pp.220—221;some associated results hadbeen obtainedpreviously bythesamewriter, Proc.London Math. Soc. xxxi.(1899), p.269. +Watson, Tram. Camh. Phil. Soc. xxii.(1918), pp.277— 308;Messenger, xlvii. (1918), pp.151—160. XNova ActaCaes.-Leop.-Acad. (Halle), 1888, pp.1—48. 5-73] MISCELLANEOUS THEOREMS 159 namely d^y ^fl-«-«- ^1-^-^' ^1-7-y ]^ d^;^\z—a z—h z—c]dz _^f«a^(g-h){a- c) _^y3/3^{h-c){b- a) _^7y(c-a)(c-6)| I2—a 2'—6 ^—cJ ^=0, (2:—a)(s'-b){z—c) weseethatthelatter reduces totheformer if a=0,a=V— IX,o!=—V— fx, whileh,c,/3,/3',7,7'tend toinfinityinsuch awaythat/3+/3'and74-7' remain finite(theirsumbeing 2/a+1)while/3/3'=77'={6-and h+c=0. Wethusobtain thescheme 0, 2^/3,-2i/3, limPIV— fi, ^, 'y,z I- J/- /i,- /^, 7, where7,7'=^+^+^[{ix+i-)-+^Q-}. Another similar scheme is limPlV-IX, ^, 7, \-v-ii, —p, 7, with thesame values of7and7'asbefore. Ascheme forJ^{z)derived directlyfromHansen's formula is r 0, 00, -4a/3,\ limpJii/,a-hv, 0,^^>. (;^::) [_ij,^ l^-\v, v+l-a-^. Olbricht hasgivenother schemes buttheyareofnogreat importanceand those which havenowbeen constructed willbesufficientexamples. Note. Ithasbeen observed byHaentzschel, Zeitschrift furMath, unciPhys.xxxi. (1886), p.31,thattheequation whose solution{%4-3)isiih^<^v (/m),maybederived byconfluence fromLame's equation when theinvariants^2andg-^oftheWeierstrassiau eUipticfunction aremade totend to zero. CHAPTER VI INTEGRAL REPRESENTATIONS OFBESSEL FUNCTIONS 6'1. GeneralisationsofFoisson'sintegral. Inthischapter weshallstudyvarious contourintegralsassociated with Poisson'sintegral (§§2'3,3"3)andBessel'sintegral (§2'2).Bysuitable choices ofthecontour ofintegration, largenumbers ofelegantformulae canbeobtained whichexpressBessel functions asdefiniteintegrals. Thecontourintegralswill alsobeappliedinChaptersviiandviii toobtainapproximate formulae and asymptotic expansionsfor/^(z)when zorvislarge. Ithappensthat theapplicationsofPoisson'sintegralareofamore elementarycharacter than theapplicationsofBessel'sintegral, andaccordingly weshallnowstudy integralsofPoisson'stype, deferringthestudyofintegrals ofBessel'stypeto§6"2.TheinvestigationofgeneralisationsofPoisson's integralwhich weshallnowgiveisdueinsubstance toHankel *. Thesimplestoftheformulae of§3"3is§3*3(4),since thisformula contains asingle exponentialunder theintegral sign,while theother formulae contain circular functions, Avhich areexpressibleinterms oftwoexponentials. We shall therefore examine thecircumstances inwhich contourintegralsofthetype 2^1e'^'Tdt Ja aresolutions ofBessel'sequation;itissupposedthatTisafunction oftbut notofz,andthat theend-points,aandh,arecomplexnumbersindependent ofz. Theresult ofoperatingontheintegralwith Bessel's differentialoperator V„,defined in§3"1,isasfollows: 'fe'-'TdA=^"+2/gu'fj(i_ ^2~)^n+(^2v+1)(>+' Te'-'^ Ttdt =t>''+ie^~^'T{t'-l)hfb a Jai2vVl)Tt-^^{T(t^-l)\dt, *Math. Ann. i.(1869), pp.473—485.Thediscussion ofthecorresponding integi-alsforIv(z} andAV(2)isdue toSchladi, Ann. diMat.(2)i.(1868), pp.232—242, though Schlafli's results areexpressed inthenotation explainedin§4-15. Theintegrals have alsobeenexamined ingreat detail byGubler, ZurichVierteljahrsschrijt, xxxni.(1888), pp.147—172, and,from theaspect of thetheory ofthelinear differential equations whichthey satisfy, byGraf, Math. Ann. xlv. (1894), pp.235—262;lvi.(1903), pp.432—444. SeealsodelaVallee Poussin, Ann. delaSoc. Sci.de Bruxelles, xxis.(1905), pp.140—143. 6-1] INTEGRAL REPRESENTATIONS 161 byapartial integration. Accordingly weobtain asolution ofBessel'sequation ifT,a,haresochosen that ^^{T{t^-l)]^{2v+l)Tt, eiztT(t^-l)=0. Theformer oftheseequationsshews thatTisaconstantmultipleof (t^— I)""*, andthelatter shews thatwemaychoose thepathofintegration, either sothat itisaclosed circuit such thate''^*(^2—1)"+^returns toits initial value after thasdescribed thecircuit, orsothat e'^{t--1)"+^vanishes ateach limit. Acontour ofthe firsttypeisafigure-of-eight passinground thepoint t=lcounter-clockwise andround t=—lclockwise. And, ifwesuppose temporarilythattherealpartofzispositive,acontour ofthesecondtypeis onewhich starts from -|-ooiandreturns there afterencirclingboth thepoints -1,-1-1 counter-clockwise(Fig.1andFig. 2).Ifwetake a,6=±1,itis Fig.1. Fig.2. necessarytosupposethatR{v+h)>0, andwemerelyobtain Poisson's integral. Tomake themany-valued function(i"—1)""* definite*, wetake thephases of^—1and ^+1tovanish atthepointAwhere thecontours cross thereal axisontherightof^=1, Wethereforeproceedtoexamine thecontourintegrals •(i+,-1-) +CCI Itissupposed that vliasnotoneofthevalues i,|,tt,^^, ...;forthen theintegrands areanalytic at±1,andboth integrals vanish, byCauchy's theorem. w.B.F. 11 162 THEORY OFBESSEL FUNCTIONS [CHAP.VI Itistobeobserved that,whenR{£)>0,bothintegralsareconvergent, and differentiations under theintegral signarepermissible. Also, both integralsareanalyticfunctions ofvforallvalues ofv. Inorder toexpressthe firstintegralinterms ofBessel functions, we expandtheintegi-andinpowersofz,theresultingseriesbeing uniformly convergentwithrespecttotonthecontour. Itfollows that /•(!+,-1-)oo,;in2:v^m r{,\+,-l-) z- ^^t(f--ly-i dt=2 r-t'^(P-1)""^dt JA m=ml JA Nowf^it^— 1)""^isaneven oranoddfunction oftaccordingasmiseven orodd;and so,takingthecontour tobesymmetricalwithrespecttotheorigin, weseethatthealternate terms oftheseries ontheright vanish, andweare then leftwith theequation /•(1+,-1-) 00(_\m^v+2m /•(!+) 2" efe*(^f--1)"-* dt=21^ i^^,f^'«(f--l)"-* dt JA m=(2m): Jo a>(_\7nf.v+2)n /(1+)=s^ /;u^>^-Hn-iy-idu, 7/1={'^lll):Jq onwritingt=\/u;inthelastintegralthephasesofuandw—1vanish when uisontherealaxisontherightoiu=\. Toevaluate theintegrals ontheright, weassumetemporarilythat i?(i/+1)>;thecontour may then bedeformed into thestraightline from to1taken twice; onthe firstpart, goingfrom to1,wehave u—l={\—u)e~''', andonthesecondpart, returningfrom 1to0,wehave u—l={\—u)6+"',where, ineach case, thephaseoil—u iszero. Wethusget f(l+)r\ ti>»-h (xi_1)^-*du={e-(-*)'r^'_ e(''-*)T'}tcn-h(1_u)"-^-du =2tcosVTTV ,—^—^^--—- . Nowboth sides oftheequation f"^'«»-» («-1)-'rf.=2icos^I<^^±ili>±i) areanalytic functions ofvforallvalues ofv;and so,bythegeneral theoryof analytic continuation*, thisresult, which hasbeenproved whenR(v+^)>0, persistsforallvalues ofv. *ModernAnalysis, §5-5.Thereader will also find itpossible toobtain theresult, when -R("+i)<0,byrepeatedly using therecurrence formula P+^.-4(u-1)"+-* du=--+J'±-±_1f"^\,"-i(«_i)v+«H du,Jo „+nA-hJwhich isobtained byintegrating theformula |^{u'"+^(u-l)''+"+il-=(m+.+n+l)u"'-^(u-l)''+«+i+(,+„+!)M«'-i(u_ !)'+«-*; theintegralisthenexpressed interms ofauintegral ofthesame typeinwhich theexponentof u-1hasapositive realpart. 6-1] INTEGRAL REPRESENTATIONS 163 Hence, forall*values ofv, Therefore, if 2/+^isnotajDositive integer, and this isHankel'sgeneralisationofPoisson'sintegral. Next letusconsider thesecondtypeofcontour. Take thecontour tolie whollyoutside thecircle [^ |=1,andthen (f-- l)"-*isexpansibleinaseries ofdescending powersoft,uniformly convergentonthecontour;thuswehave m=omil(I- /')2v—l—2rn andintheseries thephaseoftliesbetween -fttand+W. Assumingtthepermissibilityofintegrating term-by-term, wehave (-i+,i+) -2^-p(^-v+m)/•(-^+.i+).^^_^ But |(-1+.1+)f(0+) J'^i Jccexnta where aisthephaseofz(between +Itt); and,byawell-knownformula^, the lastintegral equals-27^^7^ (2m-2v+1). Hence ^i' gi2t/«_1y-idt=^^TTi{)e z L{.2V-f-vi) }^i^ ^ ,,tom\V{^-v)T (2m-2z/+1) _2-+'7rte-'"^'r(i) r(i-.)" ^-^^"^' when weusetheduplication formula^toexpressF{2m-2y+1)interms of r(i-2^+m)andF(-/'+m+1). *Ifc-4isanegative integer, thesimplest wayofevaluating theintegralistocalculate the residue oftheintegrand at«= 1. tTojustify theterm-by-term integration, observe that I' je'-'dt |isconvergent;let J<xl itsvaj«e beA'.Since theexpansion of(('--iy~^ converges uniformly,itfollows that,whenwe aregiven apositive numbere,wecanfindaninteger J/qindependentoft,such thattheremainder afterMterms oftheexpansion does notexceede/A'inabsolute value whenM>:Mq.Wethen have atonce dt <eA-i r" '^^•'|e*~^«| =e, JCX)?' andtherequired result follows from thedefinition ofthesum ofaninfinite series. Cf.ModernAnalysis, §12-22. §Cf.Modern Analysis, §12-15. 11—2 164 THEORY OFBESSEL FUNCTIONS Thus,whenR(z)>0 and 2/+-^isnotapositive integer, (2)r<'i—7/V'"rtVi «y/•(-!+. 1+)[chap.VI 27rtr(|) Thisequationwasalsoobtained byHankel. Next consider f(-i+,i+).,, J00iesp(-!to) where coisanacuteangle, positiveornegative.Thisintegraldefines afunction ofzwhich isanalytic when —iTT+ ft)<argzK^TT+0); and, ifzissubjecttothefurther condition that |argz j<^tt,thecontour can bedeformed intothesecond ofthetwocontoursjustconsidered. Hence the analyticcontinuation ofJ-t,(z) canbedefined bythenewintegraloveran extendedrangeofvalues ofargz;sothatwehave (3) J-.(^)= '^%yrnVe^^Hf--ly-Ut,ZttI L{^)Jcciexp(-iuy) whereargzhasanyvalue between—^tt+o)and^ir+(o. Bygiving wasuitable value*, wecanobtain arepresentationofJ_^(z) foranyassignedvalue ofargzbetween —ttand tt. When jR(2)>andR(v+^)>0wemaytake thecontour tobethatshewn inFig. 3, /\ Fig.3. inwhich itissupposedthattheradii ofthecircles areultimately madeindefinitelysmall. Bytaking eachstraightlineinthecontourseparately, weget J-.{z)-^"^^^s^T/:—'-'<'-")-* + +e^«e'^'(''-i)(l-«2)i/-i^^1 *If iw Ibeincreased inaseries ofstages toanappropriate value (greater thanJtt),arepre- sentation ofJ-I,(z)valid foranypreassigned value ofargzmaybeobtained. 6-11] INTEGRAL REPRESENTATIONS 165 Onbisectingthethird pathofintegration andreplacingtinthevariousintegrals byit, — t,±t, t,itrespectively, weobtain aformula forJ^v{z), duetoGubler*, which corre- spondstoPoisson's integralfor./^(s) ;theformula is (4)J-A^)=^,^J;jf [smvTTl^\-^^{l +t^^r-hdt+rcoB{zt +vn).{l-t^y-idt'],i{"+2)>K^JL J Jo J and,ifthisbecombined with Poisson'sintegral,itisfound that 2{^zr (5) n(^)= j\m{zt).{\-t^-)''-hdt-Ie-'i{\-irt'^Y-hdt\,r(v+*)r(i) aformula which was alsodiscovered byGubler, thoughithadbeenpreviouslystated by Weber tinthecase ofintegral values ofv. After what hasgonebefore thereader should havenodifficultyinobtainingaformula closely connected with(1),namely (6) inwhich itissupposedthatthephaseoft'^—\ vanishes when tisontherealaxisonthe rightofi=l. 6'11.Modifications ofHankeVs contourintegrals. TakingR{z)>0,letusmodifythetwocontours of§6"1intothecontours shewn inFigs.4and5respectivefy. Fiff. 4. Fik'. 5. Bymakingthoseportionsofthecontours which areparalleltothereal *Zurich Vierteljahrsschrift,xxxiii. (1888), p.159. Seealso Graf, Zeitschrift fiirMath, and Phys. xxxvm. (1893), p.115. tJournal filrMath, lxxvi.(1873), p.9.Cf.Hayashi, NytTidsskriftforMath, xxiii. is,(1012), pp.86—00. Theformula wasexamined inthecasei-^O byEscherich, Monatsheftc fiirMath, undPhys.iii.(1892), pp.142, 234. 166 THEORY OFBESSEL FUNCTIONS [chap.VI axismove offtoinfinity (sothattheintegrals along them tend tozero),we obtain thetwofollowingformulae: X (2)J_.U)=(1+) r(^-v)e"-(i0)(-1-) i J-1+coi 27rtT(|) (1+) Xr(i+) ,, /•(-!+) Il+ooi./—1+00} Inthe first result themany-valuedfunctions aretobeinterpreted bytaking thephaseoft^—1 tobe atJ.and tobe+ttatB,while inthesecond the phaseoff-—1isatJLcmd is—itatB. Toavoid confusion itisdesirable tohave thephaseof ^-—1interpreted inthesamewayinboth formulae; andwhen itissupposed thatthephaseof ^-—1is+TTati?,theformula(1)isofcourse unaltered, while(2)isreplaced by (3)J.A^)= X27rtr(1) (1+) e^^{t-- \y-^dt-\-e l+OOl(-1-)-vm\e^'^{t--iy-^dt -1+00j Inthelastoftheseintegrals,thedirection ofthecontour hasbeen reversed andthealteration intheconventiondeterminingthephaseofi^—1has necessitated theinsertion ofthefactor g-slv-jj^ri Oncomparing equations (1)and(3)with§3"61equations (1)and(2),we seethat (4) (5)H.^H^)=^^\-"r}a'fl''^' e^^^{t^-iy-idt, *""*^\2.'J-1+oot unless Visaninteger,inwhich caseequations (1)and(3)arenotindependent. Wecan,however, obtain(4)and(5)inthecasewhen vhasanintegral value(w),from aconsideration ofthefactthat allthefunctions involved are continuous functions ofvnear v=n.Thus ir„<i' {z)=limfr;i' (z) y^n-JTl1(t) ;1+ooi _r(|-n).(i^)^fi^^) iriVil) 1+ooie'^^if-iy-idt, andsimilarlyfor^„*^' {z). 6-12] INTEGRAL REPRESENTATIONS 167 Asinthecorresponding analysisof§6'1,therangesofvalidityof(4)and (5)maybeextended byswinginground thecontours andusingthetheoryof analyticcontinuation. Thus,if—^77<<y<fTT,wehave (6) ^.'^' (z)='^\.;ViV^"*(i'-1)"-- dt; while, if-fTT<6)<Itt,wehave (7) ^;=) (z)=^^-vwrf^I^"'(^'-1)^"-^^^' '^1' i-\2) cotexp(—!w) provided that, inboth(6)and(7),thephaseofzliesbetween —^tt+«and ^TT4-&). Representationsarethus obtained of^^'^' (5)whenargzhasanyvalue between —ttand 27r,andofH^^-^ (z)whenarg^hasanyvalue between —27rand tt. Ifo)beincreased beyondthelimits stated,itisnecessarytomake thecontours coil round thesingular pointsoftheintegrand,andnumerical errors areliable tooccur intheinterpretationoftheintegralsunless greatcare istaken. Weber, however, has adoptedthisprocedure.Math. Ann. xxxvii. (1890), pp.411—412, todetermine the for- mulae of§3-62connecting H^C^) (-z\Hj:^) (- .-)withH,m (s),HyC^) (z). Note. Theformula 2ii\{z)=Hy('^){z)-HJ~'){z) makes itpossibletoexpress 3^^(2)in terms ofloop integrals,and inthismanner Hankel obtained the series of§3-52 for Yn(3);this investigationwillnotbereproducedinview ofthegreater simplicityof Hankel's othermethod which hasbeen described in§3-52. 6"12. Integral representations offunctions ofthethird kind. Intheformula§6-11 (6)supposethatthephaseofzhasanygivenvalue between —ttand 27r,anddefine/3bytheequation argz=(o+^, sothat—Itt</3<Itt. Thenweshall write t-l= e-^^i z-^(-m). sothatthephaseof-uincreases from-tt+/3tott+/3as tdescribes the contour; and itfollows immediatelythat (1) H^^ (z)=-^^ 7,f,-. e-(-uy-^1+A du, where thephaseof1+^iulzhas itsprincipalvalue.Again,if^beagiven acuteangle (positiveornegative),thisformula affords arepresentationof i/'^<i> {z)valid overthesector ofthe2r-planeinwhich _--Itt+/3<argzk^it +13. 168 THEORY OFBESSEL FUNCTIONS [CHAP.VI Similarly*,from§611(7), (3) HJ'> (.)={^^l.^,;^I.-««-* (l+ 2i)^-. wi.«w=Q VTTTirl. -"''-K^-s)*''where/3isanyacute angle (positiveornegative)and -|7r+/3<arg^<Itt+^. Sincet,by§3-61(7),H^^^'^ {2)=e""'iT^D (^),itfollows thatwelosenothing byrestrictingvsothatR(v+^)>0;and itisthenpermissibletodeform thecontours intothelinejoiningtheoriginto00expt/3,taken twice; forthe integralstaken round asmall circle (withcentre attheorigin)tend tozero with theradius ofthecircle|. Ondeformingthecontour of(1)inthespecified manner, wefindthat where /3maybeanyacuteangle (positiveornegative)and R{v+l)>0,-Itt+/3<arg^<ftt+/3. Inlikemanner, from(2), 2\^g—Hz—ivTr—iTT)rxexp'jS/iu\''~^ where^maybeanyacuteangle (positiveornegative)and R(v+l)>0, -f7r+/3<arg^< |7r+yS. The results(3)and(4)have notyetbeen proved when 2uisanoddpositive integer. Butinview ofthecontinuity near v=n+^ofthefunctions involved (where?j=0,1,2,...) itfollows,asinthesomewhat similar work of§6"11, that(3)and(4)aretruewhen v=^, §,#,....Theresults mayalsobeobtained forsuch values ofvbyexpanding theintegrands interminatingseries ofdescending powersofz,andintegrating term-by-term; theformulae soobtained areeasilyreconciled withtheequationsof§3'4. Thegeneralformulae(3)and(4)areoffundamentalimportanceinthe discussion ofasymptotic expansionsofJ±„(z)forlargev^alues of\z\.These applicationsoftheformulae willbedealt with inChapterVll. Auseful modification oftheformulae isdue toSchafheitlin§.Ifwetake argz=l3(sothatargzisrestricted tobeanacuteangle), andthen write u=2z cot6,itfollows that ,^. rT(.^,^2"+'^" r^-cos''-*(9.e-»'<^-''^+^'^„,^,^ *Toobtain thisformula, write f-hl=e~^" 2-^(-h),t-l=2e"'"(1-^iufz). fThere seems tobenosimple direct proof that isaneven function ofv. JCf.ModemAnalysis, §12-22.§Journal furMatlu cxn-.(1894), pp.31—44. 6-13] INTEGRAL REPRESENTATIONS 169 andhence that 2^1^^ li''008"-'^$. sin(z-vO+hO)_.^^^^^g,. (7)^''(^)-r(i;+i)r(i)Josin^-'+i^-e^"rf^, These formulae, which areofcourse validonlywhenR(v+^)>0, were applied bySchafheitlin toobtainpropertiesofthezeros ofBossd functions (§§15'32—15"35). Theywere obtained byhimfrom theconsideration thatthe expressionsontherightaresolutions ofBessel'sequationwhich behave inthe appropriate manner near theorigin. TheintegralIe~"^ u^'^^{l+ii)'^~^ du,which isreducible tointegralsofthetypesJo occurringin(3)and(4)whenix=v,hasbeen studied insome detail byNielsen, Math. Ann. Lix.(1904), pp.89—102. Theintegralsofthissection arealsodiscussed from theaspectofthetheoryofasymp- totic solutions ofdifferentialequations byBrajtzew, Wm'sckauPolyt.Inst. Nach. 1902, nos.1,2[Jahrhuchiiher dieFortschritte derMath. 1903, pp.575—577]. 6*13. Thegeneralised Mehler-Sonineintegrals. Someelegantdefiniteintegrals maybeobtained torepresentBessel functions ofapositivevariable ofasuitablyrestricted order. Toconstruct them, observe that,when zispositive (=x)andtherealpartofvislessthan|,itisper- missible totake &)=Ittin§G'll(6)and totake«=—^ttin§6'11(7),so thatthecontours arethose shewn inFig.6.When, inaddition, therealpart ofVisgreaterthan—|,itispermissibletodeform thecontours(after the manner of§6"12) sothatthe firstcontour consists oftherealaxisfrom+1 to+00taken twice while thesecond contour consists ofthereal axisfrom —1to—00taken twice. N.-^.^ r,^^ / -Fig.6. Wethus obtain theformulae Ri^)(,.;)=^(i^^lM (1_e-.(-*t--) [V* {t'-1)^-^ dt, I/f^(-3) (^,)=^"^Skl^^-^ (1_e=(-5)-)^e-'-' it?-1)"-^ dt, 1 iriv(o).'1 thesecondbeingderived from§Gil(7)byreplacingtby-1. 170 THEORY OFBESSEL FUNCTIONS [CHAP.VI Inthese formulae replacez;by-1^andusethetransformation formulae given by§3-61(7).Itfollows that,whenx>0 and-l<R{v)<^, then 2 ['*e^^*dt (1) iT.W (a;)=iY{l-v)T(i).{lxy]1{t'-iy-^i' 2[*e-^'^^dt sothat (3) /.(^)=2f'"sin(a:;^).dt r(i-^)r(i).(i^)v'i (f^-i^i' 2 r°°cos(xt).di (4) Y^{x)=_- ^_^^J,^^^_^^^^^I ^^^.^_^^^^^. Ofthese results, (3)wasgivenbyMehler, Mat/t. Ann. v.(1872), p.142,inthespecial casev-0, while Sonine, Math. Ann. xvi. (1880), p.39,gaveboth(3)aud(4)inthesame specialcase. Other generalisationsoftheMehler-Sonine integralswillbegivenin§6-21, 6*14. Symbolic formulaedue toHargreave andMaodonald. "WhenR{z)>0and^(i/+^)>0,itisevident fromformula§6-11(6)that where thephaseof1—i^liesbetween and— \tt. IfDdenotes{djdz) and/isanypolynomial,then f(it).e^^^=f{D)e^\ andso,tokenv+\isapositive integer.,wehave -r(»+i)r («''+"'z When v+Jisnotapositive integer, thelastexpression mayberegardedasasymbolic representationof^^O {z\ontheunderstanding that/(Z)) (e±'^/2)istobeinterpretedas ',Ie^^'/(^dt. Consequently (1) B^K^-s(z)= ly (i+i)y-j«i!iizi^2^, andsimilarly SOthat2 6-14, 6-15] INTEGRAL REPRESENTATIONS 171 The series obtained from(4)byexpandinginascending powersofDdoesnotconverge unless itterminates; theseries obtained inasimilar manner from(3)converges only whenR{v)>\. Theexpressionsontherightof(3)and(4),with constant factorsomitted, were given byHargreave,Phil. Trans, oftheRoyalSoc.1848, p.36assolutions ofBessel's equation. Theexact formulae areduetoMacdonald, Proc.London Math. Soc.xxix.(1898), p.114. Anassociated formula, valid forallvalues ofv,isderivable from§6*11(4). Ifnisany positive integer, weseefrom theequationinquestion that '^^r(o) Ji+xi =^ ^^:A^- ^a+Z>-)" {t'^-iy-'^-he'^tdt,TTlV(2) Jl+=>Ji sothat (5)^<^>(.)="^^^^i^^^'+^'^" '^*^')"~^ ^-!-"^'^^- Asimiltir equation holds fortheother function ofthethirdkintl,andso (6) K {Z)=r(-+i-^^):(R^ (1+Z)2). |(Z,)n-.%-^_^^^„)^_ 1\V'r0) This result, provedwhenR(z)>0,iseasily extended toallvalues ofzbythetheoryof analytic continuation;itwasdiscovered bySonine, jMath. Ann. xvi. (1880), p.66,when !/=?«,andusedbySteinthal, Quarterly Journal, xviii.(1882), p..338when i/=«+|;inthe casewhen v=n-\-\ theresult wasgiven slightlyearlier (withouttheuseofthenotation ofBessel functions) byGlaisher, Proc. Camh. Phil. Soc. in.(1880), pp.269—271. Aproof based onargumentsofaphysicalcharacter hasbeen given byHavelock,Proc.London Math. Soc.(2)11.(1904), pp.124—125. 6*15. Schldjii's* integ7xds ofPoissonstypeforIu{z)andK„(z)- Ifwetake«=^ttin§6"1(3)andthenreplacezbyiz,wefind that,when Iargz I<^TT, andthephaseof^--1atthepointwhere tcrosses thenegativerealaxis is--Itt. 4> Fig. 7. IfwetakeR(v+l)>0tosecure convergence,thepathofintegration niaybetaken tobethecontour ofFig. 7,inwhich theradii ofthecircles maybemade totend tozero.Wethus findtheformulaf /_^(.)^Illi^4^;iaf):|"(i _e---) l^e-^^{f'-ir-^dt 27rir(i) +i(e-""'+e-'"^') I' e-"(1-fY'^dt, *Ann. diMat.(2)i.(1868), pp.239—241. Schliifli obtained theresults (1)and(2)directly bythemethod of§G-1. tCf.Serret, Journal deMath. ix.(1844), p.204. 172 THEORT?" OFBESSEL FUNCTIONS [CHAP.VI inwhich thephasesoft--landof1-^^areboth zero. Now, from§8-71(9), wehave andso (3) 7_,(z)-/.(z)=^^ ^^iMiy 1^" ^' that istosay* (4)/C(^)='^^|4^/V'(*-!)'-**, whence weobtain theformula (5) K,(z)=^^i^i^^ fe—'^«sinh^''^d^,^i(i/+t)Jo aresult setbyHobson asaproblemintheMathematical Tripos,1898. The formulae are allvalidwhenR{v+l)>0and [arg^|<Itt.Thereader will find itinstructive toobtain (4)directlyfrom§G'll(6). 6*16. Basset's integral forK^,{xz). When Xispositiveand^isacomplexnumbersubjecttothecondition Iarg^ I<l-TT,theintegralforH^^^ {xze^-"^)derived from§611(6)maybewritten intheform Now,whenR(v)^-l,theintegral,taken round arcs ofacircle frompto pgijTTi-iargz^ tends tozero asp^oo,byJordan's lemma. Hence, byCauchy's theorem, thepathofintegration maybeopenedoutuntil itbecomes theline onwhichR(zt)=0.Ifthenwewrite zt=iu,thephaseof—{u-jz^)—1is—tt attheorigininthe?t-plane. Itthen follows from§3-7(8)that K,{xz)=iTTie-i"-*iT^l^ (.r^e^-O 2r(i) J..V. (^^-i)"^-^ V{v+\). {Izype-^^'^du and sowehave Basset's formula ,^,_^_ V{v+\).{±zyr°°co^xu.du validwhenR{v+\)^0, x>0,\ argz\<\7r. Theformula wasobtained by Bassetf,forintegralvalues ofvonly,byregarding Kq{x)asthelimit of *Theintegral ontherightwasexamined inthecase v=byKiemann, Ann. derPhysik und Chemie, (3)xcv.(1855), pp.130—139. tProc. Camh. Phil. Soc. vi.(1889), p.H;Hydrodynamics,ii.(Cambridge, 1888), p.19. 6-16, 6-17] INTEGRAL REPRESENTATIONS 173 aLegendrefunction ofthesecond kindandexpressingitbythecorresponding limit oftheintegralofLaplace's type(Modern Analysis, §15"33). Theformula forKn(^^)isobtainable byrepeated applicationsoftheoperator— j-. Basset alsoinvestigatedasimilar formula forI^,(xz),butthere isanerror inhisresult. Theintegral ontherightin(1)wasstudied bynumerous mathematicians before Basset. Among these investigators were Poisson(see §6"32), Journal deI'FcolePolytechnique,ix. (1813), pp.239—241; Catalan, Journal deMath. v.(1840), pp.110—114(reprinted with some corrections, Mem. delaSoc.R.desSci.deLiege, (2)xii.(1885), pp.20—31);and Serret, Journal deMath. viii.(1843), pp.20,21; ix.(1844), pp.193—210; SL-hlomilch, Analytischen Studien,u.(Leipzig, 1848), pp.96—97.These writers evaluated theintegral inianite terms whenj/+|isapositive integer. Other writers whomust bementioned areMalmsten, K.Svenska V.Akad. Handl. lxii. (1841), pp.65—74 (see§7-23) ;Svanberg, Nova ActaReg.Soc. Sci.Upsula,x.(1832), p.232; Leslie Ellis, Trans. Camh. Phil. Soc, viii.(1849), pp.213—215;Enneper, Math. Ann. vi. (1873), pp.360—365;Glaisher, Phil. Trans,oftheRoyalSoc. CLXXii.(1881), pp.792— 815;J.J.Thomson, Quarterly Journal, xviii.(1882), pp.377—381;Coates, Quarterly Journal, xx.(1885), pp.250—260;andOltramare, Comptes Rendus deVAssoc.Frangaise, XXIV.(1895), partil.pp.167—171. The lastnamed writer proved bycontourintegration that rcos.Ku .du_(-)»-'TT ["(i"-'/e-->-^Vi> \-j _(-)»-i7rrc^"-! e- p=l+pY_ Theformer ofthese results maybeobtained bydifferentiating theequation cosxu .du ire~^^'^P /. andthelatter isthen obtainable byusing Lagrange's expansion. 6"17.Whittakers*' generalisations ofHankel'sintegrals. Formulae ofthetypecontained in§3'32suggestthat solutions ofBessel's equationshould beconstructed intheform zi\'e^''Tdt.Ja Itmaybeshewn bythemethods of§6'1that v,.|.»/%.«™|^ r* ( d"T dT ) andsotheintegralisasolution ifTisasolution ofLegendre's equationfor functions oforderv—\andthevalues oftheintegrated partarethesame at eachendofthecontour. *Proc. London Math. Soc.xxxv.(1903), pii.19S— 206. 174 THEORY OFBESSEL FUNCTIONS [CHAP.VI IfTbetaken tobetheLegendrefunctionQ^_j(^),thecontour maystart andendat+00iexp(— ?'&)),where wisanacuteangle (positiveornegative) providedthat zsatisfies theinequalities —Itt+w<argz<^TT+0). IfTbetaken tobePy-^{t),thesame contour ispossible; butthe logarithmic singularityofP^_i(^at^=—1(whent-—|isnotaninteger) makes itimpossibletotakethelinejoining—1to1asacontourexceptin thespecialcaseconsidered in§3'32;foradetailed discussion oftheintegral inthegeneral case, see§10"5. Wenowproceedtotakevarious contours indetail. First consider /•(-!+,1+) zie'''Q,..(t)dt, Jootexp(— /(o) where thephaseoftiszero atthepoint ontherightof^=1atwhich the contour crosses thereal axis.Take thecontour toliewhollyoutside thecircle 1^1=1andexpand Qv-^{t)indescending powersof t.Itisthus found, asin thesimilaranalysisof§6'1,that (lz\^- e-^^"^^'!'^^ r(-i+.i+) (1) J^(^)=^Hn^n— I«"*Q-i(0dt, I'^\2).xiexp (-2u)) andtherefore (2) J-A^)=-^l,^.^''Q-^-k (t)dt ""^va/ Jixiiexp (-iu>) Ifwecombine these formulae andusetherelation*connectingthetwo kinds ofLegendre functions, Avefindthat TTi (2)cosVTTJooiexp(iai)TTr(I)cosVTTJooiexp(ia.) Again,consider zHe'''Q,_^{t)dt; Jat:7AT-n (—},.\)(1+) <Kiexp (— iuj) this isasolution ofBessel'sequation, and, ifthecontour betaken tolieonthe rightofthelineR{t)=a,itisclear thattheintegralis[z^exp{—a\z \)]as z-^-\- coi.Hence theintegralisamultipleof^^"' (^).Similarly bymaking z^—00%,wefindthat /(-1+)U&'~'Q,_,{t)dt Jooiexp(—i<o)?i Tlie relation, discovered bySchlafii,is p„(.)=*^{Q„ (.)-§_„_,(.)}:TT cf.Hobson, Phil. Trans, oftheRoyal Soc.clxxxvii.(1896), p.461.2 6-2] INTEGRAL REPRESENTATIONS 175 isamultipleofiT/-' (z).From aconsideration of(1)itisthen clear that (4) H^^H^)=^ ^^p.i.I e<^Hl_,{t)dt, 7* *-V2/•00!exp(-iaj) C^z^h p-h'v+h)ni r(-l+) (5) H^^^ {z)=^Ie-'Q,_,(t) dt, andhence, by§3"61combined with Schlafli's relation, (TLz\h pk(v+h)Tvi r{-\+) (6) H.^^ (z)=l^.\^.^^^^^e^'^P._i(0dt;TT1(2/COSVTTJ rjjj-exp {-ioj) this isalsoobvious from(3). Theintegralwhich differs from(6)only b}^encirclingthepoint+1instead of—1iszero since theintegrandisanalyticinside such acontour. In(5)and(6),arg {t+1)vanishes where thecontour crosses therealaxis ontherightof—1,and, in(5),arg (<—1)is—ttatthatpoint. 6*2. Genei^alisationsofBesseVsintegral. We shall nextexamine variousrepresentationsofBessel functionsbya systemofdefiniteintegralsandcontourintegralsdue toSonine* and Schlaflif. Thefundamental formula which willbeobtained iseasily reduced toBessel'sintegralinthecase offunctions whose order isaninteger. Wetake Hankel's well-knowngeneralisation;]:ofthesecond Eulerian integral 1 1r(o+) V{v+m+l)^^i].^^"'""^*^^' inwhich thephaseoftincreases from—tttottastdescribes thecontour, and then 2\^'^ TT=^ - ,t-"-^-^ e^dt. Consider thefunction obtained byinterchangingthesignsofsummation andintegi'ationontheright;itis /•{0+) {Z-) This isananalyticfunction ofzforallvalues ofz,and,whenexpandedin ascending powersofzbyMaclaurin's theorem, thecoefficients maybeobtained bydifferentiatingwithregardtozunder theintegral signandmakingzzero after thedifferentiations§.Hence t-''-'exp\t-~[dt= SLJ_llfZ_t-''-^--U'dt, -00( 'it) „j=o'/h! J-00 *Matlmnatical Collection, v.(Moscow, 1870);Math. Ann. xvi.(1880), pp.9—29. tAnn. diMat.(2)v.(1873), p.204.Hismemoir, Math. An7i. iii.(1871), pp.134—149, sliould alsobeconsulted. Inaddition, seeGraf, Math. Ann. lvi.(1903), pp.423—432, andChessin, John Hopkins University Circulars, xiv.(1895), pp.20—21. +Cf.3IodernAnalysis, §12-22. §Cf.ModernAnalysis, §§5-32, 4-44. 176 THEORY OFBESSEL FUNCTIONS [CHAP. VI andsowehave atonce (1)^'W=y/_j-'--pK4* This result, which wasdiscovered bySchlafli, wasrediscoveredbySonine; andthelatter writer wasthe first topointout itsimportance. When Iargz\<^7r, wemayswinground thecontour about theoriginuntil itpassestoinfinityinadirection makinganangle argzwith thenegative real axis. Onwritingt=Izu,wethen find that,when |argz \<|7r, (2) ^•'^^)=2^-L„ "~''"'exp|i^(i.--j|^z^. ThisformwasgiveninSonine's earlierpaper (p.335). Again, writingu=e'",wehave (3) J,(Z)=^r—.e2Sinh«,-K!. ^^y^ validwhen jarg^[<^tt.This isthe firstoftheresults obtained bySchlafli. Inthisformula takethecontour toconsist ofthree sides ofarectangle,as inFig. 8,with vertices atoc—iri,—iri, rriandx+m. ni -TTl Fig.8. Ifwewrite t+ttiforivonthesidesparalleltotherealaxisand+idforw onthelinesjoiningto+tri,wegetSchlaflisgeneralisation ofBesseVsintegral (4) /.(2)=- f''cos(i/6'-^sin^)rf^-^H^^ ["e-^'-^^'^ht ^^^ validwhen jargz \<^tt. Ifwemakearg^-^±^tt,thefirstintegralontherightiscontinuous and, ifR(v)>0,soalso isthesecond, and/^(z)isknown tobecontinuous. So(4) isstilltruewhen ^risapureimaginaryifR(v)ispositive. Theintegrals justdiscussed wereexaminedmethodically bySonine inhis second memoir; inthatmemoir heobtained numerous definiteintegrals by appropriatemodifications ofthecontour. Forexample,ifyfrbeanacuteangle (positiveornegative) and if 6-21] INTEGRAL REPRESENTATIONS 177 thecontour in(3)maybereplaced byonewhichgoesfromx— (tt— yfr)ito 00+(tt+^|^)^. Bytakingthecontour tobethree sides ofarectanglewith corners atcc—(v—y}/)i,-(tt— -v/r) i,(tt+y}r)i,and co+(tt+-v/r)i,weobtain, asamodification of(4), g—viipfn (5) Ju{z)=e'^''*°''"-'^^*'cos(i^^-^cos-v|rsin^)rf0 e-"'* sinITTf»..,,^.,, ., TT Jo Again,ifwetake-^tobeananglebetween andtt,thecontour in(3)may bereplaced byonewhichpassesfrom oo—(|7r4 y\r)itogo+(|7r4-\|r){, and sowefindthat 1riT+^ (6) ./^(^)=1[" cos(i^^-^sin6')r/6? 1f" ^__g-zsinh«sin^-.,? g-j-^(2rCOsh tCOS"^-1I^TT- J^a/t) dt, ttJ provided that |argz \islessthanboth\\rand tt—^. WhenR{i>)>and zispositive (=x),wemaytake>|r=inthe last formula, andget* 1fi'^. 1r^ (7) ./^(a;)=- cos{yO—xsin0)dd+ e""'sin(a;cosh t—I^'tt)c/^. Anotherimportant formula, derived from(1),isobtainedbyspreadingout thecontour until itisparalleltotheimaginaryaxisontherightoftheorigin; byJordan's lemma this ispermissibleifR{v)>—1,andwethen obtain the formula (8) J.(^)=^-^^-J t-''-^exi^]t--\dt,^TTl Jc—Xi 4^1 inwhich cmayhaveanypositivevalue;thisintegralisthebasis ofmanyof Sonine'sinvestigations. Integrals which resemble those giveninthissection areofimportanceintheinvestiga- tion ofthediffraction oflight byaprism;seeCarslaw,Proc. London Math. Soc.xxx. (1899), pp.121—161;W.H.Jackson, Proc.London Math. Soc. (2)i.(1904), pp.393—414; Whipple, Proc.London Math. Soc.(2)xvi.(1917), pp.94—HI. 6-21.Integralswhichrepresent functions ufthesecond andthird kinds. Ifwesubstitute Schlafli'sintegral §6-2(4)forboth oftheBessel functions ontherightoftheequation Y^{z)=/„{z)cot v-TT-/_,.{z)cosec vtt, wefindthat firC^ nY,{z)=cotVTT COS{v6-zsind)dd-cosec pit cos(r6^+^sin^)dB J(,.' -cosVTTre-"'-''"'^'' dt-f"e"'-""'!'' dt. Jo J^ *Cf.Gubler, Math. Aiiit. xi.ix.(18',)7), pp.583—584. W. 13.V.^-^ 178 THEORY OFBESSEL FUNCTIONS [CHAP. VI Replace^bytt—^inthesecondintegralontheright,and itisfound onre- duction that 1/'" 1C" (1)Y,(2)=-sin{2sine-v6)d6- -\ (e"*+e""*cosv-n)g-^inhf ^^^ aformula, practicallydiscovered bySchlafli (who actually gavethecorrespond- ingformula forNeumann's function), which isvalidwhen [argz \<^tt. Bymeans ofthisresult wecanevaluate TTlj_x when IargzIk^tt;forwetake thecontour toberectilinear, asinFig. 9,and liy > Fig.9. write— t,id, t+iriforwonthethreepartsofthecontour;wethen seethat theexpressionisequalto ] ("00 1/"" p—vniroo andthis isequalto/„(z)+iY^{z) from formula(1)combined with§6"2(4). Hence, when |argz\<^7r, wehave (2) jy^d' (z)=—.( "'e^sinhw-w^' ^^^ (3) JY,(^)(z)= .!"' e^^'»hw-uwfi^ TTlJ -co •Formulae equivalent tothese were discovered bySomnierfeld, Math. Ann. XLVii.(1896), pp.327—357. Theonlydifierence between these formulae andSommerfeld's isarotation ofthecontours through aright angle, with acorresponding changeintheparametric variable; seealsoHopf andSommerfeld, Archiv derMath, undPhys. (3)xviii.(1911), pp.1—16. Byanobviouschangeofvariable wemaywrite(2)and(3)intheforms (4) ir,(') {z)=— . IM-"-^expU^ [u- ^^jdu, 1rocexp(-7ri)(/1V1 (5) H.^^ (z)=-A. ,^,-.-.exp |.. (u-i)|du 6-21] INTEGRAL RErRESENTATIONS 179 thecontours arethose shewn inFig. 10,emergingfrom theorigin andthen bendinground tothe leftandright respectively;resultsequivalenttothese were discoveredbySchlafli. Fig. 10. [Note.There isnodifficultyinprovingthese results forintegral values ofv,inview ofthecontinuityofthefunctions involved;cf.§6"11.] Weproceedtomodifythecontours involved in(4)and(5)toobtain the analyticcontinuations ofthefunctions onthe left. Ifo)isananglebetween —ttand -rrsuch that 1to—argzj<^tt,wehave (6) and 0)^'"'(^)=s/,»exp(tt-io)j expiu>•" ^exp -j^e (u— \\du, 1 f-Tjexp(—TT-io)i "^exp \ijZ(u—][du, TTtJoexpiu, ["\tin thecontoursbeingthose shewn inFig.11andFig.12;andthese formulae givetheanalyticcontinuations ofthefunctions onthe leftovertherangeof Fig. 11. Fig. 12. values of^forwhich (u-^tt<arg^-<w+^tt;andwmayhaveanyvalue between*—ttand tt. *If Iw Iwere increased beyondthese limits, difficulties would arise intheinterpretationof thephase ofu. 12—2 180 THEORY OFBESSEL FUNCTIONS [CHAP. VI Modifications of(2)and(3)areobtained byreplacing wbyw±^iri;itis thusfound that* (8) HJ^'H2)=^ ei2coshw-vw ^^ 2g—i'"rifao+iTT TTlJo TTlJ-oc+hni 2eigizcoshwqqqY^ J^W .dw, (9) ir^(-' (Z)=r- g-izcoshw-,'W d^ g-izcoshwqqq]^ j/|y .dw, TTl providedthat |argzIk^tt. Formulae ofspecialinterest arisebytaking^^positive (=«)in(6)and(7) and—l<R{v)<l. AdoubleapplicationofJordan's lemma(tocircles of largeandsmall radiusrespectively)shews that, insuch circumstances, wemay take (o=^7rin(6)andw=—^ttin(7). Itisthus clear, ifubereplaced by ±ie'',that (10) HJ'^Ux)= r- e^^coshi-.'^^^^ __e'-'^cosh ^cosh z^^f/«, TTlj_oo 7n J ijkvni rocOpkvrriI"ao (11) ff/"^ (x)= ^ g-ixcosht-utfli^ 1^ g-ixcosht cogh i,t .dt, TTlJ_00 "TTl J andhence, when ^>and—1<iit(i^)<1,wehave 2C'^ (12) J^,(x)=- Isin(xcosh ^—^vtt).cosh vt .dt, TTJ (13) F^(a;)=—-I cos(«cosh i—Aj/tt).cosh i/^ .c?^; and, inparticular (cf.§6'13), (15) .F.w=-?r5^*, whenwereplacecosh tby^. The lasttwoformulae areduetoMeliler, Math. Ann. v.(1872), p.142,andSoDinc, Math. Ann. xvi.(1880), p.39,respectively; audtheyhave alsobeen discussed byBasset, Proc. Camb. Phil. Soc. viii.(1895), pp.122—128. Aslightly different form of(14)hasbeen given byHardy, Quarterly Journal, xxxii. (1901), pp.369—384; ifin(14)wewritex=2J{ah),xt^au+bjw,wefindthat (16) j'%in(«u+ ^^)^^=.yo{2v/(«6)}. Note. Thereader willfind itinstructive toobtain(14)from theformula A(cos^) =^f'-sin(n+^|)(^ 77je^/{2(cos^-cos0)}"^ combined with theformula5^5-71(1).ThiswasMehler'soriginal method. *Cf.Coates, Quarterly Journal, xxi.(1886), pp.183—192. 6-22] INTEGRAL REPRESENTATIONS 181 6"22.Integrals representing /„(^) andKy(z). Themodifications oftheprevious analysiswhich arcinvolved inthedis- cussion ofI^(z)andK^(z)areofsufficient interest tobegiven fully; theyare due toSchlafli*, thoughheexpressedhisresultsmainlyinterms ofthe function F{a, t)of§415. Theanalysisof§62iseasilymodified soastoprovethat andhence, when |argz\<^'ir, (3) /,(z)=J—-V" (!"<"'"•-"• dw. -iTT% J00—17} Theformulae (2)and (3)arevalidwhenarg^:=+^ttifE(v)>0. Ifin(3)thecontour istaken tobethree sides ofarectanglewith corners at00—Tri,—iri, -rri,x+iri,itisfound that 1' ir SinVTT I"^ (4) IJ2)=- e'^'^'^cosvede ^e-'^^^^'f-"' dt, SOthat 2sin 7777r^ /_.(z)-L{z)= e-^c°**h« cosh vt .dt, andhence, when jarg^<^tt, (5) K,{z)=re-^cosht cosh jjt .dt, Jo aformula obtained bySchlafli fbymeans ofsomewhat elaborate transforma- tions. From theresultsjustobtained, wecanevaluate 27rie^coshw-uwd^ cc—ni when \argz\<^7r.For itiseasilyseen that "Irr=o-7r/r^+TTi]=_!-.J + gZCOS\MV-VW d^fj 27^^[j_^_^,- Joo-ni) 27nJ X—TTi =^.je-"^'''^''-'''dt +I,(z) "l-TTlJ _: eI/TTl 1ismvir[I_^{z)-L{z)]+L{z\ *Ann. diMat.(2)v.(1873), pp.199—205. tAnn. diMat.(2)v.(1873), pp.199—201 ;thisformula wasused byHeine, Journal Jur Math. Lxix.(1868), p.131,asthedefinition towhich reference wasmade in§5-72. 182 andhence (6)THEORY OFBESSEL FUNCTIONS^[CHAP.VI e^'''I.Az)-e-'"''I,{z)oo+wi g0COShW-vW(^,;y— 27rlj -oc-TTl Again, wemaywrite (5)intheform2isinvir (7) KAz)=^re-'^^'^"-''Ult, andhence, bytheprocessesused in§6-21, nfeeexp (-ill) 'Oexpiuj(8)1/•«=exp(-i«.)r / 1\) , ^"(^)= 9 u—^exp-iWw+-)du, when—TT<fi)<TTand—^tt+«<argz<^tt+(o. Similarly ^.s -r/xsinz^TT /'°°exp(,r-a>),-r /ix) (9)e""'/_,(^)-e-''«I,(z)= it-"-^expnz(u+-hdu ; 7^J0exp{-7r+<o)j( V ^^/j this isvalidwhen <aX27rand-^tt+&><arg^<^tt+co. Thecontours fortheformulae (8)and(9)areshewn inFigs.13and14 respectively. Fig.13. Fig.14. Further, when zispositive (=x)and—1<R(v)<l, thepathofintegra- tion in(8)maybeswunground until itbecomes thepositivehalf ofthe imaginaryaxis;itisthusfound that -^v(^)=le"^""'y"""^exp^-lixiv- sothat (10) K^(x)=le-i""je-f^sinh(-vt ^^^ and,onchangingthesignofv, (11) K^(x)=Iei""^*re-'*sinht+vt^f^ J-00Iv 6-23] INTEGRAL REPRESENTATIONS 183 From these results weseethat (12)2cosIvTT.K^(x)= [g-'^sinht^q^]^ ^^ ^^^^^ SOthat 1 f"^ (13) K^,(x)=- ^/cos(a;sinh cosh ;4 .(^^, COSjl^TTJo andthese formulae areallvalidwhen x>and—l<R(v)<i. Inparticular (14) K,[x)= cos{xsnih dt= ),/, : JoV(t"+-I) aresult obtained byMehler* in1870. Itmaybeobserved thatif,in(7),wemake thesubstitution^ze'=T,wefindthat (15) A'.(^)=I{^yj^exp {-r- £}~ , providedthatR{z^)>0.Theintegral ontheright hasbeen studied bjnumerous mathe- maticians, among whom maybementioned Poisson, Journal deVEcolePol^technique,ix. (cahier 16),1813, p.237; Glaisher, British AssociationReport, 1872, pp.15—17;Proc. Camb. Phil. Sac. in.(1880), pp.5—12; andKapteyn,Bull, desSci.Math.(2)xvi. (1892), pp.41—44.Theintegralsinwhicli vhasthespecial values\andfwere discussed by Euler,Inst. Calc. Int. iv.(Petersburg, 1794), p.415;and,when vishalf ofanodd integer,theintegral hasbeen evaluated byLegendre,Exercices deCidcidIntegral,i.(Paris, 1811), p.366;Cauchy,Exercices desMath.(Paris, 1826), pp.54—56;andSchlomilch, Journal furMath, xxxiii. (1846), pp.268—280.Theintegralinwhich thelimits ofinte- grationarearbitrary hasbeenexamined byBinet, Gomptes Rendus, xii.(1841), pp.958— 962. 6*23.Hardysformulae forintegrals ofDuBoisReyinond's type. Theintegrals r=c _ /j,2r-agO. Isin t.sin— .i""^dt, cos t .cos— .^""^dt, Jo^ Joi inwhich ./>0,—1<B {v)<1,have beenexaminedbyHardy fasexamples ofDuBoisReymond's integrals rtxtf^'^t.f-'dt,Jo cos inwhichf{t)oscillatesrapidlyas^^0.Byconstructingadifferential equation ofthefourth order, Hardysucceeded inexpressingthem interms ofBessel functions;butasimpler wayofevaluatingthem istomake useoftheresults of§§6-21,6-22. *Math. Ann. xviii. (1881), p.182. tMessenger,xl.(1911), pp.44—51. 184 THEORY OFBESSEL FUNCTIONS [CHAP.VI Ifwereplacetbyxe\ itisclear that sin ^sin— .t"-''dt=x" sin{xe^)sin(^e"*).e"*c?^ Jo ^ J-« roo j^„v I fg2ij;cosh«Ig-2ia;cosli( _gSiasinhf _g-2ia;siiilitj gf*^^ J—00 =-i^"[7rtW'""'f/'^_^* (2.2;)-TTte^"-*H^^^j,{2x) -2e-^'"^' iT^(2a;)-2ei'"^^ 7i_,(2a;)], andhencewehave (1) sinisin^.^''-irf^=-rT-^ [/.(2.t•)-J_,(2.r) +/_,(2^)-/,(2./0]- andsimilarly (2) cos ^cos^.r-i cZ^= ,,[/_.(2^)-/.,{2x)+/_,.{Ix)-1„{2x)]. When Vhasthespecialvalue zero, these formulae become (3) sinisin^.^=|7rFo(2a-) +A'„(2a.O, .' t t (4) f" cos tcos^'."^^=-IttFo(2a;)+K,{2x). Jo it 6"24.Theisinger^sextensionofBesseVsintegral. Acurious extension ofJacobi's formulae of§2*2hasbeenobtained inthecase ofJq{x) andJi{x) byTheisinger, Monatskefte fiirMath, undPhj/s.xxiv.(1913), pp.337—341;we shallnowgiveageneralisation ofTheisinger'sformula which isvalid forfunctions oforder Vwhere -h<v<\. Ifaisanypositive number*,itisobvious from Poisson's integralthat J^(x)= ,^ ^^fj ,^[""e-«'^"'«cos(A'cos^)sin2''^c?^ '"' 'r(r+A)r(i) jo^ j^—'-•f/ ^^^- (i-e-«^8ine)cos(^cos^)sin2''^<Z^.^r(r+|)r(i)jo^ ) \ I /in-Now 2" (1-e-«^sinfl) COS(.rCOS(9)sin-" <9o?^ } Ttt1_g—aarsin^=^^TT— ^— ;rssinh ixsin^-u;cos6)sin-" ^cZ^ jsnili[xsin5) __/•!1-exp (lag;.-(.-!/.)} /._-l/A^''«?2~ j_t sinhl:|.rz(2-l/2)}^^^^'V2/;2' where thecontourpasses above theorigin. Take thecontour tobetherealaxiswithan indentation attheorigin, andwritez—+tan\^onthetwopartsofthecontom*; wethus findthatthelastexpressionisequalto -^I ^^, T—r,--suiixtan^0).e""^'cot-" d) .^^ ysui(xcot(^)-sin<p +^f-"^ ^-^--^W^^-^^^t'^)sinGrtanl<^).e-"-'cot^"c^-^/o sui(.rcot0)-^^sm0 A[-'" /I 4.JLN /I 4.J,sin (.rtani<^)c?6=4 Isin(ia.rcot(/>)cos(iax cotd)—vn). '-.r-", ,cot^" d,-^-^ ,Jo- 7/ \_ -rsin (.rcot0)^sin9 *InTheisinger's analysis,aisaueven integer. 6-24-6-31] INTEGRAL REPRESENTATIONS 185 andtherefore (1)—.wis/ -^.(•»•)=" e-«^-^"'e cos(.rcos ^)sin'^" ^c^^ +2f" sin(ia.v cot0)cos(Ur cot(h-un)"*"' (f_^50) ^ot^^ <i_^^ j(I" "sni(.rcot(Ji)'^sui^ Thetransformation failswhen I'^A, hecause theintegral round theindentation does nottend tozerowith theradius oftheindentation. Theform given byTheisingerinthe case v=ldiffers from(1)because heworks with§3"3(7)whichgives (2)^-fej^I^^-^i/(*)=f'" «""'=''"^«ii(«cose)sin2''-'.^ dcos^cW , ,f-"" ,1 i.j\ /I xj ,sin'''(i^tani(i) ,,, ., ,deb+4Ism(^a.r cot0)cos(iax cot (b-vtt)——^ fJ-icot-"-- (h^-J— , joT- X- T- /sni(a-cot(/)) ^sni^<^ providedthat^<v<f. 6*3.Theequivalence oftheintegral representations ofK^{z). Three differenttypesofintegralswhichrepresent K^{z)havenowbeen obtained in§§6*15(4),6"22(5)andG"16(1),namely =fe--^'««'>*coshi/«.c?^, Jo Theequalityofthe firstandsecond wasdirectlydemonstrated in1871by Schlalii*; butPoissonprovedtheequivalenceofthesecond andthird asearly as1813, while Malmstengavealessdirectproofoftheequivalenceofthe second andthird in1841.Weproceedtodescribe thethree transformations inquestion. 6'31. Scldiifli's transformation. We firstgiveanabstract oftheanalysis usedbySchliifli, Ann. diMat.(2)v.(1873), pp.199—201,toprovetherelation r(1).{\z )^j^_,^_^y_.^^^^r e-^coshs coshvOde ^r(„+^) j,^ 'j„ which arises from acomparisonoftwooftheintegral representationsofK^ (z),andwhich maybeestablished byanalysis resembling that of^2"323. Wehave, ofcourse, tosupposetliatR{z)>tosecure convergence, and itisconvenient atfirst totaket-~h<R {v)<1. *Auearlier proofisdue toKuinmer, JournalfilrMath. xvii. (1837), pp.i-iS— 242,but itis much more elaborate than Scblatii's invesligation. tThe result isestablished forlarger values of11(c)eitlier bythetheory ofanalyticcoutinua- tionorbytheuseofrecurrence fornuilae. 186 THEORY OFBESSEL FUNCTIONS [CHAP.VI Now defineSbytheequation wherex^l; andtheu^iit=x-{x-l) u,wehave onexpanding thelastfactor oftheintegrandinpowersofuandintegrating term-by-term. Replacing xbycosh0,weseethat J,^ ^ (coshd-ty uFii) sothat,byapartial integration, ^'^^^""'"^^/""e-^eoshecoshr^c/^^ ^"^^^","'"^^1e-^^^i^esinh ^sinh r^cZ^ r(i) jo "FC*) jo r(i-.)jo = F(I^)/,i/"'"<'"-'>"'(^' rci-.-) jijo^ ^ zt" =2"re-'^f"-iy-idt; J1 theinversion oftheorder oftheintegrations presents nogreattheoreticaldifficulty, and thetransformation isestablished. 6*32. Poisson'stransformation. Thedirectproof that 2j-. ^••'r(i) jo{w^+z-^y^h isduetoPoisson* Journal deVEcolePolytechnique,ix.(1813), pp.239—241. Theequation istruewhen|arg2|<-|7r, x>0 andR{v)>— -g,but itisconvenient toassume inthe course oftheproof thatR{v)>^and |arg2 |<jtt,andtoderive theresult forother values of2and Vbyanappealtorecurrence formulae andthetheoryofanalyticcontinuation. Ifwereplace^byanewvariable defined bytheequation v^x^e'^, weseethat itis suflficient toprovethat pcos(^,70rfu _lr(l)f%x„.r_i,/.v.. „2,-i/.v^^ *Seealso Paoli, Mem. diMat. ediFis.della Soc.Italiana delle Sci.xx.(1828), p.172. 6-32, 6-33] INTEGRAL REPRESENTATIONS 187 Now theexpression ontheleft isequalto ^TTTT^^^^^T^^^rf!<= / /s"-iexp {-s(u"^+z"^)}cos.ru .dsdu ./Jo [H'+Z-)"^:- jnJn = II[exp (-su^)cosxu .du]..f"- 1exp (- sz'^)ds,Jo ./ when \vcwritet^s(u'+z-)andchange theorder oftheintegrations*. Now /cxi){-su-)eoHxu.du =hr{i)s-hes.[){-^.vys\ andsowehave =^exp{-^.(.i/'' +,<;2,-i/.).^,,"l-^^y J which establishes theresult. [Note.Itisevident that s=h.ve-yz=hvyiz. Theonlyreason formodifying 1 ^^, e-^^c"^'''-''' J< bytakingvasaparametric variable istoobtain anintegral which isostensiblyofthe same form astheintegral actually investigated byPoisson;with hisnotation theintegralis /exp (-j^>-a2.f-») dj;.] 6'33. Malmsteri'stransformation. Themethod employed byMalmstent inproving that,when R{z)>0andB{v)>-^, then T{v+i)rcos{xu)du_T_{\){ixY r {ixyTiDJo {^(^+z^y*i~ rjv+i) JIe-^--t{fl-iy-idt, isnotsodirect astheanalysisof§§6'31, 6'32,inasmuch asitinvolves anappealtothe theoryoflinear diiierential equations.Itisfirstshewn byMalmsten that thethree expressions pcosg^^^ j\-.t(i2_^y-,^t, re-^t{.c^-i^-hdt, J(,W+2 ;-JX J-X quafunctions ofx;areannihilated\bytheoperator d" ,,=, T\^->^""d^-^'-^dr'^'-^ andthatas,x'^-+x ,thethird is[f')while the firstandsecond arcbounded, provided thatR(v)>0.Itfollows thatthesecond andthird expressionsform afundamental system ofsolvitions oftheequation *Cf.Brorawich, Tlieunj ofLijinite Series, §177. fK.Svenska T.Akad. Handl. Lxn.(1841), pp.Oo— 74. 1Thereader should havenodillicultyinsupplyingaproofofthis. 188 THEORY OFBESSEL FUNCTIONS [CHAP.VI andthe first isconsequentlyalinear combination ofthesecond andthird. Inview ofthe unboundedness ofthethird asx-^+oo,itisobvious that the firstmust heaconstant multipleofthesecond sothat J{U+2')"-^JX whereCisindependentof.v.Todetermine C,makex-^0 andthen /(u^+z^) . r(.)r(^)_Cri2v)sotnat 2z^^T{v +i)z'"' andtherequiredtransformation follows, whenR{p)>0,ifweusetheduplicationformula fortheGamma function. „,,, ,, „ ,. r, ,["^cosxu .du .Animmediate consequenceofMalmsten's transformation isthat / ^„2m^^ expressibleinfinite terms;foritisequalto roc 2^-i{(n-l)!}^j,'^'^^ ^g-w «-i(2a's)'»(2/i-»i-1): Thismethod ofevaluatingtheintegralissimpler than amethodgiven byCatalan, Journal deMath. v.(1840), pp.110—114;andhisinvestigationisnotrigorousinallits stages. Thetransformation isdiscussed bySerret, Journal deMath. viii.(1843), pp.20, 21; IX.(1844), pp.193—216; seealsoCayley,Journal de,Math. xii.(1847), p.23G {Collected Papers,i.(1889), p.313.] 6*4. Airi/s integral. Theintegral rcos{t^±xt)dt Jo whichappearedintheresearches ofAiiy* "OntheIntensityofLightinthe neighbourhoodofaCaustic" isamember ofaclass ofintegralswhich are expressibleinterms ofBessel functions. Theintegralwastabulated byAiry byquadratures, buttheprocesswasexcessivelylaborious. Later,DeMorgan•(- obtained aseries inascending powersot'xbyaprocesswhich needsjustification either byStokes' transformation (whichwillbeexplained immediately)orby theuseofHardy's theoryofgeneralised integrals^. *Trans. Camb. Phil. Soc. vi.(1838), pp.379—402. Airyused theform cosIn (iv'^-mro) die, but thif? iseasily reduced totheintegral given above. fTheresult wascommunicated toAirj'onMarch 11,1818;seeTrans. Camh. Phil. Soc. viii. (1849), pp.595—599. +Quarterly Journal, xxxv.(1904), pp.22—66;2'rans. Camb. Phil. Soc. xxi.(1912), pp.1—48. 6-4] INTEGRAL REPRESENTATIONS 189 Itwasnoticed byStokes* thattheintegralisannihilatedbytheoperator andStokes alsoobtained theasymptotic expansionsoftheintegralforlarge values ofx,bothpositiveandnegative. Thereader willobserve that Stokes' differentialequation forAii-y's integral isidentical withoneofthetransformed forms ofBessel's equation (^4-3). This factwasnoticed by Stokes {loc. cit.p.187), butnospecial usewasmade ofituntilNicholson, Phil.Man.(6) XVIII. (1909), pp.6—17, expressed Airy's integral directlyinterms ofBessel functions of orders±7'..These Bessel functions havelately assumed considerable importance f;see Weyl, Math. Ann. Lxviii.(1910), p.267,andtheapproximate formulae described in§8'43. Subsequently Hardy, Quarterly Journal, XLi.(1910), pp.226—240, pointed outthecon- nexion between Airy's integral andtheintegrals discussed in5^.^6"21, 6-22,andhethen examined various generalisationsofAiry's integral (§§10"2—10'22). ToevaluateAiry's integral;]:,weobserve that itmaybewritten intheform 1r* ^Iexp{if+ixt)dt. Now consider thisintegrandtakenalongtwoarcsofacircle ofradiuspwith centre attheorigin,thearcsterminatingatp,pe"'^''andpe^'^', pe^^respectively. Theintegrals alongthese arcstend tozeroasp-*oc ,byJordan's lemma, and hence, byCauchy's theorem, weobtain Stokes' transformation /CO Yrcoexp^TTt cos{f±xt)dt= -^ exp {iV"±ixt)dt Jo2.'COexpfTTJ =lr [e^""'exp(-T^±e^'^'^t)+e-^'^'exp(-T^±e- i'^'xr)] dr;^.'0 thecontour ofthesecondintegralconsists oftvroraysemergingfrom the originandthethirdintegralisobtained bywriting re^'^^ re^'^* for tonthese rays. Now, since theresultingseries areconvergent,itmaybeshewnthat§ exp(-73±e-s"''xt)dr=S^-^t'"exp(- 7-*)dr, Jo 111=0'^>>'- .0 *Trans. Camb. Phil. Soc. ix.(185ii), pp-166—187. [Math, andPli,/s. Paper.->,11.(1883), pp.329—349.] Seealso Stokes' letter ofMay 12,1848, toAiry, SirG.G.Stoke/^, Memoir and Scientific Correspondence,11.(Cambridge, 1907), pp.159—160. tThefunctions occur inaproblem concerning thestabilityofmotion ofaviscous fluid; an account oftheproblem (with abibliography)isgiven byKayleigh, Phil.Mag. (6)xxvin.(1914), pp.G09— 619; xxx.(1915), pp.329—338. [Scientific Papers,vi.(1920), pp.266—275; 341—349.] XTheintegralisconvergent.Cf.Plardy,loc. cit.p.228,ordelaValltJe Poussin, Ann. dela Soc. Sci.deBruxelles,xvi.(1892), pp.150—180. §Bromwich, Theory ofInfinite Scries, §176. 190 andsoTHEORY OFBESSEL FUNCTIONS [chap. VI ^,^(+a;)*«cos (Im+f)TTr=^„. / ,x 7cos(f +xt)dt= S^^=—^^^^—T"'exp(-Tnc?T ^- ' ,,^^0 ml Jo -^S(±^)™sinf(m+1)TT .r{^vt+^j)/m! «Jm= =i7r(±i^) ,^=0^?^! r(?n+ This istheresult obtained byDeMorgan. When theseries ontheright areexpressedinterms ofBessel functions, weobtain Nicholson's formulae, in Avhich a;istobetaken tobepositive: (1) (2)cos{t^—Xt)dt=^TT'\/(^x)J->[/2x\/a- \Jx„/2ijc^/x\373/ '2x\/x\+J,i2x\Jx V'3V3, 2x\Jx Jf2x\/x\ /2x.\/x\ ^-^l-3V3J-^n"3V3J 6*5. Barnes'integral representations ofBesselfunctions. Byusing integralsofatypeintroduced byPincherle* andMellinf, Barnes^ hasobtainedrepresentationsofBessel functions which renderpossibleaneasy proofofRummer's formula of§4"42. Letusconsider theresidue of -r(2m-s).{izf ats=2m+r,where ?'=0,1,2,....Thisresidue is(— )'"(iz)'^"''^^'lrl, sothesum oftheresidues is(—^'n^amg-iz^ Hence, byCauchy's theorem, J,(z)e"''=(hzy5/<«+' r(2m-s).{izy ds, 00 rn=t2'Tri^=0 iX2^'".mWiv +m+l) ifthecontour encloses thepoints 0,1,2,— Itmaybeverified, byusing Stirling'sformula thattheintegralsareconvergent. NowsupposethatR{v)>— |,andchoose thecontour sothat, onit, R(v+s)>— ^.When this lastcondition issatisfied theseries r(2m-s) ~o2^"'.mir(v +m+l) isconvergentandequalto *Rend, delR.Istituto Lombardo, (2)six.(1886), pp.559—562; Atti della R.Accad. dci Lincei, ser. 4,Rendicoiiti, iv.(1888), pp.694—700, 792—799. tMellin hasgivenasummary ofhisresearched, Math. Ann. lxviii.(1910), pp.305—337. JGamb. Phil. Trans, xx.(1908), pp.270—279.Forabibliographyofresearches oninte;,Mals ofthistype,seeBarnes, Proc.London Math. Soc.(2)v.(1907), pp.59—65. 6-5]INTEGRAL REPRESENTATIONS 191 bythewell-known formula due toGauss. Iftherefore wechange theorder ofsummation andintegration*wehave Theonly polesoftheintegrandinside thecontour areat0,1,2,....When wecalculate thesum oftheresidues atthesepoles,wefindthat sothat (1) J^{z)e--= f^^r+1)1^1 ('^+I;2r+1;-2iz), which isKummer's relation. Inlikemanner, wefindthat (2) ,h{z)e^'= Y%'ll)'^'^"+i;2z.+1;2iz). These formulae, provedwhen R{v)>—\,arerelationsconnectingfunctions ofVwhich areanalyticforallvalues ofv,and so,bythetheoryofanalytic continuation, theyareuniversallytrue. ItisalsopossibletorepresentBessel functions byintegralsinwhich no exponentialfactor isinvolved. Todothis,weconsider thefunction T{-v-s)T{-s){lizy^-^^, quafunction of5.Ithaspolesatthepoints 5=0,1,2,...;-v,-v +l,-v+2,.... Theresidue ats=mis TTJ^(-)"' (2^)'"^'"' sinvir'm\T{v-{-m+1)' while theresidue at5=—i^+inis -iri-"(-)™(l2)-''+='" sinvTT'mlT(v+m+1)' sothat (3)7re-i<''+i'"' i/,'-' (z)^-^.ir{-v-s)r(-s)(lizY^'' ds, and, inlikemanner, (4)Treic+it'^'' ^,w {z)=-~[r(-v- s)r(-s)(-yzy+'' ds, where thecontours start fromandreturn to+ooafterencirclingthepolesof theintegrandcounter-clockwise. When |argiz \<hirin(3)or jarg(—iz) \<|7r in(4)thecontours maybeopened out,soastostart from ooiandendat —Xi.Ifwereverse thedirections ofthecontours wefindthat (5)7re-i"'+^'-' H/'^ (z)=^.f'^"' r(-V-s)V(-s)(^izy^'' ds, ItTI J-c-jai *Cf.Bromwich, 'Theory ofInfinite Series, §176. 192 THEORY OFBESSEL FUNCTIONS[CHAP. VI ,providedthat \argizlK^ir; and providedthat \arg(—iz) |<^tt;and, ineachintegral,cisanypositive number exceeding Il{v)andthepathofintegrationisparalleltotheimaginaryaxis. There isanintegral resemblingthese whichrepresents thefunction ofthe firstkind oforder v,but itconverges onlywhenR(v)>0 andtheargument ofthefunction ispositive.Theintegralinquestionis and itisobtained inthesamewayasthepreceding integrals;thereader willnotice that,when js |islargeonthecontour, theintegrandis{\sl"""^). 6'51. Barnes'representations offunctions ofthethird kind. Byusingtheduplicationformula fortheGamma function wemaywrite theresultsjustobtained intheform \^} ^vK^)e 2i.^/7r}^ r(s+l)r(2i. +s+l)sins7r" Consider nowtheintegral {2zy f"' T{-s)r(-2v-s)r(v +s+i).{2izy ds,2iVt. _c inwhich theintegi-anddiffers from theintegrandin(1)byafactor which is periodicins.Itistobesupposed temporarilythat2visnotaninteger and thatthepathofintegrationissodrawn thatthesequencesofpoles 0,1,2,...; —2j^,1—2v,2— 2i^,...lieontherightofthecontour while thesequenceof poles—V—^,—v~^,—''—f,•••liesonthe leftofthecontour. Inthe first place,weshallshew that, if jargiz |<|7r,theintegraltaken round asemi- circle ofradiuspontherightoftheimaginaryaxistends tozero asp^ y:); for,ifs=pe'^,wehave ,r(-.)r(-2..->)r(.+.H).(2,-.).^ p^^„/r(.+.+i).(2,>r 1(s)r(2i/+5+1)smSTTsm(2i^+s)TT and,byStirling's formula, r{v+s+i).(2izy ''^r(s+i)r{2i^ +s+i)~pe'^log(2iz)-(v+pe'^) (logp+id)+pe^^-hlog(27r) ; andtherealpartofthistends to-xwhen— -|-tt<^<^tt,because thedominant term is-pcos6logp.When 6isnearly equalto±^tt, |sins-k \iscomparable with\exp \piTjsin6 j}andthedominant term intherealpartofthelogarithm ofstimes theintegrandis pcos6log122 [—psin^ .arg2iz—pcosd\ogp +pOamO-{-pcosd—2p\^\nd\, and thistends to—xas/a—xif |argiz i<|7r. 6-51] INTEGRAL REPRESENTATIONS 193 Hence stimes theintegrandtends tozero allalongthesemicircle, andso theintegralround thesemicircle tends tozero ifthesemicircle isdrawn soas topassbetween (andnotthrough)thepolesoftheintegrand. Itfollows fromCauchy'stheorem that,when |argiz\<|7rand2visnotan integer,then r'r(-s)r(-2v-s)r (v+s+i).{2izydsJ-<x>i maybecalculatedbyevaluatingtheresidues atthepoles ontherightofthe contour. Theresidues of r(- .9)r(-2/.-s)r{v+s +1).(2/^)-^ at5=mand s=—2^+7)1arerespectively TTr(v+m+1).{2iz)'"' _TTr(-;/+m+1).(2i> )-2''+"' sin2viT rn !F(2i/+m+1)'sin^vir nilr(—2i^+m+1) andhence -Prrr{-s)T{-2v-s)T(v +s+l).{2tzYds = snr2;^r(2TTT)-^^^^+^-^^+i' -^^> g—H2^)-vr^ r(i-z.) TT^e'^ sin2t'7r Itfollows that,when |argiz |<ftt, (2) i,,..(,,=^-'"-'"<=-(^-)-(2-)" 5 77'^ -j-.i Xr(-s)r(-2/v- .9)r(i.+s+1).i2izy ds, XI andsimilarly, when jarg(—iz) \<ftt, giiz-.'Tr) cos(i^tt).(22:)" (3) F.'^' (2)= Xr(- .s)r(-2i'-s)r(r+.9+1).(-2{zyds. -XI The restriction that visnottobeaninteger mayberemoved intheusual mannerbyalimiting process,buttherestriction that 2i>must notbeanodd integercannot beremoved, since thenpoleswhich mustbeontherightofthe contour would have tocoincide withpoleswhich must beonthe left. AV.B.F. 13 CHAPTER VII ASYMPTOTIC EXPANSIONS OFBESSEL FUNCTIONS 7*1.Approximate formulae forJv{z)- InChapterIIIvariousrepresentationsofBessel functions were obtained intheform ofseries ofascending powersoftheargument z,multipliedinsome casesbylogz.These series arewelladaptedfornumerical computationwhen z^isnotlargecomparedwith 4(i^+1),4(i^+2),4(^+3),...,since theseries converge fairly rapidlyforsuch values ofz.But,when |-^ Iislarge,theseries converge slowly,andaninspectionoftheir initial terms affords noclue tothe approximatevalues ofJ^{z)and Y^,{z).There isoneexceptiontothisstate- ment;when i^+1isanintegerwhich isnotlarge,theexpressionsforJj^^ {z) infinite terms(§3'4)enable thefunctions tobecalculated withoutdifiiculty. Theobjectofthischapteristhedetermination offormulae which render possiblethecalculation ofthevalues ofafundamentalsystemofsolutions of Bessel'sequationwhen zislarge. There arereallytwoaspectsoftheproblemtobeconsidered;theinvesti- gationwhen Vislargeisverydifferent from theinvestigation when visnot large.The formerinvestigationis,inevery respect,ofamore recondite character than thelatter, and itispostponeduntilChapterviii. Itmust, however, bementioned thatthe firststeptowards thesolution of themore reconditeproblem wasmadebyCarlini* someyearsbefore Poisson'sf investigationofthebehaviour ofJq{x),forlarge positivevalues ofx,was published. TheformalexpansionobtainedbyPoisson was f1^ 1-3^5^ -Hsin(^-i,r).|^^-3J^^+.. when Xislargeandpositive. But, since theseries ontherightarenotcon- vergent,andsince Poissongave noinvestigationoftheremainders inthe series, hisanalysis (apartfrom hismethod ofobtainingthedominant term)is toberegardedassuggestive andingeniousrather thanconvincing. *Ricerche siilla convergenza della serie cheserva aliasoluzione delprohlema diKeplero (Milan, 1817). Anaccount ofthese investigations hasalready been givenin§1*4. tJournal deVEcolePohjteclinique,xii.(cahier 19),(1823), pp.350—352;see§1-6.Anin- vestigationofJv{x)similar toPoisson's investigation ofJq(^)hasbeen constructed byGrayand Mathews, ATreatise onBessel Functions(London, 1895), pp.34—38. X7-1] ASYMPTOTIC EXPANSIONS 195 Itwillbeseen inthecourse oftinschapter that Poisson's series areasymptotic;this hasbeenproved byLipschitz, Hani^el, Schlafli, Weber, Stieltjes andBarnes. Itmust bementioned thatPoissonmerely indicated thelawofformation ofsuccessive terms oftheseries withoutgiving anexplicit expressionforthegeneral term;suchan expression wasactuallyobtained byW.R.Hamilton*(cf.§P6). Theanalog-ousformalexpansionforJx{x)isdue toHansenf;andafew years later, Jacobi|obtained themoregeneralformula which isnowusually written intheform /2\*r J-ni^^'^icos(.X'—^»TT—^TT) XTTX/ I(4,^2_ 1-2)(4,„2_32^ (4n^- l'^)(4n-^-3-)(4//^-5-){^n'- 7-)] [ 2ViSoof"^ V^{^xY~ •• J These expansionsfor J,,(.r)and-Ji(*')were usedbyHansen forpurposesofnumerical computation, andacomparisonoftheresults soobtained forisolated values of.vwith the results obtained from theascendingseries ledHansen toinfer thattheexpansions, although notconvergent,couldsafelybeused forpurposesofcomputation §. Twoyearsbefore thepublicationofJacobi'sexpansion,Plana ||had dis- covered amethod oftransformingParseval'sintegralwhichplacedtheexpansion ofJo(^) onamuch moresatisfactorybasislT. Hiswork wasfollowedbythe researches ofLipschitz**,whogavethe firstrigorous investigationofthe asymptotic expansionofJo{z)with theaidofthetheoryofcontourintegra- tion; Lipschitzalsobrieflyindicated how hisresults could beappliedtoJn[z)- Thegeneralformulae forJ^{z) and Y^{z), where vhasanyassigned (com- plex)value andzislargeandcomplex,were obtained inthegreat memoirby Hankelff,written in1868. *Some information concerning W.R.Hamilton's researches willbeI'ouud inSirGeorge Gabriel Swkes, Memoir andScientific Correspondence,i.(Cambridge, 1907), pp.130—135. tErmittelung derahsoluten Storungen [SchriftenderSternwarte Seehurg], (Gotha, 1843), pp.119—123. +Asir.Nach. xxviii.(1849),col. 94. [Ges. Math. Werke, vii.(1891), p.174.] Jacobi's result isobtained bymaking thesubstitutions X-^/2.cos(,r-iH7r-i7r)=(-l)'"("+l)cos.r+(-l)^"('^-l>sinx, ^/2.sin{.V-i»7r-iTT)= (-l)i"("+l)sin .r- (-1)^"(«-^^cosx, intlieform quoted. §SeeanotebyNiemoller, Zeitschrift fiirMath, andPhys.xxv.(1880), pp.44—48. IIMem. delhiR.Accad. delle Sci.diTorino, (2)x.(1849), pp.275—292. HAnalysisofPlana's typewasused toobtain theasymptotic expansionsofJu[z)andIV{z)by MoMahon, AnnalsofMath. viii. (1894), pp.57— (51. •"*Journal fiirMath. lvi.(1859), pp.189—196. ttMath. Ann. i.(1869), pp.467-501. 13-2 196 THEORY OFBESSEL FUNCTIONS [CHAP.VII Thegeneralcharacter oftheformula forF„(z)hadbeen indicated byLommel, Studien iiber dieBesseVschen Functionen (Leipzig, 1868), justbefore thepublicationofHaukel's memoir; andtheresearches ofWeber, 3Iath. Ann. vi.(1873), pp.146—149must alsobe mentioned. Theasymptotic expansionofK^{z) wasinvestigated (andprovedtobe asymptotic)atanearlydatebyKummer*;this result wasreproduced,with theaddition ofthecorrespondingformula for/^{z),byKirchhoff f;andalittle- known paper byMalmsten:]:alsocontains aninvestigationoftheasymptotic expansionofK^{z). Aclose studyoftheremainders intheasymptotic expansionsofJ^{x\Fq{x\Iq{x) andKq{x)hasbeenmade byStieltjes, Ann. Sci.deI'Ecole norm.sup. (3)in.(1886), pp.233—252,andpartsofhisanalysis have beenextended byCallandreau,Didl. desSci. Math.(2)XIV.(1890), pp.110—114, toinclude functions ofany integral order; while results concerningtheremainders when thevariables arecomplex havebeenobtained by Weber,Alath. Ann. xxxvii.(1890), pp.404—416. Theexpansionshave alsobeen investigated byAdaraoft§, PetersburgAnn. Inst,polyt. 1906, pp.239—265,andbyValewink||inaHaarlem dissertation,1905. Investigations concerning asymptotic expaosionsoft/^[z)andF„{z),when l^iislargewhile visfixed, seem tobemostsimplycarried outwith theaid ofintegralsofPoisson'stype.ButSchlafli*[ hasshewn thatalargenumber of results areobtainable byapeculiar treatment ofintegralsofBessel'stype, while, morerecently, Barnes** hasdiscussed theasymptotic expansions by means ofthePincherle-Mellinintegrals, involving gamma-functions,which wereexamined in||6"5,6"51. 7'2.Asymptotic expansions ofHJ^^ (z)andHJ-^ (z)afterHankel. We shallnowobtain theasymptotic expansionsofthefunctions ofthe third kind, valid forlargevalues of\z\;theanalysis, apartfromsomeslight modifications, willfollow thatgiven byHankelff. Take theformula§6'12(3),namely validwhen-|7r</3<|7r and-^tt-f-/3<arg^^<3,^+^,providedthat R{v+l)>0. Theexpansionofthefactor(1+liu/z)"'^indescending powersofzis ^'^2z-^2:4:1^+•••' *Journal furMath. xvii.(1837), pp.228—242. fIbid, xlviii.(1854), pp.348—376. tK.Svenska V.Akad. Handl. lxii.(1841), pp.65—74. §SeetheJahrbuch iiber dieFortschritte derMath. 1907, p.492. IIIbid. 1905, p.328. HAnn. diMat. (2)vi.(1875), pp.1—20. **Tra)is. Camb. Phil. Soc. xx.(1908), pp.270—279. ttMath. Ann. i.(1869), pp.491—495. 7-2] ASYMPTOTIC EXPANSIONS 197 butsince thisexpansionisnotconvergentallalongthepathofintegration, weshallreplaceitbyafinitenumber oftermsplusaremainder. For allpositive integralvalues ofp,wehave* 2zJ „,=o "^' V^i2/ (p~\)l\2izJ Jo V 2tz/ Itisconvenient totakepsolargethatR(v—p—l)^0; andwethen choose anypositive angleSwhich satisfies theinequalities j/3|^|7r—S, Iarg^-—(Itt+/3)'<7r—S. The effect ofthischoice isthat,when 8isgiven,zisrestricted sothat —TTf2S^argz^2-77—2S. When thechoice hasbeenmade, then ut1- 2iz>sin S,""'^^-hut\<TT, forthevalues oftand iiunder consideration, andso 1- 2{z/]"I^e-!'<''):(sinS)«<''-^-i'=^p, say,where Apisindependentofz. Onsubstitutingitsexpansionfor(1+^iu/zy-^andintegrating term-by- term,wefindthat p£i(|-i/),».r(i^ +m+|) _^^^(j)' where (I- ^).i?(1) I<^^ I^'^-(p-iy:r(v+i)(2iz)p\]o ^Bp.\zrp,Z,mlT{v +l).{2izy Xexp i(3 (1-ty-'dt.-U I^^v+p-;^^f^ where 5^isafunction ofv,pand8which isindependentofz. Hence, whenR(v-p-l)<0andR(v+l)>0,wehave when zissuch that-tt+2S^arg2^^27r-28,8being anypositiveacute angle";andthesymbolistheBachmann-Landau symbolwhich denotes a function oftheorder ofmagnitude fofz~J^as |-^ |^^oo . Theformula(1)isalsovalidwhenR(p-p-l)>0;thismaybeseenby *Cf.BlodernAnalysis, §5-41. Theuseofthisform ofthebinomial expansionseems tobe due toGrafandGubler, Einlcitung indieTheorie derBcsseVschen Fiinktioncn,i.(Bern, 1S9S). pp.86—87. Cf.Whittaker, Modern Analysis (Cambridge, 1902), §161;Gibson, Proc. Edinburgh Math. Soc. xxxviii.(1920), pp.6—9;andMacRobert,(7;/(/. pp.10—19. tCf.Modern Analysis, §2-1. 198 THEORY OFBESSEL FUNCTIONS [chap. VII supposingthatR(v—p—^)>0 andthentakinganinteger qsolargethat B,(^i,—q—^)<.0;iftheexpressionwhich iscontained in[]in(1)isthen rewritten withqinplaceofpthroughout,itmaybeexpressedaspterms followed byq—p+1terms each ofwhich is(^"^0oro{z~^') ;andthesum ofthese q—p+1terms istherefore{z~v). Inasimilar manner(bychangingthesignofithroughouttheprevious work) wecandeduce from§6'12(4)that (2) ir,W(^)=('A'j%-^.-4--i:r)_,r=o m!(-2.-.)-+^^^ >J' providedthatR{v+\)>0 andthatthedomain of-values ofzisnowgiven bytheinequalities-277+28^argz^ir -1h. If,following Hankel, wewrite (i-v\nil+v)y, {v,m)=(-)T{v+m+1) ml m !r(i/-m+I) _{4i^^- 1^}{4i/-- 3-^}.{4i^^-(2m- 1)^} theseexpansionsbecome ,2\^ (3)TV2f/2-'^.m\ Sl=W^'^-i (4) ir;^) (^)= ^Aye-M3-A.^-i^,''"^(v,m) Forbrevity wewrite theseequationsthus :+0(z-p) (5) (6)^^WzJ ,^.0 (2t2)"' Since(v,m)isaneven function oft,itfollows from theformulae of §3'61(7),which connect functions ofthethird kind oforder i/with thecorre- spondingfimctions oforder—v,that therestriction that therealpartofv exceeds —|isunnecessary. Sotheformulae(1)—(6)arevalid forallvalues ofV,when zisconfined tooneorother oftwosectors ofangle justlessthan Stt. Inthenotation ofgeneralised hypergeometric functions, theexpansionsare (7) ^^" (^)~ (;^)'^''^--'"^"''^' ..F,U +.,l-v;i-J, (8) m^ (z)~ (A)*e-^^-i"'-^-^ •.^0(i+^,i-^;- 2^-^' ofwhich(7)isvalidwhen—tt<arg2<27r,and(8)when—27r<argz<tt. 199 7-21] ASYMPTOTIC EXPANSIONS 7'21.Asymptotic expansions ofJ^{z), J-^(z) andY^{z). Ifwecombine theformulae of§7-2,wededuce from theformulae of§3-61 (which expressBessel functions ofthe firstandsecond kinds interms of functions ofthethird kind) that (1) '/.iz)irzcos(z vir•»(-r {v,2m) ^'-^to (2#- (-)»^(^', 2//i+l)'-sm{z-h"rr-\Tr). S w=(^2r)-irt+l (2) -^A^)'^—sm(2'-iz/7r-i7r). S^^-^—^-^—' \lTZj 7H= {Izf +COS(^—Ii^TT— f7r).N\^ ^ ''' m=() (3) J-.{z) (4) F_.(^)^2 2cos{^zf /,1 1XV(-)'"•(^.2w)»i+i m-=0 • / .1 1 XV(-)"'• (^.2m+1)' %\Vl{z-^\vtt -\it).1.^-^-^^ sin(^+|z^7r- ;f7r).2m=\-"^} {-f.{y, 2m) +C0s(2'+|z/7r- :^7r).Sm=(-)»^(/',2??i+i)" (2^)-"'+i and(inthecase offunctions oiintegral order nonly), (-)"*.{n,2m)(5) Y,^z)2lT s\n{z-\mT-\'Tr). % 1)1= +co^{z—\mr— {'TT).S(2^)-"^ (-)'». (n,2m+1)' m=^^^-^ These formulae areallvalid forlargevalues of |^ |providedthat |argz\<Tr; andtheerrorduetostoppingatanyterm isobviouslyoftheorder ofmagni- tude ofthat terramultiplied byIjz. Actually, however, this factor Ijzvcaxy bereplaced byIjz- ;thismaybeseenbytakingtheexpansionsofHJ^^^ (z)and H^'--' (z)totwoterms further than the lasttermrequiredintheparticular combination withwhich wehave todeal. Ashasbeen seen in§7"2,theintegralswhich aredealt withwhen R{v)>-Irepresent H^,^^^ (z)and H^^-^ (z),but,whenR(v)>-1,theintegrals fromwhich theasymptotic expansionsarederived arethose iv/iich represent H^^'>_^{z) andH^"^^^(z).This difference inthemode oftreatment ofJ^,{z) andYt,(z)forsuch values of i'seems tohave ledsome writei'S tothink* that formula(1)isnotvalid unless R{v)>—^. *Cf.Sheppard, Quarterh/ Journal, xxiii. (188:)), p.223;Searle, Qnarterhj Journal, xxxix. (1908), p.()0.The error appearstohave originated fromTodhunter, AnElementaryTreatise on Laplace's Functions, Lame's Functions andBesseVs Functions (London, 1875), pp.312—313. 200 THEORY OFBESSEL FUNCTIONS [CHAP.VII Theasymptotic expansionofJo{2)wasobtained byLipschitz* byinte- gratinge*'^'(1-t^)~^round arectangle (indentedat±1)with corners at+1 and+1+Xi.Cauchy'stheoremgivesatonce f ei""^ Ie-"'+"»^ zr^(2-ui)"-dn=0,-1 .'0 andtheanalysisthenproceedsonthelinesalready given;butinorder to obtain asymptotic expansionsofapairofsolutions ofBessel'sequationit seems necessarytouseamethod which involves atsomestagetheloop integralsdiscussed inChaptervi. Itmaybeconvenient tonoteexplicitlytheinitial terms intheexpansions involved inequations (1)— (4);theyareasfollows : - (-y^.{v,2m)^(4.V"--1-)(iv--3^) .r^o i^^T"'2!(8^)^ "^ 4!(8^)^ ^(-yn.{v,2m +l.)_4i>--l-(4t/^-1-)(4i/--30(4i/-- 5'^) »«=o (2zym+i 1182 3!(82^)3 Thereader should notice that J,^{z)+J,^^Hz)o^2/(m), aformula givenbyLommel, Stttdien, p.67. Note. Themethod bywhich Lommel endeavoured toobtain theasymptotic expansion ofF,i(2)inhisStudien, pp.93—97,wasbydifferentiating theexpansionsofJ^^ (z)with respecttov;butofcourse itisnowknown that theterm-by-termdifferentiation ofan asymptotic expansionwithrespecttoaparameterraises various theoretical difficulties. Itshould benoticed thatLommel's laterwork, Math. Ann. iv.(1871), p.103,isfreefrom thealgebraicalerrors which occur inhisearlier work. These errors havebeenenumerated byJulius, ArchivesA'eerlandaises, xxviir. (1895), pp.221—225.Theasymptotic expansions of'-Ai(^)and T„{z) have alsobeen studied byMcMahon, Ann. ofMath. viii. (1894), pp.57—61,andKapteyn, Monatshefte fiirMath, undPhys.xiv.(1903), pp.275—282. Anovelapplicationoftheseasymptotic expansionshasbeen discovered inrecent3'ears:theyareofsomeimportanceintheanalytic theoryofthe divisors ofnumbers. Insuchinvestigationsthedominant terms oftheex- pansionsareadequateforthepurposeinview. This factcombined with the consideration thatthetheoryofBessel functions formsonlyatrivialpartof theinvestigationsinquestionhasmade itseem desirable merelytomention thework ofVoronoif andWigert Jandthemore recentpapers byHardy§. *JournalfiirMath. lvi. (1859), pp.189—196. \Ann. Sci. deVEcolc norm. sup. (3)xxi. (1904), pp.207—268, 459-534; Verh. dcsInt. Math. KongressesinHeidelberg, 1904, pp.241—245. %ActaMathematica, xxxvii.(1914), pp.113—140. %Quarterhj Journal, xlvi.(1915), pp.263—283; Proc. London Math. Soc.(2)xv.(1916), pp.1—25. 7-22] ASYMPTOTIC EXPANSIONS 201 7"22. Stokes' phenomenon. Theformula§7-21(1)forJ^{z) wasestablished forvalues ofzsuch that !arg2^|<7r.Ifwetookaxgztoliebetween and 27r(sothatarg^'g"''''lies between —ttand tt)weshouldconsequentlyhave /.,(z)=e-"^'-A(2'e~"') 2 COe" TTze' sothat,when <argz<'lir, /9cos(ire"'—Ai/TT—itt)z--^^—^- •/-HI 1X^(-)"'• (J^,2m+1)' TTZcos(^+li/7r+i7r)S(2lp Wi= —sin(^+|i^7r+Itt)S(-)'".(!/,27»+iy III=11 and thisexpansionissuperficially quitedifferent from theexpansionof §7"21(1).Weshallnowmake aclose examination ofthischange. Theexpansionsof§7'21arederived from theformula andthroughoutthesector inwhich—tt<argz<27r,thefunction HJ^^ (z)has theasymptotic expansion \m Thecorresponding expansionfor//^''-^ (z),namely (1) ^;^) (^)oo I— Ie TTZ-Hz-4— J-)I('\'"'\ is,however, valid forthesector -27r<argz<tt.Toobtain anexpansion valid forthesector <arg2^<'Iirweusetheformula of§3"62(6),namely E,^'^ (z)=2cosVTT .H,^-^ {ze-^') +e"^'F,'"{ze-^% and thisgives -('B-ii/jT-iTr) V{v,m) ..o(2izy" +2cos...(iY.^-i-^- ^(-r.(^,-)(2) H,('Hz)c^(~Ye \lT4yJ «i= \-'"^ .' Theexpansions (1)and(2)areboth validwhen0<arg^<7r;nowthe difference between them hastheasymptotic expansion 2cos^7r.(— )e''^+i-+i-> S^U'-W> and,onaccount ofthefactor e'^whichmultipliestheseries, thisexi)ressionis oilower' orderofmagnitude {when \z\\b large)than theerrorduo tostopping 202 THEORY OFBESSEL FUNCTIONS [CHAP.VII atany definiteterm oftheexpansion (1);forthiserror is(e~^^z~P~^) when westopatthepthterm. Hence thediscrepancybetween(1)and(2),which occurs when 0<arg2^<7r,isonlyapparent,since theseries in(1)hastobe used inconjunctionwith itsremainder. Generally wehave where theconstantsCi,c^have values whichdependonthedomain ofvalues assignedtoargz.And, ifargziscontinuallyincreased(ordecreased)while I^ Iisunaltered, thevalues ofCjand Cohave tobechanged abruptlyatvarious stages,thechangeineither constantbeing made when thefunction which multipliesitisnegligible comparedwith thefunctionmultiplyingtheother constant. That istosay,changesinCjoccurwhen I{z)ispositive,while changesinCgoccurwhenI{z)isnegative. Itisnotdifficult toprovethatthevalues tobeassignedtotheconstants CjandCaareasfollows : c,=le2p{>'+i)« Co=1 e2^(''+^)-^, [(2^-1)TT<argz<{2p+1)tt], Ci=|e-^+i) ^"^i^^', c,=le2?>(''+i)-^ [2p7r<arg^<(2p+2)tt], where pisanyinteger, positiveornegative. Thisphenomenonofthediscontinuity oftheconstants wasdiscovered by Stokes andwasdiscussed byhiminaseries ofpapers.Itisaphenomenon which isnotconfined toBessel functions, and itischaracteristic ofintegral functions whichpossess asymptotic expansionsofasimple type*. Thefactthattheconstants involved intheasymptotic expansionoftheanalyticfunction Jy(z)arediscontinuous wasdiscovered byStokes in(March?) 1857,andthediscovery was apparently oneofthose which aremade atthree o'clock inthemorning. SeeSirGeorge Gabriel Stokes, Memoir andScientific Correspondence,i.(Cambridge, 1907), p.62.The papersinwhich Stokesxniblishedhisdiscoveryarethefollowingt:Trans. Camb. Phil. Soc. X.(1864), pp.106—128; Xl.(1871), pp.412—425; ActaMath. xxvi.(1902), pp.:39.3-» 397. [Math, andPhys. Papers,IV.(1904), pp.77—109;283—298;V.(1905), pp.283-287.] Thethird ofthese seems tohavebeen thelastpaper written byStokes. 7'23.Asymptotic expansions ofIy,{z)andK^(z). Theformula§7'2(5)combined withequation §3"7(8),which connects K^(z)and H^^^'> (iz),shews atonce that (1)^^'W~y ''„!„(2ir CO (£)'1H h — '+ ^1!822!(8^)-. *Cf.Bromwich, Theory ofInfinite Series, §133. tStokes illustrated thechange with theaidofBessel functions whose orders areand±^, thelatter being those associated withAiry's integral (§6'4). 7-23, 7-24] ASYMPTOTIC EXPANSIONS 203 when Iarg^ [<§7r.Andtheformula /^(z)=e^"'^J„(g-*'^' z)shews that provided that—|7r<arg^^<ftt. Ontheother hand, theformula1^,(2)=e~-'"''J^{e-''' z)shews that providedthat—|7r<argzk^tt. Theapparent discrepancy between(2)and(3)when zhasavalue for whicharg^^liesbetween —^ttand^tt is,ofcourse, anexampleofStokes' phenomenonwhich hasjustbeeninvestigated. Theformulae ofthissection were stated exphcitly byKummer, Journal furMath. xvn. (1837), pp.228—242, andKirchhoff, JournalfurMath. XLViii.(1854), pp.348—376, except that,in(2)and(3),thenegUgible second series isomitted. Theobjectoftheretention of thenegligible series istomake(1)and(3)formallyconsistent with§3'7(6). Theformulae arealsogiven byEiemann, Ann. derPhysik undChemie, (2)xcv.(1855), p.135,when i'=0.Proofs aretobefound onpp.496—498ofHankel's memoir. Anumber ofextremely interesting symljolic investigationsoftheformulae aretobe found inHeavisidc's*papers, but itisdifficult todecide howvaluable such researches are tobeconsidered whenmodern standards ofrigourareadopted. Aremarkable memoir isduetoMalmst^nf,inwhich theformula r"^cosax .dx nre"^'' r>^ - jo(1+^=)"+i22«+i.ot! 'x[(2a)"+nC,{n+1).(2a)-i +,,0,{n+l)(//+2).{2af-' + isobtained (cf§6"3). Thisformula iswrittensymbolicallyintheform cosax .dx Tre"(^ 1 2ft+ .'0(1+a'')"+'2-"+i.'»![ [;^]-'J the[]denotingthat[//]""*istobereplaced by(//)_«,and this, inMalmsten's notation, means l/{(n+l){>i+2)...0i+m)}. Itwillbeobserved that thisnotation isdifferent from thenotation of§4'-i. 7"24. Theasymptotic expansions ofher{z)andbei(s). From theformulae obtained in§§7-21, 7-23, theasymptotic expansionsof Thomson's functions ber{z)andbei{z),and oftheirgeneralisations, maybe written down withoutdifficulty.Theformulae forfunctions ofanyorder have beengiven byWhiteheadj,but,onaccount oftheir complexity, theywillnot *Froc.BoijalSoc. i.ii.(1893), pp.504—529; Electromagnetic Thconj,11.(London, 1899). My thanks areduetoDrBroinwich forbringingtomynotice theresults contained inthelatter work. tA".Svend-a V.Akad. Handl. lxh.(1811), pp.05-74. XQuarterly Journal, xlii.(1911), pp.329—338. 204 THEORY OFBESSEL EUNCTIONS [CHAP. VII bequotedhere. Thefunctions ofzeroorderhadbeenexaminedpreviously by Russell*; hefound itconvenient todealwiththelogarithms!ofthefunctions ofthethird kindwhich areinvolved, andhisformulae maybewritten asfollows: iber(^)^ expg(^)cos ^^ \hei(z) V(27r^)sin^ '^^' |ker (z)^expa(-^)cos ^"'^ [kei(^) ^{-Iz/Tr) sin^^^' where _z1 25 13 °^^^^~V2"^8^V2 384^^/2 128^*•"' ^^^V2 88^v'2 162'^3842V2 Therangesofvalidityoftheformulae are |arg^-j<^irinthecase of(1)and Iargz \<|7rinthecaseof(2). These results have beenexpressedinamodified formbySavidge,Phil.Mag. (6)xix. (1910), p.51. 7*25. Hadamard'smodification oftheasrjmptotic expansions. Aresult which isofconsiderable theoreticalimportanceisdue to Hadamard:[:; hehasshewn that itispossibletomodifythevariousasymptotic expansions,sothattheybecomeconvergentseriestogetherluith anegligiblere- mainder term. Theformulae will-bestated forrealvalues ofthevariables, but thereader should havenodifficultyinmakingthemodificationsappropriate tocomplexvariables. Wetake firstthecaseof/„(«;)when v>—\.When wereplacesin|^by u,wehave ilxyU^)=rfii^V^n f'^^^''""sin- ede 1(i^+i)I(i)Jo nSm) ir'^(-'"'^>•""^'-"^'~'-'" 2(2xye''"i^-v)^[^w2.+2m exp(_2u-x)du,•Jo thelastresultbeingvalid because theseries ofintegralsisconvergent. Wemaywrite thisequationintheform _e^I(^-v\,^.y(v +m+l,2x) V(27r«)^=o r{r+l).m':{2xy-' where ydenotes the"incomplete Gamma-function" ofLegendre§. *Phil.Mag. (6)xvii.(1909), pp.531,537. tCf.thesimilar procedure duetoMeissel, which willbeexplainedin§8-11. +Bull, delaSoc.Math, deFrance, xxxvi.(1908), pp.77—85. §Cf,ModernAnalysis, §lG-2. 7-25, 7-3] ASYMPTOTIC EXPANSIONS 205 Forlarge values ofx,thedifference between is0{x''^"-'^h e~-^-)which iso(l)foreachintegral value ofn. Inthecase oftheordinaryBessel functions, wetake theexpressionfor thefunction ofthethird kind /9\-Splix-iv:r-j;nt fee,'i),\v-h sothat (2)fl!.>(,)=(A)',,-,.-i..-i„ i-^- ^';'"'_7<''+'"+.%;">+0(..-.-n andsimilarly ^ir-xj ^=r(y+I).m !(—2ta;)^ From these results itiseasytoderiveconvergentseries forthefunctions ofthe firstandsecond kinds. Hadamardgavetheformulae forfunctions oforder zeroonly;buttheextension to functions ofanyorder exceeding—|isobvious. 7'3.Formulaefortheremainders intheasymptotic expansions. In§7'2wegaveaninvestigationwhich shewed that theremainders in theasymptotic expansionsofH^,^^^ {z)and i/^<-' {z)areofthesame order of magnitudeasthe firsttermsneglected.Inthecase offunctions ofthe first andsecond kinds, itiseasytoobtain amore exact andrather remarkable theorem totheeffect thatwhen visreal* andxispositivetheremainders after acertainstageintheasymptotic expansionsofJ±^{x)and Y±t,(x)are numericallylessthan the firsttermsneglected, and,byaslightly more re- conditeinvestigation (§7"32),itcanbeprovedthattheremainders areofthe sapie signasthe firsttermsneglected. Letuswrite e"-'!(i-v£r+(i-i^.n'^-^'<-). *Wemaytakev^Owithout losing generality. 206 THEORY OFBESSEL FUNCTIONS [CHAP. VIT SOthat (1)/i,{x)={-^) [cos (oc+IvTT-Itt)P(x,v)-Sin{x^-lvir- Itt)Q{x,v)\ yrrx/ /2\* _ (2)F±^ {sc)= \—[sin{x^-lvir- ^tt)P(^, i^)+cos{x+lvir- {it)Q(x,v)]. yrrx J Nowl{v)=and, intheanalysisof§7"2,wemaytake htobe^ttsince thevariables arereal,andso^2^^=1- Itfollows that,ifpbetaken solargethat2p^v— |,there exists a number 6,notexceeding unityinabsolute value, such that [-2x) „r=o ml \.2ixJ^ {2p)l \2ixJ' and,onaddingtheresults combined inthisformula, wehave* V^2x) V2xJ ,Zo (2m)! \2ixJ^ {2p)\ \2x)' where |^o1^1 ;^"^' since 6oisobviously real,—1^^o^1. Itfollows onintegrationthat ^ '^^^Zo (2»0!i2xy^^^ (2p)!(2xypr(p+^)Jo' ' andsince rdoe-"zf+2p-* du^Ie-"if+'^P-^ du=T{v+2p^ I),\hJo weseethattheremainder afterpterms intheexpansionofP(x,v)doesnot exceed the(p+l)thterm inabsolute value, providedthat2p>v— ^. From theformula (l+"'Y""= V(i-V)m(±MV" e.Jl- v),^+,/+U\'P+^ V2x) „r=o ml \2ixJ"^ {2p+l)l\2ixJ' wefindinasimilar manner thattheremainder afterpterms intheexpansion ofQ{x,v)doesnotexceed the{p+l)thterm inabsolute value, providedthat 2p-^v-%. These results were given byHankel, Math. Ann. i.(1869), pp.491—494,andwere reproduced byGray andMathews intheir Treatise onBessel Functions (London, 1895), p.70,butsmall inaccuracies havebeenpointed outinbothinvestigations byOrr,Travis. Camb. Phil. Sac. xvii.(1899), pp.172—180. Inthecaseofif^,{x)wehave theformula *This result wasobtained inarather different mariner byLipschitz, Journal filrMath. lvi. (1859), pp.189—196. 7-31] ASYMPTOTIC EXPANSIONS 207 and and,whenj)'^v—\,thelasttermmaybewritten where <^i<$1,andso,onintegration, v^^jv, m) {v,p) ?H=(2^-)'«' (2^)pJ' where «$^o"^1whenjj^;^—|. This isamore exact result than those obtained forP{x,v)andQ(x,v) bythesamemethods;thereason whythegreaterexactness issecuredis,of course, thefactthat(1+^ut/xy~P~iispositiveanddoesnotoscillate insign after themanner of(1+liut/xy-^'i +(1-liut/xy~P-i. 7'31. The7'esearckesofStieltjesonJo{x), Y^Xx) andKq(x). The results of§7'3wereputintoamorepreciseformbyStieltjes*, who provednotonlythat theremainders intheasymptotic expansionsassociated with Jo(x),Yf, (x)andKq(x)arenumericallylessthan thefirsttermsneglected, butalsothattheremainders have thesamesignasthose terms. Stieltjesalsoexamined Iq(x),buthisresult iscomj'licated andweshall notreproduce itt. Itisonlytobeexpectedthat/q(.»•)isintractable because inthedominant expansion theterms allhave thesamesignwhereas intheother three asymptotic expansionsthe terms alternate insign. Itisevident from thedefinitions of§7"3that P{x,0)=^re--u-i{(1+|m)-i+(1-|V«)-*] dv, Q(x,0)=^ f%— w-i{(1+iuO-i-(1-im)-^}du. Inthese formulaereplace (1+^iu)~^ by 2r^-d(f) TT.1±2^^'Si^^^' *Ann. Sci. de.VEcole norm.sjyj. (3)in.(188G), pp.233—252. tThefunction/^(.r)hasalsobeenexamined bySchafheitliii, Jithre.fhericlit derDeutschen Math.-Vereinigung, xix.(1910), pp.120—129,butheappearstouseLagrange'sform for tlie remainder inTaylor's theorem when itisinapplicable. 208 THEORY OFBESSEL FUNCTIONS [CHAP.VII Itisthen evident that (1+iiu)--^+(1-Iiu)~^=- T ,.2• . ,^ - ^ ^ '^ -^ TT. 1+^ttsin-*^ =^['"{1_xu^sin^ (/>+...+(-)^-' (Art^sin"<^)?'-i +i-yiiu'sin"(/))^V(1+1'*'sin"0)} d(j>,. wherejjisanypositive integer (zero included). Now, obviously, l+^u'sm-<f)^Jo where liesbetween and1;andhence l{(l+l|u)-4+(l-lm)-4}=l-^^(iu)^+^^^^f^(in)"-... Ifwemultiply bythepositivefunction e~"*m"^andintegrate,itisevident that (^i; ^(,a;,u;i 2!(8;r)'^'^^ (2p-2)!(8^)^-^ .,.l^3^5^..(4»-l)2 where <^i<1,andpisanypositive integer (zero included); andthis isthe result which hadtobeprovedforP{x,0). Similarly,from theformula wefindthat r- ,l-.3^52,^ l-.3^5^..(4^-3)- (2)Q(^,0)-118^+3,(8^)3•••+()' (2p-l)!(8^)^P-i ,l^3^ 5^.(4p +l)- ^'^ ^ - (2^+l)!(8a;)^^+i' where <6.2<1, andpisanypositive integer (zero included) ;andthis isthe result which hadtobeprovedforQ{x,0). InthecaseoiKq{x), Stieltjestooktheformula K,{x)= ^'JJe-"^"-^(1+lu)-'- du, 2"i'^ dd)andreplaced (1+lu)~- by-.., ,;theprocedure then follows the ^ ^^ TT.o1+Jtfsm-'<|)^ methodjustexplained, andgives againtheresult of§7*3. Byaningenious device, Callandreau* svicceeded inapplyingtheresult ofStieltjesto obtain thecorrespondingresults forfunctions ofanyintegraloi'der;butweshallnowexplain amethod which iseffective inobtaining thepreciseresults forfunctions ofa,nyrealorder. *Bull, desSci.Math.(2)xiv.(1890), pp.110—114. 7-32] ASYMPTOTIC EXPANSIONS 209 7*32. Thesigns oftJieremainders intheasijmptotic expansions associated withJ„ix)andY^{x). Ithasalreadybeen seen thatJ^{x)andY^{x)areexpressibleinterms of twofunctions P{x,v) andQ{x,i') which haveasymptotic expansionsofa simpler type.Weshallnowextend theresult ofStieltjes (§7'31)soasto shew that foranyrealvalue* oftheorder v,theremainder afterpterms of theexpansionofP(a;,v)isofthesamesignas(inaddition tobeing numerically lessthanf) the{p+l)thtermprovidedthat'2p>v—|;acorrespondingresult holds forQ(x,v)Avhen 2])>v—f.The restrictions which these conditionslay onpenable thetheorem tobestated inthefollowing manner: Intheoscillatory parts oftheseries forP{x,v)andQ{x,v),theremainders areofthesamesign as,andnumericallylessthan, thefirsttermsneglected. Byaslightmodification oftheformulae of§7"3,wehave PG'P>^)=^^^l^^ f"^~"' """^Kl+2'"'O"-^'+(1-ii")"-^! du, and,exactlyasin§7'3,wemayshew that 11(1+iuo-i +(1-iiuy-^Ji(_r-(h ^'Uihuy^^ m=o {^m)l ^(-)^(l-^Wi^r-Y^-^_^^,^_,1. ,^^^liuty-^p+i-(1-^iuty-'p^i]dt Thereader willseethatwecanestablish thetheorem ifwecanprove that, when 2p>v—|,thelasttermontherightisoffixedsignand itssignisthatof {-y.{l-i'),,auypi(2p)i. Itisclearlysufficient toshew that -^1 (1-ty^-' Ii{(1+iiutY-'i>+'^-(1-1iuty-'P+i]dt 2p-v-i^o ispositive. Now thisexpressionisequal toij: I -In- typ-'~ li.I^x-v-"-^ \e-^"+=''"'-e-^'^-i'""}rf\dt {2p-v-^)T{2p-p-l)]o^^ ^ Jo^ =r/9^ ruff"(1-^y'""^""^ ^i'^^2^w^)•^-'d\dt 1{2p—p+i)JoJo =,^y^ :r. I" V^-"-^ e-^[(1-ty-"-sin(hXut)dtdX. 1{2p-v +^)Jo.'o *Asin§7-3wemaytakei-^Owithout loss ofgenerality, tThishasalready beenproveding7"3. XSince isin(hXut) \^i\ut,thecondition 2p>v-hsecures theabsolute convergence ofthe iofiniteintegral. W.B.P.^^ 210 THEORY OFBESSEL FUNCTIONS [CHAP. YJL Now (1—typ~'isamonotonicdecreasingfunction oft;andhence, bythe second mean-value theorem, anumberf,between and 1,exists such that f(1-0'^~'sin(iXut) dt=rsin(IXut)dt^0. Jo- SinceF(2p—v+^)ispositive, wehave succeeded intransforming—I—-f(1-typ-"-ii{(1+^iuty-'p^i-(1-iiuty-'p+^dt 2/5—I'— 2.' intoaninfiniteintegralinwhich theintegrandispositive, andsotheexpression 'under consideration ispositive. That istosay, 1 {(1+liuy- +(1-liuy-i] ^p^i(-r•(I-v)Uiur- (-y.(i-v\p{h^ y^ nZo i^my.^ {2p)l where 0:^0when2jw>i/—|.And ithasalreadybeen seen(§7"3)that in these circumstances |^ {^1.Consequently ^^^1;andthen, onmultiplying thelastequation bye~^^ it""^andintegrating,Aveatonceobtain theproperty stated forP(x,v). Thecorresponding propertyforQ(x,v)follows from theequation 2iK'+i"'^'-^-(1-i"')'-'i= -,!o '<J:t thedetails oftheanalysiswilleasilybesupplied bythereader. Note. Theanalysisfailswhen—h<v<^ifwetakep=0,butthen thephaseof {l±Uuy-^ Hesbetween and±h(v-^)7r, andsol{{l+liuy-h +{l-Uuy-h}hasthe samesignasunity, and, inlikemanner, ^{{l+^iu)''~i—{\-iiu)''~h}/iha,iithesame signas 2("""§)'''^^^hence P{x, v)andQ{x,v)have thesamesign asthe firstterms intheir expansions,sotheconclusions are stilltrue;andtheconclusion istrue forQ{.v,v)when i<./<f if^=0. 7*33. Weber'sformulae fortheremainders intheexpansions offunctions ofthethird kind. Someinequalities which aresatisfied bytheremainders intheasymptotic expansionsofHJ^'' (z)andJT^'^i (^^-^hg^vebeengiven byWeber*. Theseinequali- tiesowetheirimportancetothefactthattheyaretruewhether 2andvarereal orcomplex.Intheinvestigations which weshallgiveitwillbesupposedfor simplicitythat ;/isreal,thoughitwillbeobvious thatmodifications ofdetail onlyareadequatetomake themode ofanalysis applicabletocomplexvalues *Math. Ann. xxxvii.(1890), pp.404—416. 7-33] ASYMPTOTIC EXPANSIONS 211 ofV.There isnofurther lossofgeneralityinassumingthatv^O, R(z)'^0.Weshall write\z\=r,and, sincelarge values of |^^ |areprimarilyunder con- sideration, weshallsupposethat2r^i/—|. Ifi/-i>0, wehave*, by§6-12(3), ^.'^' (z) != irz)r(7Ti) .'o^'W-i 1+III 2zdu ^2\ie''*z-i'"^-j'r) 2\igUZ-iVTV-klT)e-^'ii"-^ (1+^Jdu „-p-..(i--^11"-^ du. XttzJgifZ-ii-n-\TT}i(Y 2r.ttW r(i;+i)I' 2Ni If<t-<1^,weusetherecurrence formula //<;> (^)=(2/z) (v+l)H'Xi (^)-^1+2 (^) andapplytheinequality justobtained toeach ofthefunctions ontheright. Itisthusfound that (1) andsimilarly (2) where f (3)HJ^'^{z)^G\ {l-rrz)-^e*-(^-i—i-) | ir,(^){z)^G\ {^irzyi e-i{2-i—J-) G={l G=[l-2r 2r1+2i/+2(^>*) (^<i) The results maybecalled Weber's crudeinequalitiessatisfiedbyiT^*^^ (^) andH^}^ {z').Byanelegant pieceofanalysis,Weber succeeded indeducing- more refinedinequalitiesfromthem inthefollowingmanner : Take the firstpterms oftheseries involved inHankel's twoexpansions anddenote thembythesymbols S^"' {z ;/j),S^'-' {z;p),sothat V(.; ;»="2<->•"• (^-'"^ )yi=0 (2^2)'7)1= Itiseasytoverifythat i?l+2^^+t^ 2.-(.;i.)=^''•'"''> ^~-(-2y>f-i Weregardthisasanequationtobesolved bythemethod ofvariation ofpara- meters; wethus findthat S,w{z;p)={l7rz)ie-a^-iu^-in) |^(^)^^o (^)+^(z)H,^'^ (z)], *luthethird luieofanalysis theinequality e*^ 1+x(x^O)hasbeen used, tWhen ;/<iwetake 2r>;»+ :l. 14—2 212 THEORY OFBESSEL FUNCTIONS [CHAP.VII whereA(z)andB(z)arefunctions ofzsochosen that (A'(z)^,<'' (z)+B'(z)ZT,'^' (z)=0, Y'(z)^H^^ (z)+B'(z)^H/^^ (z)=^-(iTT.)-^ e'-<-i—i'^> /(-2i^U' Itfollows that A'{z)=^7ri^TTzri e^;-J—J'^)|l^H/^^ (z), andso A(z)==A-i,tt/J{iTT(.+0}-^e^-^^^^-^-i^^ {-2i(z +l\^^"^ ^'+^>^^' where J.isaconstant. Weobtain asimilarexpressionforB(z),andhence itfollows that S;»{z;p)=[AH,^'^ {z)+5^,'-^' {z)] {^'rrz)\g-'X^-J—J'^) -h^'rrp.{v,p)l (^J l-2i(. +0P"^^^ Byconsideringthebehaviour ofboth sides ofthisequationas ^^-^+oo ,it isnotdifficult toseethatA=l and5=0. Hence wemaywrite Hankel's formulae intheforms •^ ^"'^ ^^^" (—)"^"'""^'""''"1-"'" (^ 'i')+^i'"'!' where theremainder72^*"maybedefined bytheEquation i?/'= i7rp.(.,^.)j^(^ 1_2,•(,_^,)}.^^- SinceR(z)^0,wehave\z+t]"^ \J{r^+i^),and so,byusingthecrude inequalities, weseethatthemodulus ofthelastintegrand doesnotexceed Hence Ijs^<i' I$2'-pG'p I{v,p) jr(7-'+i2)-i(p+i) dt,Jo and so,whenp^l,wehave andsimilarly These aretheresults obtainedbyWeber;and itwillbeobserved that in theanalysisnohypothesishasbeenmadeconcerningtherelative values ofv andp;inthisrespectWeber's results differ from theresults obtained by other writers. 7-34] ASYMPTOTIC EXPANSIONS 213 7*34. Approximationstoremainders intheasymptotic expansions. When theargumentofaBessel function isnotvery large*, theasymptotic expansionisnotwelladaptedfornumericalcomputation because thesmallest term init(with theremainder after thesmallestterm)isnotparticularly small;atthesame time theargument maybesufficiently largeforthe ascendingseries toconverge very slowly. Aningeniousmethod formeetingthese numerical difficulties wasdevised byStieltjesf;weshallexplainthemethod indetail asappliedtothefunction Kq{x) andstate theresults which were obtained byStieltjes byapplyingthe method. toJo(«^^)andYf^{x). Weapplythetransformation indicated in§7"31 totheformula§6'15(4), sothat e-^r e-^"du e-^V2r i'*"e-'^'^dedu TT.',).'o«*(!+I'usin^^) ITp-1 /•» rjTTp-xu- -^(- hisin'Oy^dddu + x/T ," i^—rnr dddu JoJow^l+|"sm-^) That 214 THEORY OFBESSEL FUNCTIONS [chap,vn Thedomain ofintegrationbecomes thewhole ofthe(|, 77)plane;and itis found that TT* J-00J-00 (>%s=i.^ where a0,0—2' ^,"^"24' ^0,2—8' bysome rather tedious arithmetic. Itfollows* that thedominant terms of theasymptotic expansionofR^forlargevalues ofpare R^'^2x\^e-"-^ ITJ 'p sothat (2) A~2(--V^-'c^o,2+cr2,o, 1,<^'+?V7.+ TT/J9|_2 p Itiseasytoverify byStirling's theorem that+... '{-)•V)p' sothattheerrorduetostoppingatoneofthesmallest terms isroughlyhalf ofthe first term omitted.- InlikemannerStieltjes proved that, ifP{x,0)andQ{x,0)aredefined as in§7*3,then i'(^,0)= ^X<=^9|=;^^(-)'i^;",w= .3(..o)=-s<->'"ff:r^'^(-)^ie;^'. »«=0 R^(P)r^(^\^^''' i^^'«-.TT/p1_2 Vtt/^_2 p••y(3) (4) where (5) ^«> --vwJ,L2 providedthatpischosen soastobenearly equaltox,andtisdefined to \iQ.X—p. Eesults ofthischaracter areuseful fortabulatingBessel functions inthecriticalrange; some similar formulae havebeenactuallyused forthatpurpose byAirey,Arcliiv derMath, undPhys. (3)xx.(1913), pp.240—244;(3)xxil. (1914), pp.30—43;andBritish Association Reports, 1913, 1914. Itwould beofsome interest toextend theresults, whichStieltjes hasestablished for Bessel functions ofzero order(aswell asforthelogarithmic integral andsome other functions),toBessel functions ofarbiti'aryorder. *Cf.Bromwich, Theory ofInfinite Series, §§133, 137,and174, orthelemma which willbe provedin§8'3. 7-35, 7-4] ASYMPTOTIC EXPANSIONS 215 7"35. Deductions from Schafheitlirisintegrals. Ifwereplaceaby2tan6intheformulae of§7"32,wededuce that ^.,(2^)"+*f*-sin"-* ^sin(i/-1)6/ ,,,,^ which resemble Schafheitlin'sintegralsof§6*12. Itisobvious from these results that P(.'T,V)>0, (-i< ,.<f) Q('^,v)>0, (h< i'<f) Q(*, i^)<0. (_1<^<1) Aninteresting consequenceofthese results isthatwecanprovethat Q(x,v)/F (x,v) isanincreasingfunction ofxwhen— h;<v<h:andthat itisadecreasing function ofxwhen h<v<^. Forwehave Q'{x,v)P{x,v)-P'{x,v)Q{x,v) where n//i .N (sin^sind))"-* , ., ,,,-.^^ , ix , ^^^' ^^= (cos^cos.J j^^^^^^-*^"^^'°'^""*^^''''^""^^'^' sothat F(0, cf>)+P(c^, ^)= ^^'^"^^^7^ ^Pi-(^^" ^-t-^^'^)^^^(2-^)(^- <^)- Ifweinterchangetheparametricvariables 0,cj)inthedoubleintegraland addtheresults soobtained weseethat,when—l<v< %,thedoubleintegral hasthesamesignasI—v;andthispi'ovestheresult. 7'4.Schlaflis investigation oftheasymptotic expansions ofBesselfunctions. Inamemoir which seemshardlytohave received therecognitionwhich itsimportance deserves, Schlafli* hasgivenavery elegantbutsomewhat elaborateinvestigationoftheasymptotic expansionsofthefvmctions ofthe third kind. Theintegralformulae from which hederived theseexpansionsare generalisationsofBessel'sintegral;althoughBessel'sintegralisnotsowell adaptedasPoisson'sintegralforconstructingtheasymptotic expansionot *Ann. diMat.(2)vi.(1875), pp.1—20. Theonlystandard work onBessel functions inwhich theimportance ofthismemoir isrecognisedisthetreatise byGrafandGubler. 216 THEORY OFBESSEL FUNCTIONS [CHAP.VH Jt,{z)when zislargeand visfixed, yetSchlafli's method notonlysucceeds inobtainingtheexpansion,butalso itexpressestheremainders inaneatand compactform. Schlafli'sprocedureconsisted intaking integralsofthetype ^.jH-"-exp|±i7-e- (u+ ^)Jdu, andselectingthecontour ofintegrationinsuch away that,onit,thephcifie* of ire'"(?/-2+l/w) isconstant. Hetooktwocontours, theconstants fortherespectivecontours beingand tt;and itissupposedthatrispositiveandaisreal. (I)Letusfirsttake thephasetobett;write u=l+pe*", wherepispositiveand6isreal,andthen re^'p^e-'^/il +pe'^) isnegative,and isconsequently equaltoitsconjugate complex. Hence wehave ..,, sin(a+2^) sin^.„.,,^^ Psin(a+^)'sin(a+^) Next choose anewparametricvariable<^such that * </)=2^+a-7r, andthen cos!(«-(/)) (m-1)2 -rsin-<^ (2)n= cosi(a+0)'u cos|(a—0)cos|(a+0)" Now, as^varies from—(tt— ot)to(tt—a),utraces outacontouremerging from theoriginatanangle—(tt—a)with thepositiverealaxisandpassingto infinityatanangle (tt—a)withthepositivereal axis,providedthat<a<27r. Ifthis restriction isnot laidonathecontourpassestoinfinitymore than once. Weshallnowlaythisrestriction ona;andthen thecontour isofthetype specifiedforformula§6*22(9),providedthatwegivetoandargzthesame value a,asispermissible. Itfollows that e""'/_^(re'*)-e'^^i I^(re^")=2^IZ'j'""''P 1^''''" ("+ «)1•%^'^-'2isinvir where uisdefined interms of<^byequation (2). *Thereader willfind itinteresting tocompare thegeneral methods ofthissection with the "method ofsteepest descents" which isapplied toobtain various asymptotic expansions in Chapterviii. 7-4] ASYMPTOTIC EXPANSIONS 217 Changingthesignof<^isequivalenttoreplacinguby1/u,and so,replacing theexpressiononthe leftbyitsvalue asafunction ofthethii'd kind,wehave (.S)ei""'i/,w (j-e'(«-^^-) )=Xr'or"+u")exp11/-e'""L+^]\.^^ ^^^^' d(f>. From(2)itfollows that —re"^{i(—1)-/hincreasessteadily*from to+x as </)varies monotonicallyfrom toir—a:and,ifwewrite - ?-e'"'(«-lf/u=t, sothat tispositivewhen uisonthecontour, wehave du dt _dt H"-re'"(a-Tfi()~ e-H'^-«)^(„i +u.-i) ^/{rt)' therangeofvalues ofargubeinglessthan tt. Next, byCauchy's theorem, itisconvenient totake thepoint^=1inside thecontour, but^=must be outside thecontour because theoriginisabranch-point. Itfollows that __iii+u-i i'"+•!/«+'^--Ht-1)(^^ Hence /rx rrm/,4>xe-J'"^'exp(re''^)r«r(«+.i/«+, i+)g-i^r* ^""H?- l)<^^c?^^^ "^^ ^27r'-'?;r^e*- JoJ (^- 1)-^+^^/(7-e'-) Now itisevident that 1 p-i(^_y,i^mpn (-)y l;pfP where2>isanypositive integer (zero included). Itwillbeconvenient subse- quentlytosupposethatpexceeds bothR(v—h)andR(—v—i). Onmakingthissubstitution inthelastintegrandandobservingthat 1/•(«+,1/M.+, 1+) r(i;-I-m4-A) m I(v,m) •2^j^ "^^"-^^'^^~r(v-m +i,).(2vil) (2m)! (wjththenotation of§7'2),wededuce that (9\i rj'-i(—V" (i'w) where -^^*'" 2"W(27r),'o J (T^n:p^(7^e^'M(^^^>^+T^>^}* t? sin-<* sin0(l+2cosacos0 +cos-</)) SincG — ^^" " ~>"' d(j)cosa+cos(p (cosa+cos(/>)- 218 THEORY OFBESSEL FUNCTIONS First consider[chap.VII 1/•(«+, 1/U+, 1+)^"^p-id^ Whenpissolargethat itexceeds bothR(v—^)andR{—v— -|),wetakethe contour tobeasshewn inFig.15;andwhen theradii ofthelargeandsmall circles tend toooandrespectivelytheintegrals along them tend tozero. Ifnowwewrite onthetworays(whichareallthatsurvives ofthecontour), wefindthat (-)Pcosvirn.-kP-"-*(1-xy-^'''^d.v TT JoI-tx{I-x)/{re^'') Fig.15. Now thenumerator oftheintegrandispositive (whenvisreal), andthe modulus ofthedenominator isnever lessthan 1when^tt<a<|7r;forother values ofaitisnever lessthan |sina!. Therefore do \cosvir ftd' 1<^l^7rrV(2^e-ittP-ixP-''-i(1-a:)P+''-i dxdt=6^I{v,p)I^{2r)P, ^TJ") jJ where j^o 1is1or |cosec a \accordingascosaisnegativeorpositive. When v iscomplex,itiseasytoseethat cosi^TT I6o\{R(v),p) j cosR(vtt)1~ {27')P' (7) V^ i^ 7-4] ASYMPTOTIC EXPANSIONS 219 Hence, finally,when—^tt<avgz<Ivr, where ]^j jdoesnotexceed 1or \sec(arg^)jaccordingas1{z)ispositiveor negative, providedthat visrealand'P+\>\v\. When viscomplex,themodi- fiedform oftheremaindergiven by(7)hastobeused. SinceR{\-tx{X- x)\{:re^^)\ ^0when i?(e"''^)^0,weseethat, in(8), d^ has itsrealpart positive when visrealand/(^)^0. \izbereplaced byizin(8)wefind that,whenarg^^ |<tt, (9)^''''W=(£; and,when visreal,e' (i)R(6,)^and ]^3 j<1,ifi?(^)^0, (ii)i^3 1< icosec(arg z)\,i(R{z)<0. Themodificationsnecessaryforcomplexvalues ofvare lefttothereader. (II)Wenext discuss theconsequencesoftakingthephaseof ^re''^{u-2+l/(/.) tobezero.Asbefore, wewrite w=1+pe"^, andthen re^"^p-e'-'^l{\ +pe'^)ispositive,andthereforeequaltoitsconjugate complex,sothatweobtain anewequation (1).Wethendivergefrom the preceding analysis bywriting <f>=-{2d+a) sothat (10)„=-41"+-*).-.:*. ,.e>.(ii-LZ^. ; '-^i"'^^smI{ct- (p)u sm^(a- cf))sinh(a+9) Now, as(jivaries from-atoa,utraces outacontour emergingfrom the originatanangleawith thepositiverealaxisandpassingtoinfinityatan angle—awith thepositivereal axis,providedthat aliesbetween —ttand tt. Thecontour isthen ofthetype specifiedforformula§6-22(8) if,asisper- missible, wegivewandargzthesame value a. Itfollows that,when—tt<a<tt, ii,{re'^)=icos V7T Iir"-'exp|-tre'""(u+ -JYj,f/0, where 11isdefined asafunction ofbyequation (10);andtherefore • J,/;ri fa ( ,•/ I\]dlogU,, (11) 7/,-' (re'^^-i"))=^j^^(^r"+a-^)expj-ire- (^a4-J|- ^^|#, 220 THEORY OFBESSEL FUNCTIONS [CHAP. VII andhence, ifnow t=re^''(u—lf/u, wefindthat Wehaveconsequently expressedasecond solution ofBessel'sequationina formfromwhich itsasymptotic expansioncanbededuced;andtheanalysis proceedsasinthecase of77^'^* (z),thefinal resultbeing that,when -f7r<arg^<-|7r, (13)^.'M.)=Q^--<--f:!:[i^;^^<ii; where |02. \does notexceed 1or |secarg^ ]accoi'dingasl{z)^0orl{z)^0, providedthat visrealandl)+l>\v\; andR(d^)>when I(z)^0. Ifyis complextheform oftheremainder hastobemodified, justasinthecaseof(8). Itshould beobserved that, since theintegrandsin(3)and(11)areeven functions ofv,itisunnecessaryinthisinvestigationtosupposethatR{v) must exceed —|,aswasnecessaryininvestigationsbased onintegralsof Poisson'stype. 7"5.Barnesinvestigation* ofasymptotic expansions ofBesselfunctions. Theasymptotic expansionsoffunctions.of thethird kind followimmediately from Barnes' formulae which were obtained in§§6*5,6'51. Letusconsider r{~s)V{-2v-s)V{v +s+i){-^izyds —cci—v—p (-2iz)-''-i' I' T{~s+v+p)r(-s-v+p)r{s-p +Di-2izyds. J-£i If iarg(—iz)\^%Tr—8,wehave r{-s+v+2))r(-s-v+p) r(s-p +i)(-2izycis J—Xi ^T'\r(-s +v+p)r(-s-v+p)r(s-p +i)e^''-^y''^ds, J-cci andthelastintegralisconvergentandsothe firstintegralofallis {(-2iz)-'-P\. But,bytheargumentsof§6"51, the firstintegralis—'Imtimes thesum oftheresidues atthepolesontherightofthecontour, andsoitisequalto —TT^H^^^^ {z)l{f^^~'"'^cosVTT(22^)"] plus—^Tvitimcs thesum oftheresidues at s=-V—^,—V—%,...,—V—p+\.Theresidue at—i^—??i—^is (-)"'r(1/+m+^)r(-1/+m+h) *Tram. Camb. Phil. Soc.xx.(1908), pp.273—279. 7-5,7-51]. ASYMPTOTIC EXPANSIONS 221 and so,when jarg(—t» |^f-n-—S (iy,...,-.p,;i.g„,o(..-.)- and this isequivalenttotheresult obtained in§7'2.Theinvestigationof //^'-' (2')maybeconstructed byreplacing^by-ithroughout. Thereader should notice that,althoughthedetermination ofthe07'derof magnitudeoftheremainders bythismethod istransparently simple,itisnot possibletoobtain concrete formulae, concerningthemagnitudeandthesign oftheremainders, which areultimately supplied bythemethods which have beenpreviouslyconsidered. 7*51.Asymptotic expansions ofproducts ofBesselfunctions. Itdoes notseempossibletoobtainasymptotic expansionsofthefour products J±fj,{z) J±y(z)inwhich the coefficients havesimple forms, even whenfx=v.Thereason forthis isthat theproducts i/^"' (2^)//^"^ (2^)and H^^i (z)H^'"' (z)haveasymptotic expansionsforwhich nosimple expression exists forthegeneralterm;theleading terms inthetv/oexpansionsare 2e±2*'~THA*+.'+i)-'^_2fx-+ 2p'-1 TTZI4^12 Theproducts ir^<^' {z)H^^-^ {z)and//^<-' (2)^^<"(^), however, dopossess simple asymptotic expansions ;andfromthemwecandeduceasymptotic expansions for J^{z)J^{z)+Y^{z)Y,{z) and for J^(z)F,(z)-V^(z)./,{z). Thesimplest wayofconstructingtheexpansionsisbyBarnes' method, just explainedin§7'5.Aconsideration ofseries ofthetypeobtained in §5"41suggeststhatweshould examine theintegral 27n/";r(2.+i)r(^±i;-..)r(t^-.)r(''7-.,)r(-''-;"-.)(ii.r-rf.; thecontour istobechosen sothatthepolesofF(2s+1)lieontheleftofthe contour andthepolesoftheother fourGamma functions lieontherightof thecontour;and itistemporarily supposedthat/u,,randfi±i^arenot integers,sothattheintegrandhasnodoublepoles.Tlieintegralisconvergent provided that Iarg(?2)I<|7r. 222 THEORY OFBESSEL FUNCTIONS [CHAP.VII First evaluate theintegral byswinginground thecontour toenclose the sequencesofpoleswhich lietotherightoftheoriginalcontour;theexpression isequaltominus thesum oftheresidues atthesepoles,andtheresidue at m+^{/x+v)is sin/ATT.sinv7r.sm(fi +v)'7r'm\ r{/x+')n+l)r{v+m+l)r i/x+v+m+l) Itfollows that 27rt sinfXTTsinvir\sin(,a+v)7rsin(/a— i/)tt sin(i;— yti)7r sm(/ji+v)'Tr sin{/J,+v)7r{J,(z)J,{z)+Y,(z)YAz)\ ^3g-j(M+w«|2cosit7rcosi/7r+tsm//i +i')7r} ,r/\\r^^t^/\r/\)^S^ ^—7 ^^^-—WM(z)1Az)-ru. (z)J^{z)\ TT^ [[/,(^)j,(^)+F^(^)r,(^)}2sin|(/x +v)Tr -coti(/.- I/)TT{/^ {_z)F,(^)-F^{z)J,(^)}] - 2cos|C+^)J{^-^->-^-(->^^-(->^'^-^->l +tani(/.-^)TT{./,(^)F,{z)-Y,(z)/,{z)]]. Bywriting—ifor ithroughouttheanalysis wededuce that,ifboth argiz Iand |arg(—iz) \arelessthanfvr,i.e.if jarg^:j<tt,then -cotH/*-")'^ •!-^(Z)I^.(^)-1,.(2)^.(^)l] 7-51] ASYMPTOTIC EXPANSIONS 223 and +tanJ(^-^)TT .[J,(z)r.(z)-F„(2)/.(z)]] Xr ("-^^- .)r(-^^- .).in.. .(i.rds. These results hold for allvalues offiand v(whether integersornot) provided that, inthecase oftheformeru+vandfj,—varenotevenintegers, and. inthecaseofthelatter/jl+vandfu,—varenotoddintegers. Wenowobtain theasymptotic expansionsofthefunctions onthe leftof (1)and(2)after themanner of§7'5, We firsttakeptobeanintegersolargethat theonly polesofthein- tegrandsontheleftofthelineR{s)=—2)-jarepolesofF(2s+1) ;andthen J-xi J-xi-p—\ (when eitherintegrandisinserted)isequalto27ritimes thesum ofthe residues atthepolesbetween thecontours. Since r~'~' f(s)(izrds=^0{z-^P-i), J-txi-p—\ wededuce thattheasymptotic expansions,when [argz\<ir,are (3) [J^(z)J,(z)+Y^(z)n(2)]-cotH/^-^)'r .[J,(z)Y.iz)-/,(z)Y,(z)] 11+VSm—^—TT TT^X ,^.(-rT(^%.»+i) r(^V»,+ i)r5g+,«+i) r(-'y+,.+i ~..^;7" ,..F.(^'+i. ^"+1, '-/+i. i-'^; I,-\) iTZ-smI(yu,— i^)TT V2 ^ ^ 2 2 ^^-y and .(4) [./^{z)J,iz)+Y^{z)F,(z)]+tan\{y,-v)'rr. \_J^{z)F,{z)-J,{z)Y,{z)] 2//i+IZ+l^l—V+lV—ix+l\—^—v .1 ._1 TT^COS^{iJb—v)7r//A+1/+1/A— ?-'+1 ^'—/i+11— /i.—i^ .1 ._£\ ^'^ 2'2'2'2'2'W* 224 THEORY orBESSEL FUNCTIONS [CHAP.VII Inthespecialcasewhenjx=v,thelastformula reduces to (5)JH^)+Y^^{z)^^2{1.3.5...(2m-l)l^\ and, inparticular, m JHz)+YHz)^-''~^(-r{1^3^5(2.^-l)}- (b)Jo(^)+io(^)-^^^-^ (2m)! (2.)- Formula (5)seems tohave been discovered byLorenz, K.Danske Vidensk. Selskahs Sh-ifter, (6)vi.(1890). [Oeuvres scientijiqties,I.(1898), p.435],while themore general formulae (3)and(4)v^ere stated byOrr, P)-oc. Camb. Phil. Soc. x.(1900), p.99.Aproof of(5)which depends ontransformations ofrepeated integrals wasgiven byNielsen, Hand- huch derTheorie derCylinderfunktionen (Leipzig, 1904), pp.245—247;theexpansion (5) is,however, attributed toWalter Gregory byA.Lodge,British AssociationReport., 1906, pp.494—498. Itisnoteasytoestimateexactlythemagnitudeorthesignofthere- mainder afteranynumber ofterms intheseasymptotic expansions when this method isused.Analternative method ofobtainingtheasymptotic expansion ofJy'{z)+Yy-{z)willbegivenin§13"75, and itwillthenbepossibletoform suchanestimate. CHAPTER A'lII BESSEL FUNCTIONS OFLARGE ORDER 8"1. Besselfunctions oflargeorder. Thesubjectofthischapteristheinvestigationofdescriptive properties, including approximate formulae, complete asymptotic expansions, and in- equalitiesofvarioustypesconnected with Bessel functions;andthepro- pertieswhich willbeexamined areofprimary importance when theordersofthe functionsconcerned arelarge, though manyoftheresultshappentobetrue forfunctions ofallpositiveorders. We shall first obtain results which areofapurelyformal character, associated with Carlini's formula(§1'4). Next, weshall obtain certain approximateformulae with theaidofKelvin's*"principleofstationary phase." Andfinally, weshallexamine thecontourintegrals discoveredby Debyet;these willbeemployed firstlytoobtainasymptotic expansions when thevariables concerned are real, secondly,toobtain numerousinequalitiesof varying degreesofimportance,andthirdly,toobtainasymptotic expansions ofBessel functions inwhich both theorder andtheargumentarecomplex. Indealingwith thefunction /^(sc),inwhich vand scarepositive,itis found thattheproblemsunder consideration have tobedivided into three classes, accordingasx/vislessthan, nearly equal to,orgreaterthanunity. Similar sub-divisions alsohave tobemade inthecorrespondingtheorems concerned withcomplexvariables. The trivial problemofdetermining theasymptotic expansionofJy(z),when vislarge and 2isfixed,mayhenoticed here. Itisevident, byapplying Stirling's theorem totheexpansionof§.3-1,that Jy(z)~exp [v+Vlogi^z)-{v+^)logv}.VV wheretV|=l/v^(27r);this result hasbeenpointedoutbyHorn, Math. Ann. Lll.(1899), p.359. [Note.Forphysical applicationsofapproximateformulae forfunctions oflarge order, theJi^Howingwriters maybeconsulted: Macdonald, Proc.RoyalSoc.lxxi. (1903), pp.251— 258;Lxxii. (1904), pp.59—68;xc.A(1914), pp..50—61;Phil. Trans,oftheRoyalSoc.cxx.A (1910), pp.11.3—144; Debye,Ann. derPhysik und Chcmie, (4)xxx.(1909), pp.57—136; March, Ann. derPhysikundChemie^ (4)xxxvii.(1912), pp.29—50; Rybczyiiski, Ann. der *Phil. Mcifj. (5)xxiii.(1887), pp.252—255. [Math, andPhijs. Papers,iv.(1910), pp.303—306.] Inconnexion with theprinciple,seeStokes, Trans. Camb. Phil. Soc. ix.(185G), p.175footnote. [Math, andPhys. Papers,ii.(1883), p.341.] tMath. Ann. lxvii.(1909), pp.535—558; Milnchener Sitzungsberichte,xl.[5],(1910). W.B.F. 15 226 THEORY OFBESSEL FUNCTIONS [CHAP. VIII Physik undChemie, (4)XLi. (1913), pp.191—208; Nicholson, FMl. Mag. (6)xix. (1910), pp.516—537;Love, Phil. Trans, oftheRoyalSoc.ccxv.A(1915), pp.105—131;Watson, Proc. RoyalSoc.xcv.A(1918), pp.83—99, 546—563. Theworks quotedalldealwith theproblemofthepropagationofelectric waves overthesui'face oftheearth, andare largelyconcerned withattemptstoreconcile theoretical withexperimental results.] 8'11. MeisseVsfirstextensionofCarlini'sformula. Theapproximation (§1-4)obtained byCarlini isthe firstterm ofthe asymptotic expansionofaBessel function oflarge order;subsequent terms intheexpansionwere formallycalculatedbyMeissel, Astr. Nach. cxxix. (1892),col.281—284, inthefollowing manner : Itisclear that Bessel'sequation maybewritten (1) ,-^^-_+._^_v^{l-z^)JAvz)=i); ifwedefine afunction u{z)bytheequation thenequation (1)transforms into (2)z''{u'{z)+[u{z)Y'\+ZU{z)-v-'il- z")=0. Ifnowweassume that, forlargevalues ofv,u(z)isexpansibleinaseries ofdescending powersofv,thus U(Z)=VUq+Ml+Un/v+u-s/v^+..., whereiio,u^, u.2,ih,...denote functions ofzwhich areindependentofv,by substitutingin(2)andequatingtozerothecoefficients ofthevariouspowers ofVonthe left,wefindthat Uo=Wil- z-)]/z, Ui=ir7z.— ,W2=-2(1-^^)'^ 8(1-^^)^' 4^+10^^ +^^ 64^+560^3+456^5+2,52' -U- "5.=8(1-^-^)^ 128(1-2^)- 16^+368^3+924^5+374^'+ISz^ 32(1-2-)'' Hence, onintegration,itisfound(cf.§1'4)that j\(z)dz=V jlog^^^'^^_^,^+V(l-^^)- l}-ilog(1- z-^) 2+32^J 42^+z' 24i;((1-^2)^jlQv''{l -z'-f 1 (16-15122--36542-*-375^"_ '5T60l>{ (1-^^)^ 322-^+2882"+2322«+132« 8-11, 8-12] rUNCTIONS OFLARGE ORDER 227 Hence wehave Meissel's formula (3) whereJu,z)=('^^)'exp ji/V(l- ^•^)}.exp(-F,) '""^^ e^V{v4-1)(1-^f [I+V(l-z^Y' (4)F.J^|l±^_2l^^^+^ - 1_ [16-1512^"-3654^^-:375^« 57601^^I (1_z^^ S2z'+288^''+2820«+ISz" usv'(i-z'^y"^ Itwillappearin|8'4thattheexpression givenforV^isthesum ofthe fourdominant terms ofanasymptotic expansionwhich iscertainlyvalidwhen zliesbetween and 1andvislarge. Itisstated byGrafandGubler* thatthefirstapproximationderiv^able from(3),namely /(^A^''exp{,.v/(l-f^)} isdue toDuhamel;butasearch fortheformula inDuhamel's writings hasnotbeen successful, and itseems certain that, even ifithadbeen discovered byDuhamel,his discoverywould havebeensubsequenttoCarlini's. Note. Thereader should observe that(3)mayalsobewritten intheform expI-v(a- tanh a)}.exp (-W^) (5) t/^(j/secha)= ;7r:; ;— :, ^^ .-V / ^l{2iTv tanh a) where //%N T.- coth^ a,^^,,,,coth'' a,,,^, ,. n (6)Tl„=-—(2+3sech- a)r—^(4sech^ a+sech*a) 4 ij4i' Ibi"- rofh'Jn-^^r. ,(16-1512sech^ a-3654 sech* a-375sech" a) 5/60i'-* (32sech2a+288sech^ a+232sech^ a+13sech^ a)128:/* +.... 8'12. Meissel's secondexpansion. Theexpansionobtained in§S'llobviouslyfails torepresent J„{vz) when zisreidhand greaterthanunity;forsuch values of^,Meisselfobtained two formal solutions ofBessel'sequation; and, ifwewrite z=sec^, thereader will see,bymakingsome modifications in§811(5),thatthese solutions may bewritten intheform ,/(^^)e.p!-P.±--ai, *Einleitunrj indieTlieorie derBesseVschen Funktionen,i.(Bern, 1898), p.10'2. tAstr. Nach. cxxx.(18!)2),col.363—368..' 15-2 228 THEOKY OFBESSEL FUNCTIONS [CHAP.VIII where* (1)P.=^^(4sec^y8+sec<'/3) --^—^ (32sec^yS+288sec^/3+232sec« /3-)-13sec«/3) +g"'g(768 sec^yg+41280 sec*/3+14884sec«y9+17493 sec«/3 +4242 sec^o^8+103sec^^/3) ' I' •••5 (2) Q,= z;(tan^-/3)-^||^(2+3sec=/3) ^ (16-1512 sec-/3-3654 sec* /3-375sec«yS)57601/^ -o^^-Va .(256+X8720 sec-y3+1891200sec^/3+4744640 sec«/S322od0j^* +1914210sec^yS+67599 sec^"/3) +.... Todetermine J^{ysec/8)interms oftheseexpansions, wetakeyStotend to^TT,andcomparetheresults soobtained with theexpansionsofHankel's typegivenin§7*21;weseethat, as/3—\'k, P.—©,Q.~z^(sec/S-^7r), andweinfer that (3) ir,w iysec/3)=a/(^-^)•e-^^+'^Q.'-^'"', (4) ^,*=" (i/sec/3)=a/(^-^^)•e-^--^Q-+^'''^ Itfollows that (5) J,(vsecy8)= y/'(?^^).e-P>'cos(Q,-ivr), (6) F.(^'^^^^=v/(^^)'"'''' '^^^^•'~^'^)' where P^andQ^,aredefinedby(1)and(2).Itwillappear subsequently (§8'41)thatthese formulae areactually asymptotic expansionsoft/^(ysec/3) andY^(vsec/3)when vislargeand^isanyassignedacuteangle. Formulae which arevalid forsmall values ofyS,i.e.asymptotic expansions ofJt,(z)andY^(z)which arevalidwhen zandvarebothlargeandarenearly equal,cannoteasilybeobtained bythismethod;but itwillbeseen in§8'2 that, forsuch values ofthevariables, approximationscanbeobtained by rigorousmethods from Schlafli's extension ofBessel'sintegral. *Thereader willobserve thattheapproximation hasbeen carried twostagesfurther than in thecorresponding analysisof§8*11. 8-2] FUNCTIONS OFLARGE ORDER 229 Note. Thedominant terms intheexpansions (5)and(6),whichmaybewritten inthe form (7) J^{x)=M^cos{Q^-^7r), }\ix)=M,sm{Q,-^7r), / 9 \1' where 3f„ Qv~ 'J{-'^'^~ ^'^)—^vir-\-varcsin{vjx), hadbeen obtained twoyearsbefore thepublicationofMeissel'spaper byL.Lorenz ina memoir onPhysical Optics, K.Danske Videnskahernes SelsJcahsSkrifter, (6)vi.(1890). [Oeuvres Scientifiques,i.(1898), pp.421—436.] TheprocedureofLorenz wastotake forgranted that, asaconsequenceoftheresult which hasbeenprovedin§7'51, iro:[_2 .'- 2.4 .r* ^.^L•^'-J' andthen tousetheexact equation ^'dx~ trxAI^^' which iseasily deduced from theWronskian formula of§3'63(1),toprove that Q.=x-y.-\ {^-l}^ whence theapproximation.stated forQvfollows withoutdifficulty. Subsequent researches onthelines laiddown byLorenz aredue toMacdonald, Phil. Trans, oftheRoyalSoc.ccx.A(1910), pp.131—144,andNicholson, Phil.Mag. (6)xiv. (1907), pp.697—707;(6)xix.(1910), pp.228—249; 516—537; Proc.London Math. Soc.(2) IX.(1911), pp.67—80; (2)xi.(1913), pp.104—126. Aresult concerning Q^+i-Qv, which isclosely connected with(8),hasbeen published byA.Lodge,British AssociationReport, 1906, pp.494—498. 8*2,Theprinciple ofstationary i^hase.Bessel functions ofequalorder andargument. Theprincipleofstationary phasewasformallyenunciated byKelvin* in connexion withaproblemofHydrodynamics, thoughtheessence oftheprinciple istobefound insomemuch earlier workbyStokes fonAiry's integi-al (§6'4) andParse val'sintegral (§2-2),andalsoinaposthumous paper byRiemann^. Theproblem which Kelvin propounded wastofindanapproximate expressionforthe integral ?/=-—/ cos[m[x-tf{m)}] dm, •InJ whichexpresses theeffect atplaceandtime{x,t)ofanimpulsivedisturbance atplaceand time(0,0),whenf{m)isthevelocityofpropagationoftwo-dimensional waves inwater correspondingtoawave-length 27r/??i. Theprincipleofinterference setforthbyStokes *Phil.Mag. (5)xsiii.(1887), pp.•252— 255.[Math, andPhijs. Papers,iv.(1910), pp.303—306.] tCavib. Phil. Trans, ix.(1856), pp.175,183.[Math, andPhijs. Papers,ii.(1883), pp.341,351.] tGes.Math. Werke (Leipzig, 1876), pp.400—406. 230 THEORY OFBESSEL FUNCTIONS[CHAP.VIII andRayleighintheir treatment ofgroup-velocity andwave-velocity suggestedtoKelvin that, forlarge values of.v—tf{m),thepartsoftheintegral outside therange (/x-a,n+a) ofvalues ofmarenegligible onaccount ofinterference if/aisavalue(orthevalue)ofm which makes £^[m{x-tf(,m)}]=0. Intherange (/^-a, /x+a),theexpression 7n{x—tf{m)]isthenreplaced bythe first three terms ofitsexpansion byTaylor's theorem, namely and itisfound that, if* then M'v/[-Mm/"(m) +2/'(m)]' /"cos{<,i2y'(/x) +0-2}c/o- 7rv^f-2r;Mr(M) +2/'(M)}] C0S,V/'(M) +i7r} v/[-27^^[M/"(M) +2/'(;.)}]• Inthelastintegral thelimits foro-,which arelarge eventhough abesmall, havebeen replaced by-ooand+oo . Itwillbeseenfrom theforegoing analysisthat Kelvin'sprinciple is,effectively,thatin thecaseoftheintegral ofarapidly oscillating function,theimportant partoftheintegralis due tothatpart oftherange ofintegration near xohich thephase ofthetrigonometrical function involved isstationary^. Ithassubsequentlybeen noticedJthat itispossibletogiveaformal mathematicalproofofKelvin'sprinciple,foralargeclass ofoscillating functions, byusingBromwich'sgeneralisation §ofanintegralformula due toDirichlet. Theform ofBromwich's theorem which willbeadequatefortheapplica- tions oftheprincipletoBessel functions isasfollows : LetF{x)beafunction ofxwhich haslimited totalfluctuation whenx^O; letybeafunction ofvwhich issuch thatvy^-ccasv-^ oc .Then,if—l<fj,<l, j,Mra;''-^F(x)smvx.dx^F(+0) ft''-'sint.dt=F(+0)T (fi)sin^/jltt;Jo Jo and, if<fji<1,thesinesmayhereplaced bycosinesthroughout. Themethod which hasjustbeenexplainedwillnowbeused toobtain an *This istheappropriate substitution whenm{.x-tf{m))hasaminimum atm=fjL; fora maximum thesignoftheexpression under theradical ischanged. tApersistent search reveals traces oftheuseoftheprincipleinthewritings ofCauchy. See e.g.equation (119) innote 16ofhisTheorie delapropagationdesOndes, crowned Sept. 1815, 21/<?m. presentes pardivers savants,i.(1827). [Oeiivres, (1)i.(1882), p.230.] tProc. Camb. Phil. Soc. xix. (1918), pp.49—55. §Bromwich, Theory ofInfinite Series, §174. 8-2] FUNCTIONS OFLARGE ORDER 231 approximateformula forJv{v) when vislargeandpositive.This formula, which wasdiscovered byCauchy*,is r(i) (1) JA^) ^2^1 1 2^ .3^TTV^ Thisformula hasbeen investigated bymeans oftheprincipleofstationary phase, com- paratively recently, byNicholson, Phil.Mag. (6)xvi.(1909), pp.276—277, andRayleigh, Phil.Mag. (6)xx.(1910), pp.1001—1004[Scientific Papers,v.(1912), pp.617—620];see alsoWatson, Proc. Camb. Phil. Soc.xix.(1918), pp.42—48. From§6"2(4)itisevident that andobviously smvirI g_^,j+sinhi!) ^^ Hence77 .-rrJo J^U)=- I"cos[v(6-sin6)}cie+(l/v). Now let(f)=6—sin0,andthen ,.(0 2But Imi =— , 6l-*.0 1—COSp6" andhence, if(f)'^/{l—cos6)haslimited totalfluctuationintheinterval(0,tt), itfollows fromBromzvich's theorem that "cosi/d) ,, 2r*,_a ,,,^—-a9'^—I9•"cosv(pd(p 1—cos 6^ 6'-^' =r^xr(i)cos^7r, andthen(1)follows atonce. Itstillhastobeprovedthat<^^/(l-cos6)haslimited total fluctuation;toestablish thisresult weobserve that df^s 1_(p-ismeg{d) d~6tl-cos^J~ 3(l-cos^)'-' whelT<;(^)=illl^^^3(^-sin^), sothat5f(0)=0,^(7r-0)=+oo,. ^'(^)=(1-cos^)2/(l+cos^)^0, and therefore, byintegration, g{d)^0 whenO^e^n. Consequently 0*/(l-cos^)is monotonic and itisobviously bounded. The result rpquiredistherefore proved. *Comptes Eendus, xxxviii.(1854), p.993. [Oeuvrcs, (1)xii. (1900), p.663.]Aproof by Cauchy's methods willbegiven in§8"21. 232 THEORY OFBESSEL FUNCTIONS [CHAP. VHI Bymeans ofsome tediousintegrations byparts*,itispossibletoobtain asecondapproximation, namely and itmayalsobeprovedthat (3) JJ(v)=^^^+o(v-i); anassociated formula is (4) r.w~-M). Theasymptotic expansions,ofwhich these resultsgivethedominant terms, willbeinvestigatedwiththeaidofmorepowerful analytical machinery in^8-42. 8'21. Meissel's thirdexpansion. Theintegral justdiscussed hasbeen usedbyCauchyfand Meissel:]: to obtain theformalasymptotic expansionofJnin) when wisalarge integer. Itwillnowbeexplained how thisexpansionwasobtainedbyCauchyand(in amore complete form) byMeissel; thetheoreticaljustificationofthepro- cesses employedwillbeinvestigatedin§8'42. Takingtheformula 1 f'^ Jn{n)=—cos{?i(6—sin^)|cW, letuswrite 6—sin6=^t^;itthen follows that, forsufficientlysmall values oft, /9_/_l_l/3i1/5i_V-x /2m+l.u—i-f^qI-rY4^^t'-r...—•—A„jt, ,w= and Xo=1,Xi=^\, Xa=tioo'^^~ asio^' ^^~ tt^tswuT)' \— 1213. *-5~" 7207200000'•••* Itfollows that Jn(n)=-r\ I(2m+l)X„,tAcos(Ut')^dd.'^.[m=0 JCit> When nislarge, ^nt^islargeattheupper limit, andMeissel inferred that J,,(7i)~-i(2»i+1)X„,.(?rr"cos(^nf)dt, '7r,H=o .0 *SeeProc. Camb. Phil. Soc. xix.(1918), pp.42— i8. tComptes Rendus, xxxviii.(1854), pp.990—993, 1104—1107. [Oeuvres, (1)xii. (1900), pp.161—164, 167—170.] +Astr. Nach. cxxvii.(1891),col.359— 362; cxxviii.(1891),col.145—154. Concerning formula(1),Meissel stated "Sclion vordreissig Jahren warichzufolgenden Formel gelangt." 8-21, 8-22] FUNCTIONS OFLARGE ORDER 233 whereGisthesignindicatinga"generalised integral" (§6'4); andhence, by integrating term-by-termandusingEuler's formula, Meissel deduced that (1) /.(n)--:i:X.,r(§m +f)-) cos(im-hi)7r. Meissel alsogaveanapproximationforA,„,validwhenmislarge; andthisapproxima- tionexhibits thedivergentcharacter oftheexpansion (1). Theapproximationisobtainable bythetheory developedinthememoir ofDarboux, "Sur I'approximationdesfonctions detr5sgrands nombres," Joiirnal deMath.(3)iv.(1878), pp.5—56, 377—416. AVeconsider thesingularitiesof6quafunction oft;thesingularities (where 6fails to bemonogenic) arethepointsatwhich 6^=2/-7rand ^=(12r7r)5, where/=±1,±2,±3, ...; andnear* t=±(127r)^thedominant terms intheexpansionof6are ±27r+(367r)3il +t}^ \(127r)-^i BythetheoryofDarboux, anapproximationtoX,histhesum ofthecoefi&cients of fim+ijritheexpansionsofthetwofunctions comprisedinthelastformula;that istosay that 12 1*.*.«... (2/n-i) \,„~2 .(367r)*-^'^' ^^ (2m+1)!(12,7)*'"+^ 2r(2?n,-l-§) sfm'3^r(t)r(2?rt +2).(127r and so,byStirling's formula, 1 (^) in~ 1 4 2j„ (18)^r(f)(m+l)Ml^'r)^"' This isMeissel's approximation;anapproximationofthesame character wasobtained byCauchy,loc.cit., p.1106. 8"22.Thecqyplication ofKelvins'principletoJ^,(vsec^). Theprincipleofstationary phasehasbeenapplied byRayleighftoobtain anapproximateformula forJ^{vsec/3)where/3isafixedpositiveacuteangle, andVislarge I. Asin§8*2wehave J,(vsecl3)=-fcos [v(6-sec/3sin6)}cW+Oil Iv),/^ TJ". and6—sec/3sin6isstationary (aminimum) when 6=j3. Write 6—sec13sin6^^—tan/3-t-</>,sothat^decreases tozero as6in- creases from to/3andthen increases as6increases from/3tott. *These arethesingularities which arenearest totheorigin, tPhil.Mag. (G)xx.(1910), p.1004. [Scientific Papers,v.(1912), p.620.] JSeealsoMacdouald, Phil. Trans, oftheRoijalSoc. ccx.A(1910), pp.131—144; andProc. RoyalSoc. lxxi.(1903), pp.251—258; lxsii.(1904), pp.59—68. 234 THEORY OFBESSEL FUNCTIONS [CHAP.VIII ^Now cos{v(6—sec/3sin6)]dd /"T-^+tan^" + _^tan^-/3 JO(10 COS{i^(<^+^—tan/3)}j-.dcfi. and^^dd I_1 rf</)"^V(2tany8)as^—/S. Hence, t/(^*(dd/dcf))haslimited totalfluctuationintherange0^0 f^7r,it follows from BroniwicK stheorem that pcos[v(0-sec/3sin6)]dd~2Tcos[v(</>+/3-tan/3)}//gl-^^^ .cos[v(tan^—/3)-i tt], vi/tan/3 andso ^cos{v(tan /3-^)-^tt} (1) /.(.sec^)^V(i.^tan^)' Theformula (2) F.(.secffl~ ^'°'-'<;,f^r^^:'"'•^ >'\ r-/ v(2-i''''tan;S) isderived inasimilar manner from§6"21(1). Thereader willobserve that these arethedominant terms inMeissel's expansions §8'12(5), (6). Tocompletetherigorous proofofthese formulae wehave toshew that(f)^{dd/d(j>) has limited total fluctuation. Now thesquareofthisfunction, namely {d6ld(f))^,isequalto ^-sec/3sin^-3+tan^_ (1-sec/3cos(9)2 say.But cos3cosec ^(1-sec/3cos6)"^-2((9-sec /Ssin 6-/3+tan^ cos/3cosee ^(1-sec(3cos6)^ Thenumerator, k(d),ofthisfraction hasthedifferential coefficient —cos/3cos6cosec^ ^(1-sec/3cos6)^, andsok{6) decreasessteadilyasBincreases from to^tt,andthen increasessteadily asBincreases from^ttto tt;since k{B)=when B=^<^7r,itfollows thath'{B)^0 when ^^^^and h'{B)changes sign once (from negativetopositive)intherange 0t$<9^7r. Hence |Jh(B) |ismonotonic (and decreasing) when ^^^^,and ithasonestationary point (aminimum) intherange ^<.B<tt ;since !s,fh{6) \isbounded andcontinuous when $^TT itconsequentlyhaslimited total fluctuation when ^B^ir,ashadtobe proved. 8-3] FUNCTIONS OFLARGE ORDER 235 8"3.Themethodofsteepestdescents. Adevelopmentofthetheoryofcontourintegration,called themethod of steepest descents*, hasbeenapplied byDebyeftoobtainintegral representa- tions ofBessel functions oflargeorder fromwhichasymptotic expansionsare readilydeduced. If,ingeneral, weconsider theintegral inwhich ji' |issupposedtobelarge,thecontour ischosen sothat itpasses throughapoint iUqatwhich/'(w)vanishes;andthewhole ofthecontour is then determinedbytheassumptionthat theimaginary partoff{w)isto beconstant onit,sothat theequationofthecontour maybewritten in theform If(w)=If{io,). Toobtain ageometrical conceptionofthecontour, let lu=u+iv,where u,v arereal;anddraw thesurface such that thethree coordinates ofanypoint onitare u,V,Rf(iv). IfRf{io)=z,and ifthe^-axis besupposedtobevertical, thesurface hasno absolute maxima orminimaexceptwhere /(w)fails tobemonogenic; for,at allotherpoints, Thepoints [wq, v^,i^'(^o)]aresaddlepoints,orpasses,onthesurface, sothat thecontour ofintegrationistheplanofacurve onthesurface whichgoes throughoneofthepassesonthe surface. This curvepossessesafurther propertyderived from theequationofthecontour;fortherateofchangeof f{w),atanygivenvalue oftv,hasadefinite modulus, since/(w)issupposed tobemonogenic ;andsinceIf{'w)doesnotchangeaslutraverses thecontour, itfollows thatRf{iv) mustchangeasrapidlyaspossible;that istosay,that thecurve ischaracterisedbythepropertythat itsdirection, atanypointof it,issochosen that itisthesteepestcurvethroughthatpointandonthe surface. Itmayhappenthatwehave afreedom ofchoice inselectingapassand tircn inselectingacontourthroughthatpass;ourchoice istobedetermined from theconsideration thatthecurve must descend onboth sides ofthepass; for ifthecurve ascended, Rf{w)would tend to+oo(exceptinvery special cases)as%uleftthepass,andthen theintegralwoulddivergeifi^{v)>0. *French "Methode duCol,"German "Methode derSattelpimkte." t2Iath. Ann. lxvii. (1909), pp.535—558; Munchener Sitzum/sberichte,xl.[5],(1910). The method istobetraced toaposthumous paper byEiemann, Werke, p.405;and ithasrecently been appliedtoobtain asymptotic expansions ofavariety offunctions. 236 THEORY OFBESSEL FUNCTIONS [CHAP.VIII Thecontour hasnowbeen selected* sothat theintegranddoes not oscillaterapidlyonit;andsowemayexpectthatanapproximatevalue of theintegralwillbedetermined from aconsideration oftheintegrandinthe neighbourhoodofthepass:from thephysical pointofview,wehaveevaded theinterference effects(cf.§8'2)which occur withanyothertypeofcontour. Themode ofderivation ofasymptotic expansionsfrom theintegralwillbe seenclearlyfrom thespecialfunctions which willbestudied in§§8'4—8'43, 8'6,8"61;but itisconvenient toenunciate atthisstagealemma fwhich will beusefulsubsequentlyinprovingthattheexpansionswhich willbeobtained areasymptoticinthesense ofPoiucare. Lemma. LetF{t) beanalytic when\t\r^a+8,luhere a>0,B>0;and let 00 m=l when IT I^a,rbeing positive; also, let\F(t)\< Ke^'', luhereKand bare positive numbersindependent oft,ivhen rispositive andr^a.Then the asymjjtotic expansion e-"-"F(t)dr'^:i:a,nT(m/r)v-'^"' Jo m=l isvalid inthesenseofPoincare when\v\is sufficiently largeand 1argz/ !^^TT-A, whereAisanarbitrary positive number. Itisevident that, ifMbeanyfixedinteger,aconstant K^canbefound such that M-l m=1 whenever r^whether t^aorr^a;andtherefore e-^'F{T)dT= 2 e-''^a„iT<«*/^'-^dT +i2j/, ^0 ni=l J where\Rm\^\\e-"'' \.K^r^^^'^^-^ e^^dr JO =K,T{Mfr)/{R(v)-b]^/^ providedthatR{v)>b,which isthecasewhen\v\>bcosec A.Theanalysis remains valid evenwhen 6isafunction ofvsuch thatR{v)—bisnotsmall comparedwith v.Wehave thereforeprovedthat /•"31-1 e-''F{r) dT=%a^T(m/r)v-"^''+ {y-^^l"-),Jo j)i=l andsothelemma isestablished. *Foranaccount ofresearches inwhich thecontour istherealaxis seepp.1343—1350 of Burkhardt's article intheEncyclopddie derMath. Wiss. ii.1(1916). tCf.Proc.London Math. Sac.(2)xvii.(1918), p.133. 8-31] FUNCTIONS OFLARGE ORDER 237 8'31. Theconstruction ofDebyes contours* when thevariables arereal. Ithasbeenseen in§§6-2,6'21thatthevarioustypesoffunctions associated with J^{x)canberepresented byintegralsoftheform takenalongsuitable contours. Onthehypothesisthat vandxarepositive, weshall now^examine whether anyofthecontoursappropriateforthe method ofsteepestdescents areofthetypes investigatedin§|6'2,6"21. Inaccordance with theprinciplesofthemethod ofsteepest descents, as explainedin§8*3,wehave first tofindthestationary pointsof Xsinhw—viv, quafunction of^v,i.e.wehave tosolve theequation (1) Xcoshw—v={)\ and itisatonce seen thatweshall have three distinct cases toconsider, inwhichxjvislessthan, greater than, orequalto1,respectively. Wecon- sider these three cases inturn. (I)Whenxjv<1,w^ecanfindapositive number asuch that (2)a;=z/sechct, andthen thecompletesolution of(1)is w=+a+2?i7rt. Itwillbesufficient toconfine ourattention tothestationary pointsf ±a;at thesepointstheimaginary partofxsinh lu—vwiszero,andsotheequation ofthecontour tobediscussed is /{xsinhw—viv)=0. W^rite iv=it+iv,where u,varereal,and thisequation becomes coshusinv—vcosha=0, sothat V=0,or ,vcosha (S) coshw=— -. . siU'y /-'Thecontour v=givesadivergent integral. Wetherefore consider the contourgiven byequation (3).Tovalues ofvbetween andtt,corre- spond pairsofvalues of itwhich areequalbutoppositeinsign;andasv increases from tott,thepositivevalue ofusteadilyincreases from otto+x . *Thecontours investigated inthissection arethose which were discussed inDebye'searlier paper, Math. Ann. i.xvii.(1909), pp.535—558,except that their orientation isdifferent; cf.§6-21. +The effect oftaking stationary points other than±awould betotranslate thecontour paralleltotheimaginary axis. 238 THEORY OFBESSEL FUNCTIONS [chap.VIII Theequationisunaltered bychangingthesignofvand sothecontour is symmetricalwithregardtotheaxes ;theshapeofthepartofthecontour between v=—ttandv=ttisshewn inFig.16. Fig. 16. If T=sinha—acosha—(sinhw—wcosha), itiseasytoverifythatt(whichisrealonthecurves shewn inthefigure) increases inthedirections indicatedbythearrows. Aswtravelsalongthecontour fromx—iritooc+tti,tdecreases from +00toandthen increases to+x;andsince, by§6"2(3), 27rtJoo-,ri wehave obtained acurve fromwhich wecanderive informationconcerning J^(cc) when ccand varelargeandxlv< 1.The detailed discussion ofthe integralwillbegiven subsequentlyin§§8'4, 8*5. Thecontours from—xtox+irigiveinformationconcerningasecond solution ofBessel'sequation;but thisproblemiscomplicated byStokes' phenomenon,onaccount ofthetwostationary pointsonthecontour. (II)Whena:jv>1,wecanfindapositiveacuteangle /3such that (4)a;=v sec/9, andtherelevantstationary points,which arenow roots ofthee<|uation coshw—cosyS=0, are tv—± i/3. When wetakethestationary point i/3,thecontour which weobtain is /(sinhtu—10cos/3)=sin/3—/3cos/3, sothat, replacingivbyu+iv,theequationofthecontour is sin/3+(y—/3)cos^ (5)cosh li= sin?; 8-31] FUNCTIONS OFLARGE ORDER 239 Now, forvalues ofvbetween and tt,thefunction sin^+(v—/3)cos /3—sinv hasoneminimum(v=/3)atwhich thevalue ofthefunction iszero ;forother values ofvbetween and tt, sin/3+(i'—/3)cos l3>sin v. Hence, forvalues ofvbetween andtt,equation (5)gives tworealvalues ofu(equalbutoppositeinsign),andthese coincideonlywhen v=/3.They areinfinite when t;isortt. Theshapeofthecurvesgiven byequation (5) isasshewn intheupper halfofFig.17;and if T=i(sin /3—y8cos/3)—(sinhw—wcos13), itiseasytoverifythat r(whichisrealonthecurves) increases inthe directions indicatedbythearrows. Aswtravelsalongthecontour from—x to00+iri,Tdecreases from+ootoandthen increases to+ooand sowe Fig. 17. have obtained acurve fromwhich[§6"21(4)]wecanderive information con- cerningZT;,'^' (.*•)when xand varelargeandx/v>1.Thedetailed discussion ofthe"integralwillbegivenin§§8'41, 15'8. Ifwehadtaken thestationary point —i/3,weshould have obtained the curves shewn inthelower halfofFig. 17,andthecurvegoingfrom—octo 00—TTigivesanintegralassociated with HJ-^ (w) ;this also willbediscussed in§8"41. Thetwointegralsnowobtained form afundamental systemof solutions ofBessel'sequation,sothatthere isamarked distinction between thecasex/v<1andthecasexjv>1. 240 THEORY OFBESSEL FUNCTIONS [chap.VIII (III) Thecase inwhich v=xmaybederived asalimitingcase either from(I)orfrom(II)bytakingaor/3equalto0.Thecurves now tobecon- sidered arev—and (6) coshu=v/sin v, andtheyareshewn inFig.18. /\ 8-32, 8-4] FUNCTIONS OFLARGE ORDER 241 Again,toprovethatdvjdu doesnotexceedy/S,wewrite siny' andthen itissufficient toprove that 3^/.'2(^>)-^^2(•i;) +l^0. Now theexpressionontheleft(whicli vanishes when v=^)hasthederivate =J^3''^ [(t'- ,3){sin2(.+3cos-v]cos/3+sin^ ysin^-3cosvsin{v- /3)]. ly.,a\ ,,,sin2i7sin/3-3cos vsin(i;-/a)But (v-B)cos,B-\ r--i-' ^^' ^sm^v+3cos2i; ,,, -x- J•X4sin*«;cos^ ,hasthepositive denvate -^~,—~z5-^.,,andso,since itisixjsitive when v=0,itis (sin- i'-(-3cos^ y)-•^,vi^ positive when0<v <tt. Therefore, sincey\r'(v)hasthesamesignan0-(3,itfollows that hasthesamesignasv- /3,andconsequently has v=l3foritsonlyminimum Ijetween v=and v=tt;andtherefore itisnotneo-ative. Thisprovestheresult stated. 8"4.Theasymptotic expansion* ofJ^{vsecha). From theresults obtained in^8'31weshallnowobtain theasymptotic expansionofthefunction ofthe firstkind inwhich theargumentislessthan theorder, bothbeing largeandpositive. Weretain thenotation of§8'31(I) ;and itisclear that,correspondingto anypositivevalue ofr,there aretwovalues ofw,which willbecalled u\and Wo;thevalues ofWjand w.,differonlyinthesignoftheirimaginary part,and itwillbesupposedthat l{w,)>0, /0"o)<0. Wethenhave ./,{vsecha)=^^g-*'^^--7-7 ^t, 'Itti J(dr drj where x=vsech a. Nextwediscuss theexpansionsofWiandtu.^inascending powersofr. SintJe Tanddrjdtv vanish when lu=a,itfollows that theexpansionofrin powersoflu—abeginswith aterm in{w—a.)-;byrevertingthisexpansion, weobtainexpansionsoftheform „i=oW+l- „,=o m+l* *Theasymptotic expansions contained inthis section and inSi8'41, 8-42were established byDebye, Math. Ami.lxvu. (1909), pp.535— 55B. w.B.F. 16 242 THEORY OFBESSEL FUNCTIONS [chap.VIII and,byLagrange's theorem, theseexpansionsarevalid forsufficientlysmall values of ITI.Moreover 1r(0+'0+V^wA dr a,n= 27rimwA ['d^Jri{m+l) =JLf27nJ("+)dw Thedouble circuit inther-planeisnecessaryinorder todisposeofthe fractionalpowersofr;andasinglecircuit round ocinthew-plane corresponds toadouble circuit round theorigininther-plane. From thelastcontour integralitfollows that a^nisthecoefficient ofl/{w—a)intheexpansionof ^-iiin+i) jjjascending powersofw—a;wearethusenabled tocalculate the coefficientsa,,,,. Writew—0.=IFandwehave T=—sinha(coshW—1)—cosha(sinhTT—Tf) =TFHco+CiTf+c,3F^+...), whereCo=—|sinh a,Ci=—|cosh a,c^=—^^sinh a, Therefore a„,isthe coefficient ofTf'"intheexpansionof{co+CiTf+c,W^+...)-*(™+i'. The coefficients inthisexpansionwillbecalledao(m), ai(?)i), azivi), ..., andsowehave ao(m)=Co-^("'+^), m+1Gi] "271!-^o; (1)771+1C.2(m+1)(m+S)Cf 2.1! 'co 2^2! m+1Cs (?)i+l)(m+3)2ciC2 2^2! Co' (m+l)(m+3)(/?i+5)d^ 2^3! a,{m)-c,- ^2.1! -Co 2^2! VCo' CoCoV3{' 2^3!Co-' +(m+1)(?H+3)(m+5)(m+7)Cj^ 2*.4! 'c* (2)Onsubstitution wefindthat faQ=a<^ (0)=+(—1sinha)~*, IOi=«!(1)=—(—Isinhot)~* {Icotha}, Itto=do(2)=—(—1sinh a)~^ {^- -j^coth^a}, Itta=«:;(3)=—(—Isinh a)~^ {y^^cotha—^jcoth*a}, \a,=a,(4)=+(-isinha)-t{y^g- /Jj.coth^a +^%%\ coth^a}, 8-4j FUNCTIONS OFLARGE ORDER 243 Now^^-j,^=Sa„„T'«-i CIT,„=o when IT,issufficientlysmall;andsince -j—=cosha—cosh iv,aw itfollows thatd(wi—w.^jdr tends tozero asttends to+x . Hence theconditions stated inthelemma ofS8'3aresatisfied, andso I. [dr dr) hastheasymptotic expansion ^g.,,,r(m+I) ^uhenXislarge.JH= ^ Sincearg \{%i\—o)It]-^^ttasr-*0,itfollows that, in(2),thephaseofOg hastobeinterpreted bytheconventionargao=+I'^j^iidhence (3) ^.(.secha)~ ^^^^ ^^^j^-^^^-^_-^^- • (i,tanh«)-' where fx-Io=1,^1=i-21coth- a, (^) ^^.,=yfg- ^T-Vcoth^ «+1^:^coth^ a. Theformula(8)givestheasymptotic expansionof,/^(i/secha)validwhen aisanyfixedpositive number and vislargeandpositive. Thecorresponding expansionforthefunction ofthesecond kind, obtained bytakinga contour from—ootoqo+tti,is (5)i^(,secha)~--^^^— ^^^^^ ^^2^-j^-^ -(l.tanhar'. Thepositionofthesingularitiesofd(u\-iVo);dT, quafunction ofthe complexvariable r,should benoted. Thesesingularities correspondtothe points wherewfails tobeamonogenicfunction oft,i.e.thepointswhere dr/dw vanishes. Hence thesingularities correspondtothevalues+a+2?27ri ofw,sotheyarethepointswhere /- T=2n7Ticosh a,r=2(sinha-acosha)+2nnicosh a, and nassumes allintegralvalues. Itisconvenient toobtain aformula fordto/drintheform ofacontour integral;if (wo, To)beapairofcorrespondingvalues of{tv,t),then, byCauchy's theorem, fdiv\_1/(>•«+)dwdr_J_i"(«'«+)di^ \Q?r/o27ri_/drt-t^ 2niJt—To where thecontour includes nopoint (except Wq)atwhich thasthevalue ry. IG—2 244 THEORY OFBESSEL FUNCTIONS [CHAP.VIII 8'41. Theasymptotic expansions ofJ„{vsec/8)andY^{vsec^). In§8"4weobtained theasymptotic expansionofaBessel function in which theargumentwas lessthan theorder, bothbeing large;weshallnow obtain theasymptotic expansionsofafundamentalsystemofsolutions of Bessel's equationwhen theargumentisgreater thantheorder, bothbeing large. Weretain thenotation of§8"31(II) ;itisclear that,correspondingtoany positivevalue oft,there aretwovalues ofwlyingonthecontour which passesfrom—ootogo+7^^; these values willbecalled w^andw^,and itwill besupposedthatR(wj)>0,R(wo)<0. Wethenhave HJ^^^ (vsecyQ)= -. e-*"\~ r^\dr, TTi Ji) [ar dr) Avhere x=v8ec^. Theanalysis nowproceeds exactlyonthelines of§8*4 exceptthataisreplaced throughout by i/3,andtheBessel function isofthe third kind. Itisthusfound that ^{div.Q-Xr \dr dr] „t=o af"^^ Todetermine thephaseofa^,that isof(—|tsin/3)~*, weobserve that arg [{iL\—i^)lr]-->-+5TTasT^^0,andso a„=ei'VV(ism/3). Consequently g..i(taii/3-^)-i« «r(m+|) Aj (1) ir,»>(i/sec/3)^^m ^'i^virtanyS)^.^oT{\)' {\vitan/8)-" Inlikemanner, bytakingascontour thereflexion oftheprecedingcontour intherealaxisofthew-plane, wefindthat (2)*H(^)(.secB)^ri*!-^^^!!^ ^r(m+|) A^ Inthese formulae, which arevalidwhen isafixedpositiveacuteangle and Vislargeandpositive, wehave tomake thesubstitutions : Ifwecombine(1)and(2),wefindthat (4) J^{vsecyS)-^ 2 virtan/3/.oo 1^x(-y"r(2m +|)A cos(i/tany9-^/3-|7r) Z2m m=o r(i) '(iJ/tanye)-"' +sin(.tan^-.^-l.)J/^^iM).^^^_^J^,^ 8-41, 8-42] FUNCTIONS OFLARGE ORDER 245 (5) F„(i'sec/9)~ (2-^^r•/f /D o 1 .V(-)'"r(2m +i)A^ — ; 7,sin(z/tan/3- z^/3- 1tt)S-^—=^--^ .-. ^--- - Thedominant terms intheseexpansionsarethose obtainedbytheprinciple ofstationary phasein§8*21 . 8*42.Asymptotic expansions ofBessel functions whose orderandargument arenearly equal. Theformulae which havebeen established in§§8'4,8"41obviouslyfail togiveadequate approximations when a(or/S)issmall, that iswhen the argument andorder oftheBessel function concerned arenearly equal.Itis, however, possibletousethesame method fordetermining asymptoticex- pansionsinthese circumstances, and itha]3pensthatnocomplicationsariseby supposingthevariables tobecomplex. Accordingly weshall discuss thefunctions where zand varecomplex numbers oflarge modulus, such that \z—v\is notlarge.Itwillappearthat itisnecessarytoassume that z—v=o(z^),in order thattheterms oflowrank intheexpansions maybesmall. Weshall write i'=z(l-e), and itisconvenient tosuppose temporarilythat |arg^!<|7r. Wethenhave (1) iy„^'* (z)=—. jexp [z(sinhlu-w)+zeiv] dtu, where thecontour isthatshewn inFig.18;onthiscontour sinhw—wisreal andnegative. Wewrite T=w—sinhw, andthevalues ofwcorrespondingtoanypositivevalue ofrwillbecalledlUi andW2,ofwhich w^isacomplex number with apositiverealpart,and w.,is arealnegativenumber. Wethenhave (2) H^^'' (z)=— . e-'"- jexp(zeiu^)-~-exp{zew.^-j^)dr. 246 THEORY OFBESSEL FUNCTIONS [chap. VIII Theexpansionoftinpowersofwbeginswith aterm inlu^,andhencewe obtainexpansionsoftheform exp {zeiu^)^=T-s2h,nt^"*, ttr ,«=o don'^ exp(^6m,)5^'=T-32ei('«+i'-'6,„TH (IT 7H= andthese arevalidwhen [t jissufficientlysmall. Todetermine thecoefficients b^aweobserve that 1/•(0+,0+,0+); ^^^yjS ^^ Gtti ^^P^"^^^^>-(l^)ri(m+i) ~ 67ri]^""P ^'''''^(w-sinhiv)i<'«+^)• Asintheanalogous investigationof^8'4,asinglecircuit inther-plane isinadequate,andthetriplecircuit isnecessarytodisposeofthefractional powersoft;atriplecircuit round theorigininther-plane corresponds toasinglecircuit inthe?y-plane. Itfollows that bniisequaltoigJc^+D^imultiplied bythecoefficient ofw'" intheexpansionof exp(zeiu).{(sinhlu—w)/^'}"*^"*^^). Thecoefficients inthisexpansionwillbecalled6o(//i), 6i(m), b.^im),...sothat Itiseasytoshew that fbo{m)=6^^'^+'\ (3)~2~ 60J' e^z^ (vi+l)ez 6 60 o.K^u) o 124 120^.50400 J" Forbrevitywewrite b,,{m)=m^+'^Bm{€z\ 8-42] sothat* (4)FUNCTIONS OFLARGE ORDER 247 ^B,{ez)=i-,^'z^-i-^e-^z' +^^, B,{ez)=j^^z^- ij^e'z^+«|toe^>By{ez)=ez, B,{ez)=h^'z'-j'-Eez, [Bs{0)=— -^^Sjjj, Bs{0)—jjiisooo' -'°io(0)— 655-ioOOOO-J Wethenhave diUi00 ex^(zewyp=lT-i2e^^+'^-^6''"'+'^ B,,,{ez)Ti''\dr dw„»i= 00 exp(zew.)-p=1T-'2e<'»+i)-'' 6i('»+^»5«(e^) r^'", and[exTp (zeiv).(dw/dT)]satisfies theconditions ofthelemma of§8'3. Itfollows from thelemma of58'3that i>00 (5)^,w (^)~-o^Se3<-+»'^^ B,,(ez)smh{m +l)7r.,^1'^, andsimilarly (6)H^^H^)^-^Se-3(-+i)-5,,(6^)sini(^>i-Hl)7r.-^^^^^ Wededuce atonce that (V) J.{z)1'^ram +1) (8) i^(^)~-o;l^(-r^5,„Msin4(m +l)7r.-4^^^ From theCauchy-Meisselformula§8-21(2),itistobeinferred that,when 1)1islarge, (_)m (-|)s ^^^ ^^"'^^^" r(§).(7/i +i)Hl27r)3'"' butthere seems tobenoverysimple approximateformula for5„i{ez). Thedominant terms in(7)were obtained byMeissel, inaKielProgrammi, 1892; andsome similarresults, which seem toresemble those stated in§8-43,were obtained by KoppeinaBerlin Programml,1899. Thedominant terms in(8)aswell asin(7)were alsoinvestigated byNicholson,Phil.Mag. (6)xvi. (1908), pp.271—279, shortlybefore the ap^aranceofDebye's memoir. *The values ofBo(0), />'2(0),...L'lo(0)were given byMeissel, Astr. Nach. cxxvii.(1891), col.3.59—362; apart from theuseofthecontours Meissel's analysis (of.§8-21)issubstautially thesame astheanalysis giveninthis section. The objectofusing themethods ofcontour integrationistoevade thedifficulties produced byusing generalised integrals. Thevalues of.Bo{ez), B-,[ez]andB^(ez)willbefound inapaper byAirey, Phil. M(h;. (0)xxxi. (1916), p.524. tSeetheJahrhuch ilber dieFortschritte derMath. 1892, pp.476—478. XSeetheJahrhuch iiber dieFortxcliritte derMath. 1899, pp.420, 421. 248 THEORY OFBESSEL FUNCTIONS [CHAP.VHI Wenext consider theextent towhich theconditionjarg2^|<^7r,which hassofarbeenimposedonformulae(5)—(8),isremovable. Thesingularitiesoftheintegrandin(2),quafunction oft,arethevalues ofTforwhich Wi(orw^)fails tobeamonogenicfunction oft,sothatthe singularitiesarethevalues oftcorrespondingtothose values ofwforwhich dr/dw=0. Theyaretherefore thepoints T=2u7ri, where nassumes allintegralvalues. Itisconsequently permissibletoswingthecontourthrough anyangle t) lessthan aright angle (either positivelyornegatively),andwethen obtain theanalyticcontinuation of^^'^' (z)or H^^'-^ (z)over therange—^TT— 77<arg^:<^TT— 7].Bygiving 77suitable values, wethus findthat theexpansions (5)—(8)arevalid overtheextendedregion —TT<argz<TT. Ifweconfine ourattention torealvariables, weseethatthesolution oftheproblemis notquite complete ;wehavedeterminedasymptotic expansions ofJ^(x)validwhenxand Varelargeand(i)xjv<1,(ii)x/v>1,(iii) |^- i* |notlargecompared with x^.Butthere aretransitionalregions between(i)and(iii)and alsobetween(ii)and(iii),and inthese transitionalregions xjvisnearly equalto1while\x—v\islarge. Inthese transitional regions simple expansions (involving elementary functionsonlyineachterm) donotexist. Butimportant approximate formulae have been discovered byNicholson, which involve Bessel functions oforders ±J.Formulae ofthistypewillnowbeinvestigated. 8'43.Approccimate formidaevalid inthetransitionalregions. The failure oftheformulae of§§8*4—8'42 inthetransitionalregionsled Nicholson* toinvestigatesecondapproximationstoBessel'sintegralinthe following manner : Inthecaseoffunctions ofintegralorder n, Jn(^)=-COS{nd—Xsin6)dO, and,when xandnarenearly equal (both being large),itfollows from Kelvin's principleofstationary phase (§8*2)that theimportant partofthepathof integrationisthepartonwhich 6issmall;now,onthispartofthepath, sin6isapproximately equalto^—\6'^.Itisinferred that, forthevalues of Xandnunder consideration, Jn{x)'^- rcos{nd-xd+IxO')dd IT}Q -rQos{ne-xd-^lxe^)dd, TJ"JoTTJO *Phil.Mag. (6)xix.(1910), pp.247—249; seealsoEmde, Archiv derMath, undPhys. (3) XXIV.(1916), pp.239—250. 8-43] FUNCTIONS OFLARGE ORDER 249 andthe lastexpressionisoneofAiry's integrals (§6*4).Itfollows that, when X<n, and,when x>n, (2)Jn{x)^^\^ ^^ j{/_i+J:.i, where theargumentsoftheBessel functions ontherightare^{2{x— )i)|=/a,'i. Thecorrespondingformula forF„{x)whenx>nwas alsofoundby Nicholson;with thenotation employedinthiswork itis (3) F,.(.)~-j^i^l'{J_,-J,}.3^ Thechiefdisadvantageofthese formulae isthat itseemsimpossibleto determine, byrigorous methods, their domains ofvalidityandtheorder of magnitudeoftheerrors introduced inusingthem. With aview toremedyingthis defect, Watson* examinedDebye.'s integrals,anddiscovered amethod which istheoretically simple (though actuallyitisvery laborious), bymeans ofwhich formulaeanalogousto Nicholson's areobtainedtogetherwithanupperlimit fortheerrors involved. Themethodemployedisthefollowing: Debye's integralforaBessel iunction whose order vexceeds itsargument x{=i^sech a)maybewritten intheformf J^{vsecha)=—-— .—/ e"^'' div, -iTTt JIX,-TTI where t=—sinh a(coshlu—1)—cosha(sinhw— iv), thecontourbeingchosen sothattispositiveon it. IfTisexpandedinascending powersofw,Carlini's formula isobtained when weapproximate byneglectingallpowersoflusave the lowest, —|ry"^sinha; andwhen a=0,Cauchy'sfornnila of§82(1)issimilarlyob- tained byneglectingallpowersofwsavethelowest,—\iv^. These considerationssuggestthat itisdesirable toexamine whether the firsttwoterms, namely—^w'-sinha—^vfcosh a, nlaynotgiveanapproximationvalidthroughoutthe first transitionalregion. Theintegralwhich weshallinvestigateistherefore where r=—|IF-sinha-^W'^cosh a, *Proc. Camb. Phil. Soe. xix.(1918), pp.96—110. tThis isdeducible from §8-31 bymaking achangeoforigininthe«--plane. 250 THEORY OFBESSEL FUNCTIONS [chap, vin andthecontour intheplaneofthecomplexvariableWissochosen thatris positiveonit.IfTFsU-{-iV, thiscontour istheright-handbranch ofthe hyperbola C^tanha +it^^=iF^ andthiscurve hascontact ofthethird order withDebye'scontour attheorigin. Ittherefore hastobeshewn thatanapproximationto CO+7ri /"ocexpCaTrO e-^'Uhv is e-''^dW. Theseintegralsdiffer by ro XJo) andsotheproblemisreduced tothedetermination ofanupperbound for \[d{tu —W)ldT]\. And ithasbeenproved, byexceedingly heavy analysis which willnotbereproduced here, that d{lu-W) andsodi<Sttsech a, Hence(divdW],iGtt {dr dT \ V xexp(i7r2)dW^-^,V where |^j |<1. Toevaluate theintegralontheright (whichisofthetypediscussed in §6"4),modifythecontour intotwo linesstartingfrom thepointatwhich W=—tanh aandmaking angles ±\'ttwith thereal axis. IfwewriteW—— tanha+^q-^I-^^ ontherespective rays,theintegralbecomes e^"^exp{\vtanh^a)fexp{-\v^'-\ v^e^'""'tanh-o}d^ Jo —e~^''exp(^Vtanh^a)/exp{—^v^^—Iv^e~^"^tanh'^a\d^. Jo Expandtheintegrandsinpowersoftanh- aandintegrate term-by-term —a procedure which iseasily justified—andwegetonreduction §77*tanh aexp (^i^tanh* a).[/_i{^vtanh^ a)—I^(^vtanh^*a)], andhence weobtain theformula tanha (4) Jt,{v secha)= 7r\/3exp [v(tanh a+^tanh=^ a—<x)}K^(lvtanh^a) +8^ii^~'exp {i/(tanhoc— a)}, where [^i |<1.This isthemorepreciseform ofNicholson's approximation (1). 8-43] FUNCTIONS OFLARGE ORDER 251 Itcanbeshewn that,whether Ji/tanh^'a besmall, ofamoderate size, or large,theerror isofasmaller order ofmagnitude (whenvislarge)than the approximation given bythe firsttermontheright. Nextwetake thecase inwhich theorder vislessthan theargument x{=vsecyS). Wethenhave ir^<" {vsec/3)= : e-^^dw, ITl J-cc-i^ where t=—tsin /3(coshlu—1)—cos^(sinhlu— lu), thecontour beingsochosen thatrispositiveon it. Theprocessofreasoning already employedleads ustoconsider theintegral where t=—|tW-sin/3-^Tf^cos/3, andthecontour intheplaneofthecomplexvariableWissuch that ris positiveon it.IfW=11+ iV,thiscontour isthebranch ofthecubic (U-'_V^)tan/3+^F(3^7^-7^= whichpassesfrom—go—*tan ySthroughtheorigintoxexp^Trt. Ittherefore hastobeshewn thatanapproximationto• e-^'^dv) is e-'^UlW, J—CO-/^ J-00-itan^ Thedifference ofthese integralsis 00 ./o J {drdr and ithasbeenproved that,when*fi^{tt,then ' -<12'irsecp.di Hence itfollows that -1rtc+i{n-^) \rccexpiir/ 24^' -. e-'^^dw =-~.\e-'^UlW +, w>ere |^' |<1. Toevaluate theintegralontheright, modifythecontour intotwo lines meetingatW=- itan^andinclined atangles4ttand ttrespectivelytothe real axis.Onthese lines, write Tf=-itany8-^,-itan/3+^e^''^ *Theimportantvalues of/3are,ofcourse, small values. If^isnotsmall, Debye's formulae of§8-41 yieldeffective approximations. Thegeometrical propertyofDebye'scontour which was provedin§8-32 isused intheproofofthetheorem quoted. 252 THEORY OFBESSEL EUNCTIONS [CHAP.VIII expandtheintegrandsinpowersoftan-/3,integrate term-by-term,and itis found that /OOeXp jTTt g-^""dW— ^-rritan/8exp(—ivitan*/Q)—00—/tail;3 X[e-i-' J_j (ii/tan*/S)-he*'^'/i(^i^tan*yS)] Tre^-^tan^ ^^^^_^^^.^^^^^,^^,^,^^^^,^^3^^_V3 Onequatingrealandimaginary parts,itisatoncefound that (5) J^{vsecyS)=itan/3cos[v(tan /S-^tan* /3- /3)}.[/-j-f/j] +:3-Han/3sin{i/(tan /3-1tan*/3- yS)|.[/- j-J{\+24^^/^', (6) F^(i^sec13)=itanyQsin[v(tan yS-1tan*/8- /3)}.[./_1-HJ^.] -3-*tan /3cos{1/(tan /3-1tan*/S- /9)}.[/_ .-J^]+24^3/1^, where theargumentofeach oftheBessel functions J±iontherightis 1^1/tan*^;and |0.^and j^glareboth lessthan 1.These arethemoreprecise forms ofNicholson's formulae(2)and(3);andtheygiveeffective approxima- tionsexceptnear thezeros ofthedominant terms ontheright. Itishighly probablethat theupperlimits obtained fortheerrors are largelyinexcess oftheactual values oftheerrors. 8*5.Descriptive properties* ofJ^{vx)luhen <a;^1. The contourintegral,which wasobtained in§8"31(I)torepresent Jv{vsecha)wasshewn in§8*4toyieldanasymptotic expansionofthefunction. Butthecontourintegralisreallyofmuchgreater importancethanhashitherto appeared;foranintegralisanexactrepresentationofafunction, whereas an asymptotic expansioncanonly give,atbest,anapproximate representation. Andthecontourintegral (togetherwith thelimitingform ofitwhen x=1) ispeculiarlywelladaptedforgiving interestinginformationconcerning /„{vx) when Vispositive. Inthecontourintegraltake vtobepositiveandwrite %u=log {?'e'^|, sothatu=logr,v=0. With thecontour selected, Xsinhw—w isequaltoitsconjugate complex,andthepathofintegrationisitsown re- flexion inthereal axis.Hence 1foc+-rri T qv(Xsinhw—w)^y *Theresults ofthissection areinvestigatedinrather greaterdetail inProc.London Math. Soc. (2)XVI.(1917), pp.150—174. 8-5] FUNCTIONS OFLARGE ORDER 253 Changingthenotation, wefindthattheequation ofthecontour is 1 2^ r+-— rXsin6 sothat x^md{ V\ 6- and,when thissubstitution ismade for/•,thevalue of(lu—x'sinh w)is log^\.^'--cot^ .V(^'-*'sm^6).Xsin This lastexpressionwillinvariablybedenotedbythesymbol* F(6,x), sothat (1) J,(vx)=-re-^'P^''-^Ue, andbydifferentiatingunder theintegral sign (aprocedure which iseasily justified)itisfound that (2) .//(«)= ifV-».. '^p^^ldff. TTj X\J{U'—X-sm-u) This isalsoeasilydeduced from theequation 'liriJ^'{vx)=^-.\e"^•'^''"'^ ''-"''sinh lodw. Beforeproceedingtoobtain further resultsconcerningBessel functions, it isconvenient tosetonrecord variousproperties!oiF(6,x). Thereader will easily verifythat (4)4^,o,.)=-^-^.o. SOthat (5) F{e,x)^F(0,x)^F(0,l)=0; andalso .„, ^n/n \ ^—^'sin6cos6 Nextweshall establish themore abstruseproperty (7) F(e,x)^F(0,x)-\-^(e^--x''sm'e)/^(l^x').' Toprove it,weshall firstshew that ,a N^—a;^sin^COS^ *This function willnotbeconfused with Schlafli's function defined in§4'15. tItissupposed throughout thefollowing analysisthat0<.r^l, 0^0 ^tt. 254 THEORY OFBESSEL FUNCTIONS[CHAP.VIII Itisclear that g(7r,x)=l <^/(l+x% sothat, ifg{d,x), quafunction of6,attained itsgreatest value at ortt, that value would belessthan \J{1+X'). If,however, g{6,x) attained its greatestvaluewhen 6hadavalue 6qbetween andtt,then 1— A'-cos2^0 (^0—^'sin ^oCos ^o)^ /^ (6,^-X-sin-^o)^ (^o'-^sin^d^f andtherefore g{6,x)^g{do,x)=\/{l-af-cos2^o)^\/(l+^), sothat,nomatter whereg(d,x)attains itsgreatest value, that value does notexceed V(l +-'^'')- Hence dF(0,^) _ d-x^sinlcos.e andso whence(7)follows atonce. Another, butsimpler, inequalityofthesametypeis (8) F(d,x)^F{0,x)+^d'^{l-x'). Toprove this,observe that ^-^^^^>^{0'-x^sin^e)^e V(l-^), andintegrate;then theinequalityisobvious. From these results wearenow inapositiontoobtain theoremsconcerning J^{i>x) and//(px)quafunctions ofv. Thus, since dJ^(vx) _1=-irF(d, x)e-"^^''""^ dd^0, TTJO dv theintegrand being positive by(5),itfollows thatJt,(vx)isapositivede- creasing function ofv\inlikemanner, // (I'a;)isapositive decreasing function ofv. Also, since OV TT.' theintegrand being positive b}^(5),itfollows that e"^^^-^' J^{vx)isadecreasing function ofv;andsoalso, similarly,ise"^'*^-*)//(i/a;). 8-51] FUNCTIONS OFLARGE ORDER 255 Again,from(8)wehave Q—vF[0,X) j-ir g-vF{0,X) j-ir J,(vx)^ exp[-i z;(9-V(1-^')1cie "^ J{} rCO exp[-^ve\/(l-x-)]dd,Ag-vF(0,x) /-co< SOthat g—vF(0.x} (1- a;^)* \/{2'7rv) The lastexpressioniseasily reduced toCarlini'sapproximate expression (§§1*4, 8-11)forJ^{vx); and soCarlini'sexpressionisalwaysinerrorby excess, forall*positivevalues o^v. Thecorrespondingresult forJJ{vx)isderived from(7).Write e--x-^in-e=G{d,x), andreplace G{6,x)byGforbrevity. Then 2xJ: {vx)=^ fV^^(^--''Ei^ [G(6,x)]-icW Tr.'o cL" g-vFtO,:r) fv^ expI-^vG/'^il +x'^)].G-^dG ^T. g-vF{0,x}rao exp[-ij.{?/^(l +^2)1.Q-hdG, andso (10) xJJ{vx)^e-^^'f.^) (1+,^2)V^/(27^I^). Theabsence ofthefactor\/(l—a;-)from thedenominator isremarkable. Itispossibletoprovetheformulaf inaverysimilar manner. This concludes theresults which weshall establishconcerningasingle Bessel function whoseargumentislessthan itsorder. 8"51.Lemmaconcerning F{6,x). Wfishallnowprovethelemma that,when <:a;^l and0^0 ^tt,then ^^^-16-^-^^^^' "^-^^^'^^^^^-^^sin-^^^- Thelemma willbeusedimmediatelytoproveanimportanttheorem con- cerningtherateofincrease ofJ^{vx). *Itisevident fromDebye's expansion thattheexpressionisinerrorb^-excess forsiificientUj large values ofv. tCf.Froc. London Math. Soc.(2)xvi. (1917), p.157. 256 THEORY OFBESSEL FUNCTIONS [CHAP.VIII If\J{&^—X-sin-6)=H{6, x),weshall firstprove that dOIdd isanon-decreasingfunction of6;that istosaythat (l-^cot^)- +6?--A'-sin-^ 6—ofsin6cos6 isanon-decreasingfunction of6. The differential coefficient ofthis lastfunction of6is {6-X-sin6cosd)-^ [{d^-cosec^O-l-^sin^d)(1-a^) +2(^2cosec- 6*-^='cot^cosec'^ 9-isin^6){\- x-) +2a;2(1_^cot6){dcosec ^-cosOf+sin-(9(1- a,-'^)-], andevery groupofterms inthisexpressionispositive (orzero)inconsequence ofelementary trigonometrical inequalities. Toestablish thetrigonometrical inequalities, wefirstobservethat,when $^^tt, (i)(9+sin^cos (9-2(9-1 sin'-(9^0, (ii)6+sin (9cos (9-26-cot <9^0, (iii)sin^-^cos^-^sin^'^^O, because theexpressionsonthe leftvanish when 6=andhave thepositivedifferential coefficients (i)2(cos^-<9-isin^)2, (ii)2(cos^-(9cosec^)2, (iii) sin (9((9-sin^cos^), andthen (92cosec2 6-6'^ cot6cosec-6-1sin^6 ={6-cosec2 ^-1)(1-^cot^)+cosec 6(sin6-6con6-1sin-"*6)^0, ^2cosec2^-l-Jsin2^=6cosec^ 6{6+sin6cos6-26'^ sin^^)+cosec 6(sin^-^cos^-^sin^5)>0, sothattheinequalitiesareproved. Ithasconsequentlybeen shsAvn that re\m^"'de where thevariables areunderstood tobo6and x,andprimes denote differ entiations withregardto^.Itisnowobvious that d[FHJ rjd{F')^ dd and, ifweintegratethisinequalityfrom to6,weget F'He _ Since F'andHJH'vanish when ^=0,thisinequalityisequivalentto andthetruth ofthelemma becomes obvious whenwesubstitute thevalue ofH{6,x)inthelastinequality. 8-52J FUNCTIONS OFLARGE ORDER 257 8"52. Themonotonic property ofJ^(vic)fJ^(v). Weshallnowproveatheorem ofsomeimportance,totheeffect that,ifoc isfixed,and ^if^1,thenJ^{ikv)/J^ (v)isanun-increasing function ofv,when Vispositive. [Theactualproofofthetheorem will lievalidonlywlien S$./<!, (wliere8isan arbitrarilysmallpositive number),sincesomeexpressions introduced inthejjroofcontain an X-intheir denominators; butthetheorem isobvious when ^.r^8since e"^*"- ^)J^(i/.r)and e-''F(o,ce)iJ^ (^)arenon-increasingfunctions ofpwhenxissufficientlysmall ;moreover, as willbeseen inChapter xvii, thetheorem owes itsrealimportancetothefactthat itis true forvalues ofxintheneighbourhood ofuniiT/.] Itwill firstbeshewn that (1)'JV{yx)--, -, -^ ^>0, dpooc ox Ov Toestablish this result, weobserve that, withtheusual notation, J,{vx)= ^Te-'Fi^'^^dylr, dJ^,{vx)V f'^ [G^l^>^')}"^^^—e-"^"'^'-)dx Stta-Jo'dd and,whenwedifferentiate under theintegral sign, dJ^iix)1dO, dv n"F{ylr,x)e-''^^*'''^dAlr, cKjivx) ^1p j_.d_G{ex)^_^,^,^^^ ^^^ dvdx 27rxjo' ^ ^'dO F{0, x){G(6,x)]-i—fj^^^"''^^''"' <l^ 27rxj V TTX J{G(e,x)]i^^l^-^,F{e,x){G{e,x)]-^^_,dG{d,x) dd .xe-'"f«'.«(W ifweintegrate bypartstheformer ofthetwointegrals. Hence itfollows that 'dvdx dx dv ztt^xJoJo where n{e,ir)=2{Gid,x)]i' dF(e,x) F(ylr,x)-F (0,x)dG{9 ,^ dd^ 2G(e,x) dd :^2[G{d,x)\^ ^0,,[dF{d, x)F(0, x)-F(e, x)dG(0,x) dd '>G{d,x) dd byusingtheinequality F{-\\r, x)^F{0, x)combined with thetheorem of^8"51. W.B.F. l? 258 THEORY OFBESSEL FUNCTIONS [chap.VIII SinceO(0, yjr)isnotnegative,therepeated integral cannot benegative; that istosay,wehaveprovedthat T/ .d'~Ju(vx) dJv{vx)dJ^(vx)- sothat Integratingthisinequality between thelimits xand 1,weget dJy(vx) /r/1^ /^dx dv J^ivxyr ^0. dv sothat dJy(vx) SinceJ^{i'x) andJ^„(t')arebothpositive,thisinequality maybewritten intheform 3 (2)dv[J,(vx)/J,{v)}^0, and thisexhibits theresult which wastobeproved, namelythatJ^(vx)/J^(v) isanon-increasingfunction ofv. 8"53.Properties ofJ^(v)andJJ{v). If,forbrevity, wewriteF{6)inplaceoiF{d, I),sothat (1) F(e)^ log-^^r—-™'^^(^'-^'°'">• theformulae* forJv{v) andJv'{v)are (2),/„«4/>-....,.;(.)= lj7^^^^ia|^^-..<.>rf.. The firstterm intheexpansionofF{6)inascending powersof9is 4^7(^ \^'^)''^^^ ^^'^shallproveaseries ofinequalities leading uptothe result thatF{6)16^isanon-decreasingfunction of6. Weshall firstshew that dd\e-^P^- Toprovethisweobserve that F'id)_\{\-e cote)id^Y,.-,ua,,i„.Q , *Itistobeunderstood that Jl,'(v)means thevalue ofdJ„ {.r)ld.r when .rliastheparticuhir value V. 8-53] FUNCTIONS OFLARGE ORDER 259 andthat de\ &'J~ l^' d^{s/{e-'-Qm'd )\^_(6'^cosec'^ +^cot^-2)sin-6> de\e-I e\'{e^~^n^• Hence itfollows that (/iF'id))6*-sin^cos 6^,,, ,^ ^ ^^, X{$+sin cos--26-cot6) byinequalities provedin§8"51. Consequently (3) eF"{d)-2F'{e)^0, that istosay^^[OF'(d)-SF(6)]^0. Ifweintegratethisinequalityfrom to6weget (4)OF'(6)-SF(0)^0, and this isthecondition thatF{6)16^should beanon-decreasing fanction of6. Itfollows that ^(''•l>.lnn^<'')-' andtherefore6'^ e-^o &' 9V-3' 1 ;•-(4,ve-') TTJJ,{u)<-Iexpj-^^jc^d* 2f3^TTz;*' SOthatCauchy's approximationforJAv)isalwaysinerrorbi/excess. Aninequality which willberequired subsequentlyis (5)- 2(^2_i^iii2e)F'{6)-3(^-sin6cos^)F(6)>0. Thetruth ofthismaybeseenbywriting theexpressiononthe leftintheform {&'-2sin'-^e+6sin6cjs6)F'(0)+{6-sin6cos6)[BF' {6)-?>F{d)\, inwhich eachgroupofterms ispositive (cf. J^8-51). [Note. Aformula resembling those which have justbeen established is p 1 23 (6) J^(vi)dt seePhil.Mag. (G)xxxv.(1918), pp.304—370.] 17—2 260 THEORY OFBESSEL FUNCTIONS[CHAP.VIII 8"54. Monotonicproperties ofJ^{v)and J^'{v). Ithasalreadybeen seen(§8"5)that thefunctions J^(p)andJJ(v)are decreasingfunctions ofv.Itwillnowbeshewn that both v^J^, (v)andv^JJ{v) aresteadily increasing*functions ofv. Toprovethe first result weobserve that dv StTJ TTj "SttOe-^P^^) +^I.W (^)-3^(^)l^""^'^'^ dd <J7'.' >0, since theintegrated partvanishes ateach limitand(§8"53)theintegrandis positive. Hence v^Jv(v)isanincreasingfunction ofv;andtherefore (1) v^J,{v)<lim{i.-^/,(!/)]=r(i)/(2^3^7r)=0-44731. Inconnexion with thisresult itmaybenoted that Ji(1)=0-44005, 2^8(8)=0-44691. Toprovethesecond result, byfollowingthesamemethod wefindthat d\v^j;(v)]^2i'-^ dv Stte--ne) ^(^2_sin20) by§8*53(5),andsov^J^iv)isanincreasingfunction ofv. Hence (2)v-^J:{v)<lim{v^JJ(v)]=3^r(|)/(2* tt)=0-41085. Itistobenoted that Ji'(1)=0-32515, 4^8' (8)=0-.38854. 8"55. Themonotonicproperty ofv^JJ {v)jJ^{v). Atheorem which isslightly more recondite than thetheoremsjustproved isthat thequotient isasteadily increasing function ofv. *Itisnotpossibletodeduce thesemonotonic propertiesfrom theasymptotic expansions. If, asv-^-Qo, f(v)~ (p{v),and if(j){v)ismonotonic, nothing canbeinferred coucerning monotonia properties of/(j')intheabsence offurther information coucerning /(j*). 8-54, 8-55] FUNCTIONS OFLARGE ORDER 261 Toprovethis result weusetheintegrals alreadymentioned in§§8"53, 8*54 forthefourfunctions dv dv Takingtheparametricvariable inthe firstandthirdintegralstobe-^in placeof6,wefindthat where n,{6,^)=IF'(0)V(^-'-sin^0)-l^~^^^F(6) ^-sin^cos^ , , ,XT// ,X CT/ ,M >M^_-_^_^)(,r(6)-F(e)]-'j-^^\ifr(f)-F(n. by§8'51. The function n^{d, -yfr)does notseem tobeessentially positive (cf§8"52) ;toovercome thisdifficulty, interchangetheparametricvariables andyjr,when itwillbefound that Now, from theinequality justproved, n,(d,f)+Hi(>/.,d) ^|^^+i/rsini|rcosx/r-2sin--^n^^p'/zdn P/Z3M ^2+^sin^cos6^-2sin-^ ,. ,-p,.,.ti,,\) ^^V(^--sm-.)^*^^ ^^^~^^^^^• Since 0-^^J{0--sin-0)and40i?"(^)-F{0) areboth(§8-53) increasingfunctions of0,the ftictors ofthe firstterm inthesumontherightarebothpositiveor bothnegative; and,by§§8-51,8-53, thesecond and third terms arcboth positive. Hence Hi{0,~^)+i\{^,0)ispositive,andtherefore which establishes theresult stated.> 262 THEORY OFBESSEL FUNCTIONS[CHAP.VIII 8'6.Asymptotic expansions ofBessel functions oflarge complexorder. The results obtained(§§8-31—8'42) b}-Debyeinconnexion with./^(.r) andY^{x) where vandxarelargeandpositive weresubsequent!}^extended* tothecase ofcomplexvariables. Inthefollowing investigation,whichis,in somerespects,more detailed thanDebye's memoir, weshall obtainasymptotic expansionsassociated with J^{z)when vandzarelargeandcomplex. Itwill firstbesupposedthat jargz<^Tr, andweshall write v=zcosh'y=zcosh(a+1/3), where aand/3arerealand7iscomplex. There isaone-onecorrespondence between a+i^andvjzifwesupposethat/3isrestricted toliebetween f andTT,while amayhaveanyrealvalue. This restrictionprevents zjvfrom lyingbetween —1and 1,butthiscasehasalready (§8"4)beeninvestigated. Theintegralstobeinvestigatedare H,^'^{z)= —.r^^\-'-f^'^'^ dw, TTlJ _oo -trcc-H 1rcc i7;2) (^x=-A e-zf{w) ^^^=_— ./ e^/(«"dw, wheref(iv)=tocosh7—sinh ic. Astationary pointoftheintegrandisat7,andweshall therefore in- vestigatethecurve whoseequationis If(w)=Ifij). Ifwereplace wbyk+iv,thisequation maybewritten intheform (v-/3)coshacos^+(u— a.)sinhasiny8-coshusinv+coshasin/3=0. Theshapeofthecurve near(a,/3)is {(u-ay-{v- /8)-}coshasin/3+2(u-a){v- y8)sinhacos/3=0, sotheslopesofthetwobranchesthroughthatpointare Itt+^arctan(tanh acot/3), —jTT-!-^arctan(tanh ocoty8), where thearctandenotes anacuteangle, positiveornegative;Rf{iu)in- creases aswmovesawayfrom7onthe firstbranch, while itdecreases asw movesawayfrom7onthesecond branch. Theincrease (ordecrease)issteady, andRf(w)tends to+x(or—oc)astumoves offtoinfinityunless thecurve hasaseconddouble-point J. *Miinchener Sitzungsherichte, xl.[5],(1910);theasymptotic expansions of/^(.r)andAV(.r) were statedexplicitly byNicholson, Phil.Mag. (6)xx.(1910), pp.938—943. •j-That istosay0</3<7r. +Aswillbeseenlater, this istheexceptional case. 8-6,8-61] FUNCTIONS OFLARGE ORDER 263 If(i)and(ii)denote thewhole ofthecontours ofwhich aportionare marked with thosenumbers inFig'. 19,weshall write TTi .(i)"*J(ii) andbyanalysisidentical with that of§8'41(exceptthat2/3istobereplaced by7),itisfound that theasymptotic expansionsofS^^^^ (z)andS^,'-' (z)are given bytheformulae (1) >SV'^' (z) (2) S,-> {z)v/(- hv-witanh7)„,1T(^)' {hvtanh7)'"' -_y s/{- h,v-rritanh7),"„T(|)' (-11'tanh7)"^' •wherearg(—|viritanh7)=argz+arg(—isinh7), andthevalue ofarg(—isinh7)which liesbetween — ^ttand^ttistobetaken. (i) Fig. 19. Thevalues of^4o, A-^, A.^,...are (3) 1^0=1, ^i=i-Acoth^7> -4.=tIs- 5¥ffcoth=7+iff^coth^7, Itremains toexpress if^,'" {z)and ff^'^' (^)interms of>S^'i' (2)and >SV'"(^); andtodothisanintensive studyofthecurve onwhich If{^)=^/(7) isnecessary. 8-61. Theform ofDebyescontours when thevariables arecomplex. Theequationofthecurve introduced inthelastsection is (1)- {v— /3)cosh acosy8+{u—a)sinhasin|8. —coshusinv+coshasin /3=0, where{u,v)arecurrent Cartesian coordinates and0</3<tt. Since theequationisunaltered byachangeofsigninboth uand a,we shall firststudythecase inwhich a^ ;andsince theequationisunaltered whenTT-v and tt-y8arewritten forvand /S,weshall also atfirstsuppose that0</8<i7r, though manyoftheresults which willbeprovedwhen (3is anacuteangleare stilltruewhen^isanobtuseangle. 264 THEORY OFBESSEL FUNCTIONS [chap.VIII Forbrevity,theexpressiononthe leftin(1)willbecalled(f)(u, v).Since d(f){u,v).,.^.,--~ =sinhasmB—smh asm v,ou itfollows that,when visgiven, dcp/duvanishes foronlyonevalue ofu,andso theequationinu, (p(w,v)=0, hasatmosttworealroots;andoneofthese isinfinite whenever visamultiple ofTT. When <?'<TT,wehave* (f>{-X,v)=—OC, (f){+x,v)=—CC, (f)(a,v)=cosha{{v—/3)cosfS—sinv+sin/3|>0^ andsoonerootoftheequationinu, 4>(%v)=0, islessthan aandtheother isgreaterthan a,bothbecoming equalwhen v=0. Byconsideringthefinite root oftheequations (^(w,0)=0, <^(;^,7r)=0, itisseen that, ineach case, thisroot islessthan a,sothelargerroottends to+OCasvtends to+ortott—0,and forvalues ofvjustlessthan orjust greaterthan tttheequation (f)(u,v)=hasalarge negativeroot.Theshape ofthecurve isthereforeroughlyasshewn bythecontinuous lines inFig.20. Next consider theconfiguration when vliesbetween and—tt. When Vis—^,d(f>{u,v)ldu vanishes at li=—o,andhence^(v/,—/3)hasa minimum value 2cosh asin /8(1—/3cot/3—atanha) atIt=—a.There arenowtwocases toconsideraccordingas 1—/3coty8—atanha is(I)positiveor(II)negative. *Since d(p(a,v)ldv=cosha.(cos^-cosv),and thishasthesame sign as i>- j3,(p(a,r)hasa minimum value zero atr=/J. 8-61] FUNCTIONS OFLARGE ORDER 265 Thedomains ofvalues ofthecomplex y=a+t/3forwhich 1—/3cot/3—atanh a ispositive (inthestrip 0"$/3^7r) arenumbered 1,4,5inFig.21; inthe domains numbered 2,3,6a,6b,la,7btheexpressionisnegative;thecor- respondingdomains forthecomplex v/z=cosh(a+^yS)have thesamenumbers inFig.22. N^6/> 6c? m Fig.21. Fig.22 (I)When 1—y8cot/3—atanh aispositive, <^(u,—/S)isessentially positive, sothatthecurve never crosses theline v=— /3.Theonly possibilitytherefore isthatthecurve aftercrossingtherealaxisgoesoffto—xasshewn bythe upperdotted curve inFig.20. (II)When 1—/3cot/3—atanh aisnegative,theequation (f)(—a,v)= hasnorealrootbetween and^—27r,for d(f){—a,v)jdv=cosha(cos /3—cosv). Therefore(f)(—a,v)hasasingle maximum at— /3,and itsvalue there is negative,sothat<^(—a,v)isnegative when vliesbetween and /3—27r. Also -h(a,/3—27r)hasamaximum ata=a,and itsvalue there isnegative, sothatthepurve (p(u,v)=doesnotcross v=jB—'lir;hence, aftercrossing thereal axis,thecurve mustpassofftox-irl,asshewn bythedotted curve ontherightofFig.20. Tjiis completesthediscussion ofthej^artofthecurve associated with ^V'>' {z)when a>0,</3$Itt. Nextwehave toconsider what happenstothecurve aftercrossingthe line w=+TT. Since</>(a,v)=cosha[{v—/3)cos/3—sinv+sin/3}, andtheexpressionontherightispositive when /'^(3,thecurve never crosses thelineu=a.;also {u,iitt)={u— a.)sinhasin/3+(mr-jS)coshacosy8+coshasin/5, 266 THEORY OFBESSEL FUNCTIONS [chap.VIII andthis ispositive when u>a,sothatthepartsofthecurve whichgooffto infinityontherightmust lieasshewn inthenorth-east corner ofFig.23. When 1-atanh a+(tt- /3)coty8>0, i.e.when(a,/3)liesinanyofthedomains numbered 1,2and3inFig. 21,itis found thatthecurve doesnotcross v=27r—^,andsothecurve aftercrossing v=7rpassesoffto—x+ttiasshewn inFig.23byabroken curve. Fig.23. Wenowhave toconsider whathappens when(a,/8)liesinthedomain numbered 6ainFig.21.Insuchcircumstances 1—atanh «+(tt—/S)cot/3<; and (f)(—a,v)hasamaximum atv=2tt—^,thevalue of<})(—a,'Itt-/3)being negative. Thecurve, aftercrossingv=tt,consequentlyremains ontherightof u=—auntil ithasgotabove v=27r— y3. Now <f)(—a,v)isincreasingintheintervals (;g,27r-/3), (27r+/3,47r-)S), (^tt+y8,Gtt-/3),...; letthe firstofthese intervals inwhich itbecomespositivebe (2il/7r+/8,2i/7r+27r-yS). Then^(u,^Mir+2ir—^)hasaminimum atm=—a,atwhich itsvalue is positive, and sothecurve cannot cross thelinev=2M'Tr+2Tr—^ ;itmust therefore goofftoinfinityonthe left,andconsequently goesto -00+(2M+l)7ri; itcannotgotoinfinitylower than this, forthen thecompletecurve would meet ahorizontal lineinmore thantwopoints. 8-61] FUNCTIONS OFLARGE ORDER 267 When{a,/3)isinGa,thecurveconsequently goestoinfinityat -oG+{2M+ \)7ri, whereMisthesmallestintegerforwhich ]—a.tanh a.+[(M+1)tt—ySlcot/3 ispositive. Wecannowconstruct atable ofvalues oftheend-pointsofthecontours foraS„<^' (z)and<S^*-' (z),andthence wecanexpresstheseintegralsinterms of Hy^^> (z)and ZT^*-' (z)when(a,^)liesinthedomains numbered 1,2and6ain Fig.21;andbysuitable reflexions weobtain their values fortherest ofthe complete stripinwhich </3<tt.Thereader should observe that, sofaras thedomain 1isconcerned, itdoesnotmatter whether ^isacute orobtuse. IfAIisthesmallestintegerforwhich 1-atanh a+{{M-I-1)tt- ^8}cot/3 ispositive when cot/3ispositive,and ifNisthesmallestintegerforwhich 1-atanh ct-(i\V+/3)cot/3 ispositive when cot/3isnegative,thetables ofvalues of/SV'^'(z)and*S^'-'(z) areasfollows : Regions 268 THEORY OF'BESSEL FUNCTIONS [CHAP.VIII Thereader willfind itinterestingtoprove that, inthecritical case/3=|7r,thecontours passfrom—ootocc+iriandfrom—qc+7^^'togo,sothattheexpansions appropriateto theregion1arevalid. Note. Thedifferences between theformulae fortheregions 6aand66andalsoforthe regions laand76appeartohavebeen overlooked byDebye, andbyWatson, Proc.Royal Soc.xcv.A,(1918), p.91. 8*7.Kapteyns inequality forJ^(nz). Anextension ofCarlini's formuhi(§§8"11, 8'5) toBessel coefficients in which theargumentiscomplexhasbeen effected byKapteyn* who has shewn that,when zhasany value, real orcomplex,forwhich z-—1isnot arealpositive numberf ,then z''eKp{n\/(l -z^)} (1) \Jn(nz)\^ {1+V(l-2')]^ Thisformula islessprecisethan Carlini's formula because thefactor{irrn)"^ (1—z-)* does notappearinthedenominator ontheright, butnevertheless theinequalityissufficiently powerfulforthepurposesforwhich itisrequired;]:. Toobtain theinequality,consider theintegralformula 1 /'<"+' J,^(nz)=^-^. r'*-iexp{Inz {t-1/0} dt, inwhich thecontour isacircle ofradius e",where uisapositive number to bechosensubsequently. Ifwewrite t=e"^'^,weget J^(nz)=^rexp[n[^z(e^e'"-e-"e-'«)-u-iO]]dO. Now, ifMbethemaximum value of Iexp[\z(e»e'^-e-«e-'^)-u-id] \ onthecontour, itisclear that IJn{nz)I<M-. But if^= /je'",wherepispositiveandaisreal,then therealpartof \z{e^e'^-e-"e-'^)-«-i9 is |pfe"cos{a+ 6)—e~"cos(a-6)]—u, andthisattains itsmaximum valuewhen tan6=—cothutan a, and itsvalue isthen pV(sinh^ u+sin-a)—u. *Aim. Sci.deI'Kcole norm sup. (3)x.(1893), pp.91^120. tSince both sides of(1)arecontinuous when zapproachesthereal axis itfollows thatthe inequalityisstilltruewhen z--1ispositive:forsuch values ofz,either signmaybegiventothe radicals according totliewayinwhich zapproachesthecuts. JSeeChapterxvii. 8-7J FUNCTIONS OFLARGE ORDER 269 Hence, forallpositivevalues ofa, IJji(npe"^)j^exp[npv/(sinh- u+sin-a)—nu]. Wenowchoose usothattheexpressionontherightmaybeassmall as possibleinorder togetthestrongest inequalityattainablebythismethod. Theexpression p/v/(sinh-//+sin-a)—a hasaminimum, quafunction ofm,when uischosen tobethepositiverootof theequation* sinh ucosh u 1 \/(sinh- u+sin-a)p' With thischoice ofuitmaybeprovedthat 2\/(l— •s^O•sinh ucosh u=±(cosh2ii— e-''^), and,bytakingztobereal,itisclear thatthepositive signmust betaken in theambiguity.Hence 2[1+x/(l- z"")]sinhucoshu=e-"'-e•-'^ andso logzexpV(l-2^)^_i^^„2V(sinh-u+sin^a). \exp v'(l- •2")' 1+V(l-2')=logg2M_g2la /h(n^) I^_sinh^ u+sin-a sinhucosh li =pV(sinh- u+sin-a)—u, and itisnow clear that izexp \/(l— •2")I (1+v(i-^T' Aninteresting consequenceofthisinequalityisthat ;Jn(nz)i^1solong asboth 1-J I$1and \zexp \/(l— z'^) ^— ^^ '<1 I1+V(i-^') Toconstruct thedomain inwhich thelastinequalityissatisfied, write as before z=pe'",anddefine ubytheequation sinhucosh u 1 \/(sinh-u+sin-a)p' Theprevious analysisshews atonce that,when zexpv(l—2') thenl+^/(l-z') p\/(sinh-u+sin-a)—u=0. This equationisaquadraticinsinh- uwithonepositiveroot. 270 THEORY OFBESSEL FUNCTIONS [chap.VIII Itfollows that^ 2u p-= sinh2u' sin^a=sinhw{ucoshu—sinhu). Asuincreases from to1-1997 ...,sin-aincreases from to1andpde- zexp \/(l—^^)creases from 1to*0-6627434 .... Itisthen clear that ^1 inside andontheboundaryofanovalcurvecontainingtheorigin.Thiscurve Fig.24.Thedomaiu inwhich |J„(nz) jcertainly doesnotexceed unity. isshewn inFig.24;itwillprovetobeofconsiderableimportanceinthe theoryofKapteynseries(Chapter xvil). When theorder oftheBessel function ispositivebutnoti-estricted tobeaninteger we takethecontour ofintegraiiontobeacircle ofradius 6"terminated b\-tworaysinclined +TT-arctan(cothwtana)tothereal axis. Ifwetake |^ |=6"ontheserays,weget cosh(u+v)-cos2acosh(v-u) \J^{vz)\4:M''-{-sini/TT =$3/" 1+/" r 1' Iex-^ {—V{v-u)]dv\^ \Ju )vvdo andso Thisvahie isgiven byPlummer, Dynamical Astronomy (Cambridge, 1918), p.47. CHAPTER IX POLYXOxMIALS ASSOCIATED WITH EESSEL FUNCTIONS 9'1.Thedefinition ofNeumannspolynoynial On(t). Theobjectofthischapteristhediscussion ofcertainpolynomials which occur invarioustypesofinvestigationsconnected with Bessel functions. The first ofthesepolynomialstoappearinanalysisoccurs inNeumann's* investigationoftheproblemofexpandinganarbitrary ;inalyticfunction/(^) intoaseries oftheformXa^.Jni^)- ThefunctionOn{t), which isnowusually called NeumannsjJolynomial,isdefined asthecoefficient ofenJn {z)iuthe expansionofl/(^—z)asaseries ofBessel coefficientsf,sothat (1) .-^=Jo(z)0,(t)+2.1,(z)0,(t)+2J,(z)0,(t)+ ... =i^n-Jn{z)Onit). From thisdefinition weshall derive anexplicit expressionforthefunction, and itwillthenappearthat theexpansion (1)isvalid whenever\z\< ^t. Inorder toobtain thisexpression, assume that|2;|<|^|and, afterexpanding l/{t—z)inascending j)owersofz,substitute Schlomilch's series ofBessel coefficients(§27)foreachpowerof 2'. Thisprocedure gives 1 1^^_ 1» -2M-(s+2m).(s +m- l)l )=7-e2m'J2m{z)+ i-f^j:i — Js^2m{2)l.Ini=0 S=lf'(./n= '"iJ Assumingforthemoment thattherepeatedseries isabsolutely convergent :|', ,*Tluorif dcrBeascVschen Functioiicn(Leipzig, 1867), pp.8—lo,83;seealsoJournalfiir Maih. Lxvii.(1867), pp.310—'6\\.Neumann's procedure,after assuming; theexpansion (1),isto deriv^tlie differential equation which will begiven subsequently (§!)-12) and tosolve itin series. fInanticipationof§1611,weobserve thattheexpansionofanarbitrary function isobtained bysubstitutingforl/((- :)intheformula {z-^)f[t)(lt XCf.Pincherle's rather more general investigation, Reiuliconti Jt.Lst.Loiiibardo, [2)xv.(1882), pp.224—225. 272 THEORY OFBESSEL FUNCTIONS [CHAP. IX weeffect arearrangement byreplacingsbyn—2m,andtherearrangedseries isaseries ofBessel coefficients;wethusget 1 1^ ^..^ |<i(.-i)2»-^r«-i n.(n-m-l)l) ^,, =-^f2m^2m (^)+Se„^>.,n-2m+i'—, I^n{z) Accordinglythefunctions On{t)aredefined bytheequations ^x ^ /X1It**" •(»-m-1): (3) 0.(0=l/«. Itiseasytoseethat (4) e.On(t)= ^^;^-^i+ 2(2„_2)+ 2 .4 .(2«-2)(2n-4)"^ •••r andtheseries terminates before there isanypossibilityofadenominator factor beingzero ornegative. Wehavenow toconsider thepermissibilityofrearrangingtherepeatedseries for \l{t— z).Asufficient condition isthattheseries "2»J- {s+27n).{s+m-l)\ \ ,r7i»+"i 1\ ^\'•'''+2''' f.^/If should beconvergent. Toprovethat this isactuallythecase,weobserve that,by§2*11 (4),wehave ,„!or,i !I''**2-^'>^,„!o-n^Hs+^ir^l)!^^^iil^l) <2(^l2i)'+2"'{exp(i|0P)}/(2m)! <(i|.i)«exp(il^|2). Hence "2'f" (.s+2?w).(s+OT-l)!, ,,,1" \z\' ,,, „, ^,|7^^ ^r~^l«^..2m(^)|U2^iexp(i|0p) s=lI'' I \,m=0"' •js=l I' I ^Ig Iexp(^IzP) I^Ki^l-MI)• Theabsoluteconvergenceoftherepeatedseries istherefore established under thehypothesisthati^|<!i{j. And sotheexpansion (1)isvalidwhen \z\< \t\,andthecoefficients oftheBessel functions intheexpansionare defined by(2)and(3). Itisalsoeasytoestablish theuniformityoftheconvergenceoftheex- pansion (1)throughouttheregions \t\^R, |^^ |^r,whereR>r>0. 9-1] ASSOCIATED POLYNOMIALS 273 When theseinequalitiesaresatistied, thesum ofthemoduli oftheterms does not exceed i-^^ Since theexpressionontherightisindependentofzand<,theuniformityofthe convergencefollows from thetestofWeierstrass.(g+2m).(g+m-l)! (^r)« -^2'»exp (|-/-2)1exp(ir^) Thefunction 0„(Owascalled byNeumann aBesselfunction ofthesecoyid kind*; butthistei'm isnowused(cf.§§3"53, 3*54)todescribe acertain solution ofBessel'sequation,andsoithasbecome obsolete asadescriptionofNeumann's function. Thefunction 0„(0isapolynomialofdegreen+1inl/t,and itis usuallycalledNeumanns polynomial oforder n. Iftheorder oftheterms inNeumann'spolynomialisreversedbywriting \n—mor\{yi—\)—m formin(2),accordingasniseven orodd, itisat oncefound that (5) 0„it)= 5S^(l,_,.)!(iO--^^'^'"^ =--I 1^^H^ ^— + .. (6) /,)_l^^V^^,Jii(iZi+"^-i)' (nodd) nV(n~-in nCtf--1••^)(n--3^ t^ t' f These results mavbecombined intheformula Theequations (5),(6)and(7)weregiven byNeumann. Bythemethods of§211,itiseasily provedthat (8) \enOn{t)\^i .{n\).{^\t\)—^ exipil\t\% (9) enOn(t)=i^.(nl).{H)—^{l +d), (n>l) where" |^ j^[exp (^\tf)-l]/(2/;-2). From these formulae itfollows thattheseries SanJn(^)On(t)isconvergent whenever theseries-an{zjt)"'isabsolutely convergent;and,when zisoutside thecircle ofconvergenceofthelatter series. anJn{^)0,i(t) does nottend to zero ,as^7i-^oc ,andsotheformer series does notconverge. Again,itiseasy toprove that, as *;-^oo, enJn{Z)On{t)= ^"Sl|l"^^+ (/^-^)|, *Byanalogy with theLegendre function ofthesecond kind, y„{t),which issuch that '"^71= Cf.ModernAnalysis, §15"4. W.B.F. 18 274 THEORY OFBESSEL FUNCTIONS[CHAP. IX andhence itmaybeshewn* thatthepointsonthecircle ofconvergenceat which either series convergesfareidentical with thepointsonthecircle at which theother series isconvergent.Itmayalsobeproved that, ifeither series isuniformly convergentinanydomains ofvalues ofzandt,soalso is theother series. Since theseries ontherightof(1)isauniformly convergentseries of analyticfunctions when\z\< \t\,itfollowsbydifferentiationJthat ,.r.. (-)i.{p +qV._»dPJn(2) d<iOn(t) ^^{t-zf+9+i ^^/n ^^p ^^q' wherep,qareanypositive integers (zero included). Itmaybeconvenient toplaceonrecord thefollowing expressions: 0„(t)=1/t, 0,(t)=l/f; 0,(0=V^+4/^^ Os(t)=S/t'+24>/t*, 0,(0=l/t+IQ/t'+192/i^ 0,{t)=5/t'+120/^+1920/^". The coefficieuts inthepolynomial 0„(0,forn=0,1,2,...15,have been calculated by Otti,Bern Mittheilunyen, 1898, pp.4,5. 9*11 .Therecurrenceformulae satisfied hyOn(t). Weshallnowobtain theformulae (1) (»-1)0.^,(0 +{n+1)0,Ut)-^ ^"\~^'' Onit)=^^^«^B!W, (,,,^1) (2) 0,^,(0-0„+,(0=20,/ (t), (n^1) (3) -0,(t)=Oo'(t). The firstofthesewasstated bySchliifli, Math. Ann. iii.(1871), p.137,andproved b}- Gegenbauer,^VienerSitzungsberichte,LXV.(2),(1872), pp.33—35,buttheother twowere proved someyearsearlier byNeumann, Theorie derBesseVschen Functionen(Leipzig, 1867), p.21. Sinceearly proofsconsisted merelyofaverification, weshall notrepeat them, butgiveintheirplaceaninvestigation bywhich therecurrence for- mulae arederived inanatural manner from thecorrespondingformulae for Bessel coefficients. Taking |^ |< |^|,observe that,by§91(1)and§2-22(7), {t-z)^ enJn {z)On(0=1=-e„COS*|n7r.Jn{z),«=0 n=0 *Itissufficient tousethetheorems that,if2?)„isconvergent, soalso is"Zbjn, andthatthen llbjn^isabsolutely convergent. tThiswaspointedoutbyPincherle, Bologna Memorie, (4)iii.(1881—2),p.160. XCf.Modern Analysis, §o*33. 9*11] ASSOCIATED POLYNOMIALS 275 andhence 05 00 Z1CnJn {Z)On(0=-^n'^n {z)[tOn (t)-COS^|n7r}n=0 M=0 =2enJn (z){tOn (t)-COS^|w7r}, 71=1 sincetOo{t)=l.Ifnowweusetherecurrence formula forJn{z)tomodify theexpressionontheright,weget XenJn {Z)On(t)=I{Jn-l {z)+Jn+l{z)][tOn(t)-COS^l/lTrj/n. n= 71=] Ifwenotice thatJn+i(z){tOnit)—cos-^n7r]/ntends tozero asn-^x, it isclear onrearrangementthat /.(2)10.(t)-to,(t)]+7,(2)120, (()-itO, (t)+i] +2/.(.){20,.(0-'«^f-^'>+?^"li-l=0. n=2 { n+l n—ln^—l) Nowregard2^asavariable, while tremains constant;ifthecoefficients of alltheBessel functions onthe leftdonotvanish, the firstterm which does notvanish canbemade toexceed thesumofalltheothers inabsolute value, by taking \z\sufficientlysmall. Hence allthe coefficients vanishidentically* and,from this result, formula (1)isobvious. Toprove (2)and(3)observe that -—]^-0 dtdzj t—z and so,\z\beinglessthan j^ j,wehave ienJn {Z)On{t)+56„./„' {z)On(t)=0. Byrearrangingtheseries onthe leftwefindthat ienJn (Z)On'(t)=/,(Z)Oo(0-S\Jn-l (z)"</„+: (z)}0.(t) n=0 n=l 00=-/o(Z)0,(t)-SJn(Z){0„+, (0-0,,_, (t)]. «=1 that istosay, ^0(Z)[O: {t)+6,(01+iJniz){20„' {t)+On+, (t)-On-, {t)}=0. »=1 Onequatingtozero thecoefficient ofJn{z)onthe left,justasinthe proofof(1),weobtain(2)and(3). *This istheargument used toprove that,ifaconvergent powerseries vanishes identically, then allitscoefficients vanish(cf.Modern Analysis, §8-73). Theargumentisvalid herebecause thevarious series ofBessel coefficients converge unifoimly throughout adomain containingz=0. 18—2 276 THEORY OFBESSEL FUNCTIONS[CHAP. IX Bycombining (1)and(2)weatonce obtain theequivalent formulae (4) ntOn-x (t)-{n--l)Onit)=(n-l) tOn (t)+nsin-Inir, (5) ntOn+^ (t)-(n^-1)0„(0=-(n+l)tOn (t)+nsin^ Ititt. If^bewritten for t{djdt),these formulae become (6) (w-1)(^+n+1)On(0=^i{iOn-, (0-sin'i?i7r}, (7) (/I+1)(^-n+1)On(t)=-n[tOn+^ (t)-sin^l^iir]. TheNeumannpolynomialofnegative integralorderwasdefinedbySchlafli* bytheequation (8) 0.n(t)=i-rOn(t). With this definition theformulae(1)—(7)arevalid for allintegral values ofn. 9*12. Thedifferential equation^ satisfied hyOn(t). From therecurrence formulae§9'11(6)and(7),itisclear that (^+?i+1)(^-n+1)On(0=r(^+n+1){-ntOn+, {t)+nsin''|n7r} ft j~X =^(^+n+2)On+i (t)+nsin^^mr =—t{tOn (t)—cos^2^7r}+nsin^^nir, andconsequently On(t)satisfiesthedifferential equation (^+1)^On(t)+(t^- 71')On(t)-tcos-^/iTT+7?sin-|??7r. Itfollows thatthegeneralsolution ofthedifferentialequation d^y Sdy /, n-—1\ cos^hiir nsin-hiir <i>-d-'-tI+V--^)y=—^^-^^ is y=On{t) +t-'Wn(t), and sotheonlysolution of(1)which isexpressibleasaterminatingseries isOn(t). Itissometimes convenient towrite(1)intheform ^-^^dt'^tdt^V f-r^"^^' where /ox ^/A_i^/^'(**®^^") ^"^ ^""^^^~ [nit'. (71odd) *Moth. Ami. Tii.(1871), p.138. tNeumann, Theorie derBessel'schen Functionen(Leipzig, 1867), p.13;Journal fiirMath. Lxvii.(1867), p.314. 9-12, 9-13] ASSOCIATED POLYNOMIALS 277 Another method ofconstructing thedifferential equationistoobserve that andso i^"""^*^"^^^^"^^^=f' (J^+^ a;^ '"'}r^2<-2 2z^ z 22 (<-2)3 («-2)2 «-Z Now 1=2f2«'^2« (2),2=2e2n+l(2^i+l)«^2n +l(4 00 andhence t+z=fi'2e„,9'„(0»/«(2). »i=0 Therefore 2e„,./™ (2) 7(=0^'5"•^4+^*"^'" ""'j^"^^^~ ^'•^" ^^^]=^• Onequatingtozerothecoefficient oiJ^iz) ontheleft-hand side ofthis identity, just asin§9"11,weobtain atoncethedifterential equationsatisfied byOn(t). 9'13. Neumann's contourintegralsassociated with On{z). Ithasbeenshewn byNeumann* that, ifGbeanyclosed contour, (1)\0^{z) On{z)dz=0, {m=nandmi^n) Jc (2)!J^{z)On{z)dz=0, (m?^n') Jc (3)[Jn(z)On(z)dz=27rikl€n, where kistheexcess ofthenumber ofpositivecircuits ofthecontour round theoriginoverthenumber ofnegativecircuits. The first result isobvious fromCauchy's theorem, because theonlysingu- larityofOm{z) On{z)isattheorigin,andtheresidue there iszero. Thethird result follows inasimilar manner;theonly poleoftheinte- grandisasimple poleattheorigin,andtheresidue atthispointisl/e„. Toprovethesecond result, multiplytheequations /- V,,J^{z)=0,V,{zOn (z)}=z'gn (z) byzOn(z)and/,„(z)respectively,andsubtract. IfU{z)bewritten inplaceof ,.d{zOn(z)}_^ (.dJjAz)^""^^^dz'''^^dz' theresult ofsubtractingassumes theform Z'U'(Z)+ZU{Z)+{W?- ?i^)Zj,n {Z)On{Z)=Z'gn {z)Jm(z), *Theorie derBesseVschen Functionen (Leipzig, 18G7), p.!!>. 278 THEORY orBESSEL FUNCTIONS [CHAP. EX andhence [zU{zyic+(w'-rr)\J^{z)On{z)dz=\ z^^g^ {z)J^(z)dz. Jc Jc Theintegrated partvanishes because U(z)isone-valued, andtheintegral ontherightvanishes because theintegrandisanalyticforallvalues ofz;and hencewededuce(2)when m^^n^. Two corollaries, duetoSchlafli, Math. Ann. iii.(1871), p.138,arethat 1/(0+ ) (4)/(0+ ) ^-./Jn{X+y)0^(y)dy=Jr,-m (*)+(-)'" ^n+m(^), (5) — ./ 0^(^+y)JnLv)dy=J^,_,,(^)+(-)"J^^„{x). The first isobtained byapplying (2)and(3)totheformula§2-4(1),namely •4(«+.y)= 2Jn+p{x)J_j,{y),P=—QC andthesecond followsbymaking anobvious changeofvariable. 9*14. Neumann'sintegral forOn{z). Itwasstated byNeumann* that (1) 0.(.)=/;'"^^<"'+ ^^'^^-^<"'+^'>'"e-'du. Weshallnowprove byinduction theequivalentformula /•ooexpia (2) On{z)=l\ [[t+V(l+«^)}"+[t- V(l+r-'))"]e-'d^, where aisanyanglesuch that |a.+arg2^ |<^tt ;onwritingt=u/z,thetruth of(1)willthenbemanifest. Amodification ofequation (2)is (3) On(z)=If"" '"{e""+(-)"«-"*}e-^sinhfl coshOdd. Jo Toprove (2)weobserve that foaexpia feeexpia Ooiz)= e-''dt, Oi(^)=•te-'^dt; Jo Jo and so,byusingtherecurrence formula§9'11(2),itfollows thatwemaywrite Tooexpia On(z)= <\in{t)e-''dt, J where (4) <\>n+, it)-2t<Pn (t)- (t>n-i(0=0, and (5) <^o(0=l. <PAt)=t. *Theorie derBesseVschen Functionen (Leipzig, 1867), p.16;JournalfiirMath, lxvii.(1867), p.312. 9-14]ASSOCIATED POLYNOMIALS 279 Thesolution ofthedifferenceequation (4)is 4>,(t)=A[t+^{t^-+1)}"-\-B\t- V(l+^0}". whereAandBareindependentofn,though theymightbefunctions of t. The conditions (5)shew, however, thatA=B=\\andtheformula (2)is established. ThisproofwasgiveninasymbolicformbySonine* whowrote(^„(Z)).{\lz)where we /ocexpia- have written / ^„{t)e~"' dt,Dstandingfor{djdz). '/: Acompletelydifferent investigationofthis result isdue toKapteynf, whoseanalysisisbased ontheexpansionof§9"1(1),which wenowwrite in theform z—^„=o When j^ i< !2^ i,wehave -r,=- \exp \^ii\du 2J(»=-oo ifpbesochosen that =K^-^)-w 1 z Itfollows that ^-r=r[ii^i±4^:±i2i!.^„(f)e~"du. Weshallnowshew thattheinterchangeofsummation andintegrationisjustifiable;it willbesufficient toshew that, foranygivenvalues of^and z(suchthat |C1< I^|), n=N^\ J andsoZ"' canbemadearbitrarilysmall bytakingNsufficiently large |;now \u±s/{u^ +z^)\^^{u +\z\), |2!"Jul *Math. Ann. xvi. (1880), p.7.Forasimilar syinbalic investigationsee§G-l-l supra. t^rm. Sci.deVEcole norm. sup. (3)x.(1893), p.108. JCf.Bromwich, Theonj ofInfinite Series, §176. 280 THEORY OFBESSEL FUNCTIONS [CHAP. IX Therefore, since !f|< i2 |,wehave r{u±J{u^ +z')Y. I, IC^+^Iexp {Iz \+iIC?}2 andtheexpressiononthe leftcanbemadearbitrarilysmall bytakingNsufficiently large when zand(arefixed. Hence, when |^ j< |^^ |,wehave •3b«=-<» .'0 ^ =ienJniOOniz), «= .where 0„(i^)isdefined bytheequation and itiseasytoseethat0„{z),sodefined, isapolynomialin\jzofdegree ?i+ 1. When theintegrandisexpanded*inpowersofzandintegratedtermby term, itiseasytoreconcile thisdefinition ofOn{z)with theformula§9'1(4). 9'15.Sonine'sinvestigation ofNeumannsintegral. Anextremely interesting andsuggestive investigationofageneral type ofexpansionofl/(a—z)isdue toSoninef; from thisgeneral expansion, Neumann's formula(§9*1)with theintegralof§9'14canbederived without difficulty.Sonine'sgeneraltheorem isasfollows: Leta/t(w)beanarbitrary function ofw;and,ifyjr{w)=cc,letlu—^(x), sothat-//visthefunctioninverse toyjr. LetZnandA^bedefined bytheequationsl Then ^=^^nAn, itbeing assumed that theseries ontherightisconvergent. Supposethat foranygiven positivevalue o{x,\w\>\-^{x)\on o.closed curveCsurroundingtheoriginandthepoint z,and\'w\<\^{x)\onaclosed *Cf.Hobson, PlaneTrigonometry (1918), §264. tMathematical Collection(Moscow),v(1870), pp.323—382. Sonine's notation hasbeen modified slightly, butthesymbols \f/and//iarehis. XThis isconnected with Laplace's transformation. SeeBurkhardt, EncyclopadiederMath. Wiss. II.(Analysis) {m&), pp.781—784. 9*15, 9-16] ASSOCIATED POLYNOMIALS 281 curve csurroundingtheorigin butnotenclosingthepointz.Then 'n-=OJ JC 1 '^,f^'f W'^ 27riJo[JCJr] tu— -/fi{x) — I I QZ>il(W\—aiX 2'rriJ J w—jf^{x) Jo providedthatR{z)<R (a) ;andtheresult isestablished ifitisassumed thatthevarious transformations arepermissible. Inorder toobtain Neumann'sexpansion,take yfr{iv)=1(w-1/w),^(x)^x± ^(x"+1), andthen "^11=-'X QO=y1, M= Since ^„+(-)"^-n=re--""[[x±^(ar+1)]"+(-)« [x±V(*'-'+1)}-"] dx, weatonce obtain Neumann's intesfralo' Sonine notes(p.328)that sothat theexpansionofl/(a-2)converges when |2 |< ja j;and inthelater partofhis memoir hegives furtherapplicationsofhisgeneral expansion. 9*16. Thegenerating function ofOn{z). The series 2(-)"e«i!"0„ (2),which isagenerating function associated with0„(2), doesnotconvergeforanyvalue oftexceptzero. Kaptejn* however, has"summed" the series after themethod ofBorel, inthefollowing manner : '•" , , /^ ,N1"^ '"n.(n+m-l)\t~'' »i=0'^n=()»(=o \i''—inj.(^2*/ _*"{n+^).{n+m )lt^"-^'^ '''^_11+r^" ^n.{n+m-l)\t-'"' _1-{2m)If^"'(l+t^) _\'^ (2TO+1)!i!^'"+'(l+i'^) Nieuw Archief voorWiskunde(2),vi.(190.5), pp.49—55. 282 THEORY OFBESSEL FUNCTIONS [CHAP. IX -Tj— -57———- ,andthisintegral isconvergentsolongas(1-fi)zjtisnotnegative. 00 There isnogreat difficultyinverifyingthattheseries 2(—)"f„^"0„(3)isanasym-n=o ptotic expansionoftheintegralforsmallpositive values oftwhen |arg2 1<tt,andsothe integral mayberegardedasthegeneratingfunction ofOn(2).Kapteyn hasbuiltupmuch ofthetheoryofNeumann's function from ihis result. 9"17.Theinequality ofKapteynstypeforOn{nz). Itispossibletodeduce fromNeumann'sintegralaninequalitysatisfied byOn(n^;)whichcloselyresembles theinequalitysatisfied byJ^inz) obtained in§8-7. Wehave ^"(^^^)=2^ f" 1^^^^^^'^^^'^^'^^"+{w-V(w^+^-)}«]e-""-dw, thepathofintegration beingacontour inthew-plane,andso where that value oftheradical istaken whichgivestheintegrandwith the greatermodulus. Now thestationary pointof is\/(l— •2^),andso where thepathofintegrationisoneforwhich theintegrandisgreatestatthe stationary point. Ifasurface ofthetypeindicated in§8*3 isconstructed overthety-plane, thestationary pointistheonly passonthesurface;andbothw=and ta=+ccareatalower levelthan thepassif (2)zexpV(l-Z-) Hence, since acontourjoiningtheorigintoinfinitycanbedrawn when (2)is satisfied, andsince theintegralinvolved in(1)isconvergentwith thiscontour, itfollows that,throughoutthedomain inwhich(2)issatisfied, theinequality 1+V(l-^')""' (3)On(nz)<r^^ zexp \/(l—2^) issatisfied forsome constant value ofA ;andthis isaninequalityofthesame character astheinequalityof§8"7. 9"17, 9-2] ASSOCIATED POLYNOMIALS 283 9'2.Oegenhauers generalisation* ofNeumanns'polynomial. Ifweexpand z^^t—z)inascending powersofzandreplaceeachpowerof zbytheexpansionasaseries ofBessel functionsgivenin§b%wefindon rearrangement that z"_^z"-^' t-Z.,=^^+1 n=[m-0in--.m+, ^^1 therearrangement hasbeen effectedbyreplacingsbyn—2m,and itpresents nogreater theoretical difficulties thanthecorresponding rearrangementin§9-1. Wearethus ledtoconsiderGegenbauer's polynomial An,^(t),definedby theequation (1)^...W=2::;i^)?I>±^(W:=0 ni. thisdefinition isvalidwhenever visnotzero oranegative integer;andwhen I^1<1^1,wehave (2) ,—--XA^,^(t)J,^,,{z). I'—Z„=o Thereader should havenodifficultyinprovingthefollowingrecurrence formulae : (3) {v+n-l)A,^,,,(t) +{v+n+l)A,,_,^^(t)-^^^''^f~^'AnAt) 2"(v+7l)\(V+nY-1|r(i;+1 ;,-1 ).= ^TIFTf)-sm-l.vr, (5){v+n)tAn-,,,{t)-{n+1){p+n-l)A„^At) —{v+n-\.)tA n,V\t) -\^^ryi——it^m-gnir, 1{^^n+2/' (6){v^n)tAn+,,At)-(v +n+l){2v +n-l)An,At) / ix..' /.N 2''(v-\-n){v +n+\)r{v +hi+l). .,, 1\^n+^) (7) A,,^{t)=2''T{v+l)lt. *Wiener Sitzungsberichte, lxxiv.(2),(1877), pp.124— 1;^0. 284 THEORY OFBESSEL FUNCTIONS[CHAP. IX The differentialequationofwhich J.„„(<)isasolution is where Thegeneralsolution of(8)isAn,„(t)+1"-''^^+„ (0- Ofthese results, (3), (4),(8)and(9)areduetoGegenbauer;andhealso provedthat (10)— . jAn, .(t)e-*dt=2"i-r(v).(v+n)C."(z), where Cn{z)isthecoefficient ofa"intheexpansionof(1—2az -\-a^)'" \this formula iseasily proved bycalculatingtheresidue of(t^^^)"* An,v{t)attheorigin. Thecorrespondingformula forNeumann'spolynomialis 1 ;*«>"•"* (11)-— .IOn(t)e^^^dt=i^cos\narccosz],"tti J Thefollowingformulae mayalsobementioned : (12)IAjt,„{z)Any{z)dz=0, (m=nandm^n) Jc' (13) [2-''J,^,n(z)An,Az)dz=0, (m^ :^w=) Jc (14) (z-''J,+niz)An,,{z)dz=27rik, .'c whereCisanyclosed contour,vi=0,1,2,...,andkistheexcess ofthenumber ofpositivecircuits overthenumber ofnegativecircuits ofCround theorigin. The firstandthird oftheSe lastresults areproved bythemethod of§9'13; thesecond isderived from theequations ^i'+mdv+m{z)= 0,^v+n{z^"An,v\Z)\=Z^ gn,v\Z), whence wefindthat (m-n){2v\-m+n)z-"J^+ni (z)An,u{z)dz= \z''-"gn,v{z)J"^+,« {z)dz=0. Jc Jc 9*3. Schldfli's polynomial Sn(t). Apolynomial closelyconnected withNeumann'spolynomial 0„(t)was investigated bySchlafli. Inview ofthegreater simplicityofsome ofits properties,itisfrequentlyconvenient touse itrather thanNeumann'spoly- nomial. 9-3] ASSOCIATED POLYNOMIALS 285 Schlafli's definition* ofthepolynomialis (1) S,„(t)=t"^:^^l^l^)}(lt)--^-m^ (u^l)«=o nil^ (2) So(t)=0. Oncomparing (1)with§9-1(2),weseeatonce that ('3) |/iSn(t)=tOn(t)-cos2hmr. Ifwesubstitute forthefunctions 0,j(t)intherecurrence formulae§9-11(1) and(2),wefindfrom theformer that (4) Sn+^ it)+.S;_i {t)-2)H-'S,,(t)=U-'cos-1 n-TT, andfrom thelatter, i(n-l)Sn-^ (t)-^(n+l).9„+,(0=nS^' (t)-nt-'S,,(t)-2t-'cos^i/^tt. Ifwemultiplythisby2andaddtheresult to(4),weget Theformulae(4)and(5)may,ofcourse, beproved byelementary algebra byusingthedefinition ofSn{t), withoutappealingtothepropertiesof Neumann'spolynomial. The definition ofSchlafli'spolynomialofnegative order is and,with thisdefinition, (4)and(5)aretrue forallintegralvalues ofw. Theinteresting formula, pointedoutbySchlafli, iseasilyderived from(3)and(4). Other forms oftherecurrence formulae whichmaybederived from(4) and(5)are (8) tS„_, (t)-nS„ (t)-tS,,' (t)=2cos-^Imtt, (9) tS,,+, (t)-7lS,, (t)+tSn (t)-2COS^iUTT. Ifwfcwrite^for t(d/dt),these formulae become (10) {^+77)Sn(t)=tSn_,(0-2COS'-^i 7l7r, (11) (^-n)Sn(t)=-tSn+, (t)+2COS-1 WTT. Jt-follows that (^--n-)Sn{t)=t('^+l-n)Sn-i (t)+-Incos^|mr =— ^'-*S'„ {t)+2tsin'^^UTT 4-2ucos-inir, andsoSn{t)isasolution ofthedifferentialequation (12)t'^i^^+t'',+{P- H-)u^-ltsin^\mr+2ncos-1?i7r. *Math. Ann. in.(1871), p.138. 286 THEORY OFBESSEL FUNCTIONS[CHAP. IX Itmaybeconvenient toplaceonrecord thefollowing expressions: S,(t)=2/t+16It', S,(0=8/t'+96/f, S,(t)=2/t+48/t'+768/t', S,(t)=12/t'+384/^^+7680/^. Thegeneral descending series, given explicitly byOtti, are ^(|n+m-l): ,=1(ln-m)l{ity(13) Sn{t)= 2,il_.^.i..,,n (neven) 2n2n{n"-2'-) 2n(n- -2-)(n- -4^-) t- f f' _22{ii"-1^) 2(/?.^-1-){n?-3-). -^+ ^^+ ^i+•••• The coefi&cients inthepolynomial »S'„{t),for»=1,2,...12,havebeen calculated byOtti, Bern Mittheilungen, 1898, pp.13—14;Otti's formulae arereproduced (with some obvious errors) byGrafandGubler, EinleitungindieTheorie derBesseVschen Funktionen,ii.(Bern, 1900), p.24. 9"31. Formulaeconnectingthepolynomials ofNeumann andSchldfli. Wehavealreadyencountered twoformulaeconnectingthepolynomialsof Neumann and Schlatli, namely \nSn(t)=tOn (t)—cos'^|?l7r, Sn-^(t) +S,^^(t)=Wn{t), ofwhich theformer isanimmediateconsequenceofthedefinitions ofthe functions, andthelatter follows from therecurrence formulae. Anumber of other formulaeconnectingthetwofunctions aredue toCrelier*; theyare easilyderivable from theformulaealready obtained, andweshallnowdiscuss themoreimportantofthem. When weeliminate cos-|w7r from§9"3(3)andeither§9"3(8)or(9),we findthat (1) Sn-At)-Sn'{t)=20nit), (2) S„^,{t) +S,;{t)=20n{t). Next, onsumming equationsofthetype §9'3(5),wefindthat (3) .Sf„{t)=-2^^T'^ *Sf'„_„„_i (t)+sin^iUTT :S,it), andhence (4) Sn{t)+S,,_,(0--2's'^n-m-i (t)+S,it).m= *Comptes Rendus, cxxv.(1897), pp.421—423, 860—863; BernMittheilungen, 1897, pp.61—96. 9-31, 9-32] ASSOCIATED POLYNOMIALS 287 Againfrom§9"3(7)and(5)wehave 4[On-, it)+On^. {t)]=Sn-, (t)+28,, (t)+S,,^, (t) ={S,^2 (t)-S„(t)]- \Sn(t)-Sn^, (t)}4>Sn (t) SOthat (5) Sn" (t)+S,(t)=0„_, {t)+0,,+, (t). This isthemostinterestingoftheformulae obtainedbyCrelier. Again,onsummingformulae ofthetypeof§9'11 (2),wefindthat (6)On{t)=-2S0'„_.>,„_!(0+sin-^1nir .0,{t)+cos^|n-Tr .Oo(0, -.11= andhence (7) 0„(t)+0n-At)=-2t0',,_,,,_,(t) +0,{t)+0,{t).m= 9'32. Graf's expression ofSn(z)asasum. Thepeculiar summatoryformula (1) Sn(z)='rri[Jn(^)ym(2)-J>n(z)Yn(^)} III=-n wasstated byGraf* in1893, theproof being suppliedlater inGraf and Gubler's treatisef.Thisformula ismostreadily proved byinduction; itis obviouslytruewhen 7i=0,and also,by§3'63(12), when n=l.Ifnow the sumontherightbedenotedtemporarily by(f}n{2),itisclear that nfl n+1 4=ITJn+i {Z)t Y,n{Z)-TTYn+i {z)SJ,„{z)m-—?i-l »ft=-«-l n— 1 n-l +irJ.n-,{z) SY,,{z)-'7TYn-,{z) 1J,,{z) m,=—n+l m=—)i+\ -{2n7rlz)Jn(z) SY,,(z)+('In-rr/z) Yn{z) I/,«(4 in=—n m=-n Nowmodifythesummations ontheright bysuppressingorinsertingterms atthebeginningandendsothat allthesummations run fi'ora—nton;and wethen seethat thecompletecoefficients ofthesums!£t/„i(2) and2F,„(2) vanish. Itfollows that <^„+, (2)+</)„_! {z)-{2n\z) (f)n(z)^=7r/„+, {z) \F„+, {z)+F_„_j {z)]-ITF„+i (2){./„+j {z)+/_„_, {z)\ -TTJ„_j {z) [F«{z)+Y_,{z\\+'TT IV, ^z)[J,,(z)+J_n {z)] =_^|i^(_i)«}!/„_^ (^)r,^(^)_/„ (.)F,_^ (2)} =4^~^cos--^?i7r, by§3'63(12);andso(^n{z)satisfies therecurrence formula which issatisfied bySn{z), andtheinduction that<i>n(z)=Sn{z)isevident. *Math. Ann. xliii.(1893), p.138. tEinleitiuig indieTheorie dcrBesserscli'm Funkliou<>n,11.(Bern, 1900), pp.34—41. 288 THEORY OFBESSEL FUNCTIONS[CHAP.IX 9'33. Creliersintegral forSn{z). Ifwetaketheformula§9'14(2), namely rXexpia On(^)=i I[[t+V(l+t^)Y+{t- V(l+P)Y'\e-^'dt, andintegrate byparts, wefindthat On{z)=''I^'+o-(e-^^[{^+v(l+t^r+{t- V(l+t^r]dt z Hence itfollows that1z\ dt "_f"exp/a ;^+V(l+^^)]»-|^_^(l+f2)j« _^^ "^2^Jo V(l+«0^ (1)'^-^^^-Jo v(i+^=')rfi. Thisequation,which wasgivenbySchlafli, Math. Ann. iii.(1871), p.146, intheform (2) Sn{z)= j"{enB-^-y^e-ne^e-ZBinhdcld^ isfundamental inCrelier's researches* ofwhich weshallnowgiveanoutline. Wewritetemporarily Tn={t+x/(l+t^)}^-{t- V(l+t')Y\ T—^fT—T — -*71+1-t -*-n-*n— 1— '-')andthen sothat in+i-.. - 1=2t+ andtherefore-*n/J-n—i ^^=2^+11 1 ^;^'^2^ +2^+...+2r thecontinued fractionhavingnelements. Itfollows that Tn^^jT^isthe quotientoftwosimplecontinuants^ sothat Tn+i_ K{2t,2t, ...,2t}n Tn" K{-lt,2t,...,2t)n-,' thesuffixes n,n—\denotingthenumber ofelements inthecontinuants. Itfollowsthat| TJK(2t)n-iisindependentofn;andsince r,=2vXl+nK{2t\=\, wehave T,,=-l^{\+t^).K{2t)n-,, *Comptes Rendus, cxxv.(18!)7), pp.421—423, 860—863;Bern Mittheilungen, 1897, pp.61—96. tChrystal, Algebra,ii.(1900), pp.494—502. XSince alltheelements ofthecontinuant arethesame, thecontinuant may beexpressed by thisabbreviated notation. 9-33, 9-34] ASSOCIATED POLYNOMIALS 289 andhence /•a3expia (3) Sn{z)=2 K{2t)n-^e-''dt.Jo From this result itispossibletoobtain alltherecurrence formulae for Sn(z)byusing propertiesofcontinuants. 9*34.Schldfiis expansion ofSn{t+z)asaseriesofBesselcoefficients. We shallnow obtain theresult due toSchlafli* that,when\z\<' t\, Sn(t+z)canbeexpandedintheform (1) Sn(t+z)^ IS,_„,{t)J,,(z).m=—00 Thesimplest method ofestablishingthisformula forpositivevalues ofn isbyinductionf.Itisevidentlytruewhen n=0,forthenboth sides vanish; when 7i=1,theexpressionontherightisequalto m=1 =20o (t)./„{-z)h I{^„,,_i(0+'SV, (t)}J,,(-z)m=l =21e„,0,n{t)Jra{-z)m= =2l(t+z)=8,(t+z), by§9-1(1) and§9-3 (7). Now, ifweassume thetruth of(1)forSchlafli'spolynomialsoforders 0,1,2,...n,we-have S,,+, {t+z)=S^_, {t+z}-2S,: (t+z) m=—X w=—Qo =?{Sn-,a-^ it)-2.S"«_„, {t)\ J,„,(Z)m=—X andtheinduction isestablished;toobtain thesecond line intheanalysis, wehave used theobvious result that s,:{t+z)=^^s,,(t+z). *Math. Ann. iii.(1871), pj:).139—141; tlieexamiuation oftheconvergenceoftheseries is lefttothereader(of.§9-1). tTheextension tonegative values ofnfollows ontheproof forpositive value?, by§9-3(6). W.B.F. 10 290. THEORY OFBESSEL FUNCTIONS [CHAP.IX Theexpansion wasobtained bySclilafli byexpanding every term ontherightof(1)in ascending powersofzanddescending powersof t.Theinvestigation givenhere isdueto Sonine, Math. Ann. xvi.(1880), p.7;Soniue'sinvestigation wasconcerned with amore general class offunctions than Sehlafli'spolynomial, known ashemi-cylindrical functions (5510-8). When wemake useofequation §9'3(7),itisclear that,when [^^ |< |^|, (2) On{t+z)=iOn-rn{t)J,niz).m=—00 Thiswasproved directly byGegenbauer, WienerSitzungsberichte,Lxvi.(2),(1872), pp.220—223,whoexpanded 0„{t+z)inascending powersofzbyTaylor's theorem, used theobvious formula[cf.j^9'll (2)] (3)2"—^'= ^^2^(-)-,C„,.0„_p+2,H {t),dtp,„=( andrearrangedtheresulting double series. Itiseasytodeduce Graf's*results(valid when\z\<\t\), 00 (4) Sn{t-z)=^ 1S„^,„(t)J.Az), (5) On{t-z)= iOn^,n(t)J,n(z).m=^00 9'4.Thedefinition ofNeumann'spolynomial iln{t)- Theproblemofexpandinganarbitraryevenanalyticfunction into a series ofsquaresofBessel coefficients wassuggestedtoNeumannfbythe formulae of§2'72,whichexpress anyevenpowerof^asaseries ofthistype. Thepreliminary expansion, correspondingtotheexpansionofl/(t—z) givenin§9"1, istheexpansionof\l{t-—z'^)] andthefunction n„(i)will bedefined asthecoefficient ofenJn{z)intheexpansionofl/(f-—z^),sothat (1),T^=Jo'(^)^0(0+^J^ (^)-^1(0+2/2^ (z)a(0+... »i= Toobtain anexplicit expressionforn„(^), take^'j< |i|,and, after ex- panding l/(^^— z"^^inascending powersofz,substitute foreachpowerofzthe *'Math. Ann. xliii.(1893), pp.Ill—142; seealsoEpstein, Dievierliechnung<operationen milBesseVschen Functionen(Bern, 1894). [JuJirbuch ilber dieFortschritte derMath. 1893—1891, pp.845—846.] tLeipzigerBerichte, xxi.(1869), pp.221—256. [Math. Ann. in.(1871). pp.581—610.] 9-4] ASSOCIATED POLYNOMIALS 291 series ofsquaresofBessel coefficientsgiven byNeumann(§272). Asin §9'1,wehave 1_^z""' _1V rv.^4- ^2"M 1V(-2^+2m).(2^+77.-1)! ] m= ,rif-'*+^*(2s)! u=o 1» cc 912-« rS'V I"??+s—1V when werearrangetheseries bywritingn—sforni ;thisrearrangement presentsnogreatertheoretical difficulties than thecorresponding rearrange- ment in§9'1. Accordinglythefunction Cla(t)isdefinedbytheequations (2)1-n.(n+s-l)l{ sir- ^"^^ ^^"^^^~ 4,ro(n-s)!(2s)! (i0'"^'' (B) n,(t)=i/t\ Onreversingtheorder oftheterms in(2)wefindthat _1^n.(2n-m -1)1 {(71-m)lY(n^1) (4) n.(04,„=om I(2n-^2m)I(|i)2«-2m+2(ri^l) while, if(2)bewritten outinfull, itassumes theform 114/1^ 1.24?2^(4n--2-)1 .2 .34?r-(4/t--2-)(4n--4^) (5)"n(0= ,.+2^+3f4 ,« Also 2-"(w.!)-i (6) e«n,,(i)= ^271+ :1+®.t'+ +4.5.6+. e,t* I^"^ 1 .(2?i-1)"^ 1 .2.(2w-1)(2n-2)+ . where (7)(h)—+ ©4=) 1.2...w.(2%-l)(2n-2)...?ij' (2n-l)(2n-S) a=•Zn 2n{2n-2) {2)i-l)(2n-S){2n-o) 2n(2/?-2)(2n-4)' ^^(2m-1)(2»-3)...1 '-"2?i(2n-2)...2 Since <©271<1,itiseasytoshewbythemethods of§2-11that (8) i€nn„(t) 1^2- 11 1-^"-^(n !)•-'exp (I«\'), and,when n>0, (9) e,n,(0-2-i-'^»'-^ (niy (1+^), where\d\% [exiplty' -l}/{2)i -I). 19—2 292 THEORY OFBESSEL FUNCTIONS[CHAP.IX Byreasoningsimilar tothatgivenattheendof§91, itiseasytoshew thatthedomains ofconvergenceoftheseries Sa^J^{z)n,i{t)and 2a,i{zjty^ arethesame. Thereader should havenodifficultyinverifyingthecurious formula, due toKapteyn*, (10) n„(o= -2^/;'o..(J|-,)rf^. 9*41. Therecurrenceformulae f07'n„{t). Theformulaecorrespondingto§9"11 (2)and(3)are (1)|n„'(o=^H^-5^-??4>,(»^2) t n—\ 71+1 Tn}—\ (2) (2/0n/(0-^iHo {t)-in,(0, (3) (2/0 Ho'(0=-2n,(0+m,(0- There seems tobenosimple analogueof§9*11(1).Themethod bywhich Neumannfobtained these formulae isthatdescribed in|9'11. Take thefundamentalexpansion §9*4(1),andobserve that andthat,byHansen'sexpansionof§2*5, 2J"o{z)J:(z)=-zi {J\-, (z)-/Vi (z)]/"' w=l Wefindbydifferentiations withregardtot,andwithregardtoz,that n= 2z/{t'-zj=2Jo(z)/„'(z)Ho{t)+zl[J\_, {z)-J\^, {z)\n„(0/n =zl[J^^.,{z)-J\+,{z)]{n^{t)-n,{t)]|n. Oncomparingthese res'dts, itisclear that t-^ie„Jn'{z)^n{t)+ i\J\^,{Z)-J\^AZ)\ .[Unit)- ^o{t)]ln=0. n=0 M-1 Onselectingthecoefficient ofJn{z) onthe leftandequatingittozero (cf.§9'1),weatonce obtain thethree stated formulae. *Ann. Sci.deVEcole norm. sup. (3)x.(1893), p.111. tLtipziger Berichte, xxi.(1869), p.251.[Matli. Ann. iii.(1871), p.606.] 9-41, 9-5]ASSOCIATED POLYNOMIALS 293 9*5. Oegenbaner's generalisation ofA^eumannspolynomial I^nCO- Ifweexpand z'^'^^l{t—z)inascending powersof^andreplaceeachpower ofzbyitsexpansionasaseries ofproductsofBessel functionsgivenin§5"5, wefindonrearrangement (byreplacingshyn—2m)that _"2'"+"+^ i|,r(ju+|^+l)r(»/ +^g+l){fM+ i'+s+2m)V in-Vv+s+m)^ s^ot'-"' (m=o~ r{fi+v^s+l) ml XJfjL+^s+m\^)"f+As+w\^) f 00 I<A»9/x+f+n—2m n=()(w=0«i— 2?n+i (ft+1/+//)r(/A+i?i-??i+1)r(i/+-^n-m+l)T(fj,+ v-{-n- m) m\V{iJi,^-v +n-'Im+l) itissupposedthat |^ |< |i |,andthen therearrangement presentsnogreater theoretical difficulties than thecorresponding rearrangementof§9"1. Weconsequentlyareledtoconsider thepolynomial 5„.^_^(^),definedby theequation (1) Bn,^,,{t)=-^3 ^''V{lx+\n-m+l)V{v+\n-m+l)V{fi+v+n-m) Thispolynomialwasinvestigated byGegenbauer*;itsatisfies various recurrence formulae, none ofwhich areofasimplecharacter. Itmaybenoted that (2) B,n.,.At)=^nt^n{t). Thefollowing generalisationsofGegenbauer'sformulae areworthplacing onrecord.Theyareobtained byexpandingtheBessel functions inascending series andcalculatingtheresidues. (3) ^.\^'''^\-^ J,{^t%m6)B,„ +,,^,At)dt=i). (4)^.^t-"J.{2tsin(l>)B,n.,,, At)dt^ _2'^+"(/i+t;+2/0r(/x+1)r(/A+1/+w)sin" (f>~ n\r(/i+i^+1) X3^2(-n,^l^\,ii.^V'^n\\^l\\v +\,\^JL^\v^r\\sin-</)). Inthespecialcase inwhich/x= i',thisreduces to (5)^r"V''/,(2«sin(^)i^,„;...(0rf^=2-=''('+»)r(^)siii''<^C',/(cos2(/.).LlTl J Thisformula maybestillfurtherspecialised bytaking </>equalto^ttor^tt. *Wiener Sitzungsberichte, lxxv,(2),(1877), pp.218—222. 294: THEORY OFBESSEL FUNCTIONS [CHAP.IX 9'6. Thegenesis ofLommel's*'polynomial Rm,v{z). Therecurrence formula .7^+1 {z)={2vlz) J^(z)-/,_i (z) may obviouslybeused toexpress Jv+m (-2^)linearlyinterms ofJ^(z)and J„-i (z) ;andthecoefficients inthis linear relation arepolynomialsin1/z which areknown asLortimeVspolynomials. Weproceedtoshewhow to obtainexplicit expressionsforthem. The result ofeliminating .Z^+i {z),J^+2 i^),•••Jv\-m-\ i^)from thesystemof equations 7^+^+1 {z)-{2(v+p)lz] J,+p {Z)+./,+p_i {Z)=0,(p=0,1,...771-1) iseasilyseen tobe Jv^m{z\ -2z-'^{v+m-\\ 1, 0, 1, -22-i(i/ +ffi-2), 0, 0, 1, 0, 0, 0,1 Jy{z\ 0, 0, -22-i(v+l) J,_i{z)-{2v/z)J,(z), 0, 0,1 Byexpandingincofactors ofthe firstcolumn, weseethat thecofactor of Jv+m (2)isunity;andthecofactor of(—)"'~^J„(z)is -2?-i(j/ +m-l), 1, 0, 1, -2z-^{p+m-2), 1, 0, 1, -23-i(i/ +m-3),0, 9-6,9-61] ASSOCIATED POLYNOMIALS 295 that istosay Itiseasytoseethat*R,n^^{z)isthenumerator ofthelastconvergentof thecontinued fraction 2z-'(p+ ni-1)2z-'(v+m-2)-2z-'(i'+m-8)- ...-2^-' v' The functionR,n,^^,(z)wasdefined byLommel bymeans ofequation (1). Hethen derived anexplicit expressionforthecoefficients inthepolynomial byasomewhat elaborate induction;itis,however, simplertodetermine the coefficientsbyusingtheseries fortheproductoftwoBessel functions inthe waywhich willbeexplainedin§9'61. Ithadbeenobserved byBessel, Berliner Abk., 1824, p.32,that, inconsequenceofthe recurrence formulae, polynomials ^b_i (2),At-i (z)existsuch that where[cf.§9-62(8)] ^n-l (z)B,,{z)-A,,(z)B„_i{z)=^, ^^^ ^,^^^_^y,g^^• Itshould benoticed thatGraft andCrelier|useanotation which differs from Lommel's notation;theywriteequation (i)intheform ,/^(^0=P""^ (x)Jix)-P" ,ix)J,{x\ 9'61. Theseries forLommeVsliolynoviial. Itiseasytoseethat(—)'"J_^_,« (^),quafunction oftheinteger m,satisfies thesame recurrence formulae as/„+,„ {z) ;andhence theanalysisof§9'6also shews that (1) (-)'» ./_,_,^ {Z)=./_,{Z)R,n.,.(z) +./_,+, (Z)Rra-l,.+l (z). Multiplythisequation by./,_i (z)and§9-6(1)by./_^+i (z),andaddtheresults. Itfollows that (2)J,+„ (z)./_,+! (z)+(-)- /_,_,„ (z)/,_! (z) =R,n,u{z) [Ju(z)/_,+! (z)+./_,(z)/,_, (z)] 2sinVTT TTZRm,v{z), *Cf.Chrystal, Algebra,11.(1900), p.502. tAnn. diMat.(2)xxiii.(1895), pp.45— fi5;Einh'ilang indieTheorie derBesscVschen Funk- tionen,11.(Bern, 1900), pp.98—109. tAnu. diMat.(2)xxiv.(1896), jip.131—163. 296 THEORY OFBESSEL FUNCTIONS [CHAP.IX by§3-2(7). But,by§5-41,wehave n^Qn\V{-V-in+?i+1)r(i/+n) when Avereplaceninthe lastsummationb}-m+p+1.Now itisclear that {V+})m+p+^ ^(??t+2j)+1)!^(m+jj +2)p and so,whenwecombine theseries fortheproductsoftheBessel functions, we findthat 2sinvir„ /x_vizT^H-jn +«)„(i^)-'"+^^-i _sinvir<|"'(-)"(m-n)\r{v+ m-n)(|^)-^+^*-^ , ~V~„to w !(m-2n)ir{v+n)' theterms forwhichn>^m.vanish onaccount ofthepresenceofthefactor (—on+n)ninthenumerator. When Visnotaninteger, weinfer that n^ 7? f.\-^s"(-rOn-n)lr(v+m-n)(|^)-'"+^» ^d; ii,,,,{z)- ^^^^ n\{m-2n)\r(v +n) -^^"(^Y c^(^+"^- ''"> (1,)-^n^-^n n=o 1(t*+«; Buttheoriginaldefinition ofR,n„iz), bymeans ofadeterminant, shews thatR^^{z)isacontinuous function ofvforallvalues ofv,integralornot; .and so,byanobviouslimiting process, weinfer that(3)isavalidexpression forRjft„(z) evenwhen visaninteger. When z/isanegative integeritmay benecessarytoreplacethequotient ^(^+^n-n) ^^^_^„,n-v-n +l) r{v+n)^^^r{-v-m +n+l) inpartoftheseries. The series(3)wasgiven byLommel, Math. Ann. iv.(1871), pp.108—111;anequi- valentresult, inadifferentnotation, had,however, beenpublished byhimtenyears earlier, Archiv derMath. 7mdPhys.xxxvii.(1861), pp.354—355. Aninteresting result, dependingontheequivalenceofthequotients just mentioned, was firstnoticedbyGraf*, namelythat (4) R,n,.{z)=(-)'"Rm,-.-«,+! {z). *Ann. diMat.(2)xxiii.(1895), p.56. 9-62] ASSOCIATED POLYNOMIALS 297 Inthenotation ofPochhammer(cf.|§4*4,4*42),wehave (5) 72,„_^(2)=(v)„i (Iz)-'".oF,(I-Im,-\m; v,-in,\-v-m;- z"-). SinceR,„^v{z)/2isalinear combination ofproductsofcylinderfunctions of orders v+mand i>—l,itfollows from15"4that itisannihilatedbytheoperator [^^-2{{v +mf+(v-If]^'^+{{i'+my-{v- If}-']+4>z^(^'^+3^+2); where^=^(d/dz);andsoi?„, „(2^)isasolution ofthedifferentialequation (6) [(^+m)(^+2v+m-2)(^-2i;-m)(^-,u-2)]y +4^-^^(^+ 1)^=0. Anequation equivalenttothiswasstated byHurwitz, Math. Ann. xxxill.(1889), p.251;andalengthy proofofitwasgiven byNielsen, Ann. diMat.(3)vi.(1901), pp.332—334;asimple proof, differing from theproof just given,maybeobtained from fornuila(5). 9*62. Various'properties ofLommeVspolynomial. Weproceedtoenumerate some theoremsconcerning Rm,i'{z),which were published byLommel inhismemoir of1871. Inthe firstplace, §9'6(1)holds iftheBessel functions arereplaced byany other functionssatisfyingthesame recurrence formulae;and, inparticular, (1) };,+„, {z)=F,{z)R„,,{z)-r,_i (z)E„,_>, ,+,(z), whence itfollows that (2)7,+,, (z)o\_, {z)-J,^,, (z)F„_i (z) =R,n,A^) [l\{z) /,_, {z)-J,{z)F,_,(2)|=-2R,nA^)l{irz). Next, in§9'61(2),takemtobeaneveninteger;replacemby2//<,and vhyV-m.Theequationthenbecomes (3)J,+,n(z)Jm+i-.(z) +J-.-,. {Z)j_,n.l+.{Z)-2(-)'"sinVIT.R.^,n,.-m{z)l(-7rz), and, in'thespeciafcasev= }2,weget (4) J%n^, (z)+J^_,„_i {z)=2(-)- i?,,,,i_,„ iz)l(7rzl that istosay 2^,(2^r--""(2m-n)!(2m-2»)! (5) J^„+. {z)+./-_,„_> (z)=—^^—— - .,,,,.• This isthesj)ecialcase oftheasymptotic expansionof§7-51when the order ishalfofanoddinteger. Inparticular, wehave J%(z)+JU(z)=--, (6)irz 2/, 1 X2/, 645 225, 298 THEORY OFBESSEL FUNCTIONS [CHAP. IX Formula(5)waspublishedin1870byLommel* who derived itatthattimebya directmultiplicationoftheexpansions (§3"4) ^...iWT(-)».-^-,-iw-(i)*(T.T^-."-;i /'"-;^;.)f-j^g. followed byasomewhat lengthy induction todetermine thecoeflQcients intheproduct. Asspecialcases of§9*6(1)and§9"61(1),wehave / /2\* /2\5 (7)2\^ /2\^ (-)»^ J_rn-i (2)={~)cos^ .R„t, J(^)+(^^jsinz .R.^-i, I(z). Bysquaringandadding wededuce from(4)thatf (8) R%n, ^(2)+R\n-:,^, (z)=(-)'" R^m, i-,H(^). Finally, if,in§9*61(2),wereplacembytheoddinteger 2m+1andthen replacevhyv—m,weget (9)J^^m+i \Z)J-v+m+i \Z)—J_i,_^_i i^z)J^^jn^i (z)=2(-)""•sinv7rR2,n+i,„-,n {z)l{'rrz). Aninteresting result, pointedoutbyNielsen, Ann. diMat.(3)v.(1901), p.23,isthat n ifwehaveanyidentityofthetype2/m(2)«/i,+m(2)—0>where thefunctions/,„(2)are m=o algebraicinz,wecanatonce infer thetwoidentities n n 2f,n{z)-Rm.v{z) =0,2/„,(2)/i!„.-l,„ +l(2)sO, 7rt=0 m=0 bywriting thepostulated identityintheform 2/,„(Z){J^{z)R,n,v{l)-J.-\{z)Rm-l,v^i (2)}=0, 7)1=0 • « andobserving that,by§4-74combined with§3*2(3),thequotient J^-i {z)lJv (2)isnotan algebraic function. Nielsenpoints outinthismemoir, and itssequel,ibid.(3)vi.(1901), pp.331—340,that thisresult leads tomany interesting expansionsinseries ofLommel's polynomials ;some ofthese formulae willbefound inhisHandhuch derTheorie der Cylinderfunktionen (Leipzig, 1904), butthey donotseem tobeofsufficientpractical importancetojustifytheir insertion here. 9'63.Recurrenceformulae forLommel'spolynomial. Inthefundamental formula Jf+m (2)=Jv{z)Rm,Az)- Jv-\ {z)Rm-\ v+i(z), replacemand vbym+1and i^—1;oncomparingthetwoexpressionsfor Jv+m {z),weseethat Jy-X {z) [R,n_,^ y+,(z)+i?^+i_ ^_i(z)}=[Jy(z)+J"^_2 (^)l R,n, u(z). *Math. Ann. 11.(1870), pp.627—632. tThis result wasobtained byLommel, Math. Ann. iv.(1871), pp.115—116. 9-63] ASSOCIATED POLYNOMIALS 299 Divide by«/^_i (z),which isnotidentically zero,and itisapparentthat z Toobtain another recurrence formula, wereplacemin§9'62(2)bym+1 andm—1,andusetherecurrence formulaconnectingBessel functions of orders v+m—\,v+mand v+7n+I ;itisthen seen that (2) R,a-^,A^) +^m+.,.(^)=^^"'^'''^ Kr.A^),Z andhence, bycombining (1)and(2), 2(7/1+1) (3) il!,rt_i „(5)+XL,,^^.] „(2')— il„i,_,^+i (^r)—it„j+] „_i(2)= -lijny{z),z Again,write§9"62 (2)intheform -^—^ E,,,,{z)={^-'— /,+„ {z)\ [z^-^ F,_, {z)\-{^-'— F,+,, (^)l[z^-^/,_! {z)\, IT anddifferentiate it.Wededuce that in+2 (4) R'm, V{z)=Rm, V{z)+Rni+i,v-l{z)-Rm+\,v(^),z and so,by(3),(1)and(2), (5) -R',„, ,(^)=R,n,V{Z)+i?„,_i, ^{Z)-Rnx-i, .+1{Z), 2j/ -|-/^^ (6) R\„Az)= Rn,^Az)-R,n-i,^^i(z)- R,n+^,Az),z z Themajorityofthese formulae were given byLommel, Math. Ann. iv.(1871), pp.113— 116,but(6)isduetoNielsen, Ann. diMat.(3)vi.(1901), p.332;formula(2)hasbeenused byPorter, AimalsofMath.(2)iii.(1901), p.66,indiscussingthezeros of/?„,,v{z). Itisevident that (2)maybeused todefine R,„^y(z\when theparameter miszerooranegative integer; thus, if(2)istohold forallintegralvalues ofm,wefind insuccession from theformulae Z" z that (8) i^o,.(^)=l, i?-i,.(^)=0,i^-2,.(^)=-l, ancHiencegenerally, byinduction, (9) it;_,„, .{Z)=(-)»'-' i^m-2,2-. {Z). Thisformula wasgiven byGraf, Ann. diMat.(2)xxiil. (1895), p.59. Ifwecompare (9)with Graf's other formula, §9-61 (4),wefindthat (10) Rra,u{z)={-)"'-' i?_„,-o,2_. {Z)=-i^_„,_o .+,«,+! {Z)=(-)'" /l'.,_.-mfl C^)- When thefunctions ofnegative parameteraredefined byequation (9),allthe formulae(1)—(7)aretrue fornegativeaswell aspositivevalues ofm. 300 THEORY OFBESSEL FUNCTIONS [CHAP.IX 9"64. Three-term relationsconnecting Lommelpolynomials. Itispossibletodeduce from therecurrence formulae aclass ofrelations which hasbeen discussed byCrelier*. Therelations wereobtained byCrelier from thetheoryofcontinued fractions. First observe that§9-63(2)shews thatJ^^m{z)and i^,„ „{z),quafunctions ofm,satisfy preciselythesame recurrence formula connectingthreecontiguous functions; andsoarepetitionoftheargumentsof§9*6(modified hyreplacing theBessel functions bytheappropriate Lommelpolynomials)shews that (1) Rm+u,i'{z)=Rm,^{z) Rn,v+m (^)"J^m-i.f (^)Rn-i.f+m+i (^)- Next in§9-63(2)replacembym-1and vbyv+l, andeliminate 2{nt+v)/zfrom thetwoequations;itisthenseen that —Rm—\,v \Z)tl'm-\,v+\ \^)~Rm,v \^)-f^'m— 2,i'+i K^/y andsothevalue ofthefunction onthe left isunaffected bychanging minto m—1.Itisconsequently independentof?«;and, since itsvaluewhenm= isunity, wehave Crelier's formula (2) Rm,Az) Rm.f+l (^)-Rm+hA^) Rm-l,^+l (^)=1, aresultessentiallydue toBessel (cf§96)inthespecialcase v=0. Moregenerally,ifin§9"63(2)wehadreplaced mbym—nand i/byv+n, weshould havesimilarlyfound that Rm,V\Z)R^n-n+i, v+n\^)~Rni+i.v\Z)Rm-n,f+n\Z) =Rjn-A, v\^)Rm—n, v+n\^)~Rm,v\^)tlm—n—\, v-\-nV'Jy andsothevalue ofthefunction onthe left isunaffected bychanging minto m—1.Itisconsequently independentofm;andsince itsvaluewhenm=n isRn-,,,{z), wefindfrom§9-63(10)that (3) R,n, V(•2')Rm-n+\, v+n{^)~J^m+i, v{^)Rm-n, v+n\^)—Rn~i,v\Z), aresultgiveninadifferent formbyLommel f. Replacemandnbym-1and?n-1inthisequation,and itisfound that (4) Rin-'i,v{z) l^m-n-\,v+n^i{z)—Rm,v{z) Rm-n-'i,v+n+\{z)=Rn,v\2')- Ifwerewrite thisequationwithpinplaceofnandeliminateRm-\,v{z)be- tween thetwoequations, weseethat Rn, V(^)Rm-2}--i,v+p+\ {^)~ Rp,V{z)Rm-^n-i, v+n+i \^) =R,n^^{z) \^Rm-p-2,v+p+\ (^)Rm-n-\,v+n+i (^)""Rm-n-i,v+n+i {z)Rm-p-i,v+p+i (^)J ^J^m,v \Z)iin—p—i, v+p+i \Z), by(3).Ifwetransform thesecond factor ofeachtermbymeans of§9'63(10), weobtain Crelier's result{loc.cit.p.143), (5) Rn, V(z)Rp-m-i, v+m+i {z)—Rp,viz)Rn-^m-i, v+m+i {z) =Rm,v\Z) -tl^;>-ji-i, f+TH-l \Z)- *Ann. diMat.(2)xxiv.(1890), p.136 etseq. f^^atJi. Ann. iv.(1871), p.115. 9-64J ASSOCIATED POLYNOMIALS 301 This isthemostgenerallinear relation ofthetypes considered byCrelier; it connectsanythreepolynomials R,„^^{z), Rn,,(z), Rp^^{z) which have thesame parametervandthesame argumentz.Theformula maybewritten more symmetrically that istosay (7) SRn,A2)Rp-m-^,pJrm+\{z)=^ 1)1,n,p Asimilar result maybeobtained which connectsanythree Bessel functions whose orders differbyintegers.Ifweeliminate '/„+m-i (z)between theequations* ]''v+n\^)^^'Jv-\-tii\^)i^n—m,v+m \^)'Jv+in—i \^)ti'n—m—i,v+tn+\ \^)> 'Jv+p\Z)='Jv+in\^)^^p—}n,v+7ii \^)'Jv+m—\ \^)-'^p—m—i,i'+m-\-i \^)> wefindthat "v+n\Z)iip—),i—\,v+m+\ (-2')—•'v+p\^)^n—m~l,v-\-m+i \^) ^^'Jv+m\^)V^n—m^u+in, \^)-t^p—m—-i,v+m-\-\ \^)~J^p—m^f+ni \^)^n—m—i,i'+vt+i \^)\ ='Jv+m\^)-ti'p—n—l.i'+n+i \^)i thelastexpressionisobtained from aspecialcaseof(5)derivedbyreplacing m,n,»,Vby0,n—m,p—m,i>+mrespectively. Itfollows that m,n,p andobviously wecanprovethemoregeneral equation (9) S'^^+n{2)Rp-jn-i,i'+m+i(^)=^, m,n,p' where ^denotes anycylinderfunction. The lasttwoformulae seem never tohave been previouslystatedexplicitly, though GrafandGubler hint attheexistence ofsuch equations, EinleitungindieTheorie der BesseVschen Funktionen,ii.(Bern, 1900), p}).108, 109. [Note.Ifweeliminate J^-i {z)from theequations W^+m-l {^)=Jv{Z)R,n-\, V(Z)-Jv-\ {z)Rm--2. v+1{z), anduse(2)tosimplifytheresulting equation, wefindthat J„(2)=—J^ 4.„j(z)Rin~->, v+1 (^)+»^i/+m-1 (z)I^m~\,v +\\Z)i and/SO, replacingvhjv-m,wehave Jv-m\Z)= —^'v (z)Rm-2, v-m +l(-)+"i'-l (-)"?n-1,v-m+\{Zj- Byusing §9'G3(10),wededuce that •^v-m {z)—Jv(z)R-m, v(z)-J^-\ (z)R-m-l, v+1 (^)> that istosaythat theequation i^9-6(1),which hashitherto been considered onlyfor positive values oftheparameter ??;,isstilltrue fornegative values.] *Itissupposed temporarilytliatmisthesnmllest oftheintepiers ?;(,»,p;butsince thefinal result issymmetrical,thisrestriction mayberemoved. Seealsothenote attlieeiul ofthesection. 302 THEORY OFBESSEL FUNCTIONS [CHAP. IX 9'65. Hurwitz limitofaLommelpolynomial. Weshallnowprovethat (1)lim(iy-^^".-i(^)=j^(^). This result wasapplied byHurwitz, Math. Ann. xxxiii. (1889), pp.250—252,todiscuss therealityofthezeros oiJy{z) when vhasanassignedrealvalue(§15"27). Ithasalso beenexamined byGraf, Ann. diMat.(2)xxiii.(1895), pp.49—52,andbyCrelier, Bern Mittheilwngen., 1897, pp.92—96. From§9-61 (3)wehave r(i/+7?i+l) „=o?'!r(z' +?i+l)'(m-2w)!r(y +m+l)' Now write {m—n)\r(//+m—n+1)_. {m-2n)ir{v +m+l)=^^"'' '''' sothat (m—n){m—n—1)...(m—2?/+1)6(m,n) {v+m){v+m—1)...{v+m—n+1)' IfnowNbethegreatest integercontained in\v\,then each factor inthe numerator of6(m,n)isnumericallylessthan thecorrespondingfactor inthe denominator, providedthatn>N. Hence, when 7i>N,andm>2N, I6{m,n) \<1, while, w^hen nhasanyfixed value, . lim{m,n)=1, in-*-X Since i<-);<^^>"'°, isabsolutely convergent,itfollows fromTannery'stheorem* that m-*x>n =«!r(l/+7l+l)^ ' „=oW !T(j/+/I+1) andthetheorem ofHurwitz isestablished. Again,since theconvergenceof2\'^^—isuniform inanybounded domain t ofvalues ofz(bythetestduetoWeiei'strass),itfollows thattheconvergenceof (i0)''+'»i?,„,,+i(2)/r(v+'/» +i) toitslimit isalsouniform inanybounded domain ofvalues ofz. *Cf.Bromwich, Theory ofInfinite Scries, §49. tAnarbitrarily smallregion ofwhich theoriginisaninternal point must obviously be excluded from thisdomain whenR[v]^0. 9-65, 9-7] ASSOCIATED POLYNOMIALS 303 From thetheorem ofHurwitz itiseasytoderive aninfinite continued fraction fort/^_i {z)IJ„ {z). For,when ./„{z)^0,wehave =inn VI-*- y.2pz-'-1-^'m, v+i\Z) by§9-63(1).Oncarryingouttheprocessofreduction andnoticingthat JXo ,/_1-'»h—1\Z)^ . I wefindthat—^2{v+m-l)z• -''i.v+m v^y9^-1 (I'+m) andhence=2vz-'-1 2(i.+1)^-^-2(2/+2)^-1-...-2(v+m)^-1' (2)=2i.^-i-1 2(17+1)^-1-2(i.+2)^-1- This procedure avoids thenecessityofjn-oving directly that,when wi-s-oo, the last element ofthecontinued fraction J.-^{^--2vz-^--•^V+7(i+1(2) maybeneglected;themethod isduetoGraf, Ann. diMat.(2)xxirr.(1895), p.52. 9"7.Themodified notationforLommelpolynomials. Inorder todiscusspropertiesofthezeros ofLommelpolynomials,itis convenient tofollow Hurwitzbymakingachangeinthenotation, forthe reason thatLommelpolynomialscontainonlyalternatepowersofthevariable. Accordingly wedefine themodified Lommelpolynomial gm,v{z) bythe equation* {-yV {v+m-n+l)z''' (1) sothat (2)gm,v\Z)—-^m-iv^n n= r{v+n+l) Bymakingtherequisite changesinnotation in§§9"63, 9"64, thereader willeasilyobtain thefollowingformulae : (3) g.n+,,Az)={v+m+\)g„r,.{z)- zg„,_,^^{z), [§9-63(2)] (4)^ gm+^,.-l {Z)=vg,n,Az)- ^5''m-i,.'+i (-)> [§^'OS(1)] (5)^^{^''i/,.,.(^)|=^^.-..(^)+i/».-i-.,.-:(4 [§9-(^'K7)] (6)ym+-2 dz{z~"'-' g,u,.(z)}=g,n+,, .-1{z)- grn-,:, .(z), [§9-63(4)] (7)gm,Az)g,„+,„+, {z)-g,„^.^^{z)g,n-i,y+i {z)=z'''go,v(z)g,^^+„,+i (z). [Aspecialcase of§9'64(5).] *This notation differs inunimportantdetailf* from thenotation used byHurwitz. 304 THEORY OFBESSEL FUNCTIONS [CHAP. IX These results willberequiredinthesequel;itwillnotbenecessarytowrite down theanaloguesofalltheother formulae of|§9*6—9'64. The result ofeliminatingalternate functions from thesystem (3)isof someimportance.Theeliminant is {if+m)gm+2,u {2)=C,n{z)ffm,u{z)-{v+iU+2)Z'g„t_o^^(z), where c,n(z)={v+m+I)[(v+m){v+m+2)—1z]. Wethusobtain thesetofequations: ^ (i.+2)g,^,{z)=c,(z)g,^,{z)-(u +4>)z'g^,.{z), {v+4)^G,. (^)=Ciiz) g,^,{z)- (i/+6)z''g.2,^{z), (8)< (v+2s) (/2S+2,^(z)=c.^{z)g^^^(z)-(v +2s+2)z^g^^o,„{z), ^(v+2m-2)g^^^{z)=c.^rn-^ (z)g^_^^^{z)-{v +2m) z'^g.^^n-i, v{z). 9*71. Thereality ofthezerosofg.2m,v{z)u'hen vexceeds —2. WeshallnowgiveHurwitz'proofofhistheorem* that luhen v>—2,the zeros ofgzm,v(z)cl^'^cdlreal; and also thattheyareallpositive, except when —1>1/>-2,inwhich caseoneofthem isnegative. After observingthatg-2m,v{2)isapolynomialinzofdegree m,weshall shew thatthesetoffunctionsg.>m,v{z), <7m-2,.'(^),•••g2,v{z), ga,v{z)form aset ofSturm's functions. Sufficient conditions forthistobethecase are(i)the existence ofthesetofrelations§9"7(8),combined with(ii)thetheorem that therealzeros oig^m-^^viz)alternate with those of(/2ot,v(^)- Toprovethatthezeros alternate, itissufficient toprovethatthequotient g2m,v{z)lgim-2,v{z)isamonotonic function oftherealvariablez,exceptatthe zeros ofthedenominator, where thequotientisdiscontinuous. Wehaveg'^m~2,u(z)-r-l/'"''"tA=~^^2m,2m-2,dz[g.an-2,v{z)) where W^r,..= g,-,u(z)g's,y{z)- g^,u(z)g'r,„{z) ; andfrom|9'7(3)itfollows that ^^>m,2,n-2-g-2m~2,A^) +(v+2m) 5l21,„i-l.2»n-2, ^512R.,,„_i.2m-2=Z"S(?B2m-3,m-4 +(r+2»<-2)g%m-^,y{z), SOthat ^^m,^n-2=g'^-2A^) +{v-^2m)1.{v+2r)z'-^'^^-'g%r-^,.{z), r=l and therefore, ifm^1,^^2m,-im-2isexpressibleasasum ofpositive terms when V>—2. *Math. Ann. xxxiii.(1889), pp.254—256. 9-71, 9-72] ASSOCIATED POLYNOMIALS 305 Themonotonicpropertyistherefore established, and itisobvious from a graphthattherealzeros ofg.2m-2,i>(z) separatethose of^a^,.-(^)- Itfollows fromSturm's theorem that thenumber ofzeros ofg^m^vi^)on anyinterval oftherealaxis istheexcess ofthenumber ofalternations ofsign inthesetofexpressions go.m,v{z), g2m-2,ui^),•••.^'o,.- (2')attheright-handendof theinterval overthenumber ofalternations attheleft-hand end. Thereason whythemimber ofzeros istheexcess andnotthedeficiencyisthat the quotient g^m, v{^)/g2m-2, v(^)isadecreasing function, andnotanincreasing function ofz, asintheusual version ofSturm's theoi-em. SeeBurnside andPanton, Theory ofEquations^ I.(1918), §96. Thearrangementsofsignsforthesetoffunctions when zhasthevalues —X,0,ocareasfollows : 2m 306 THEORY OFBESSEL FUNCTIONS [CHAP.IX Bymeans oftheformulae quotedin§9'7,itisclear that —9'^^-^,''\^) 1^5'aOT-i, •'('2^)+5'2w+i,i'— i(^)f~9im,v\Z) \g2m-\,v-A^)~ 9'2m-\,v\^)\ =(v+2m)g\n_-^^^{z)^-go,n^2,v{^)92^n^^,v-i{z)-g2m~i,v-i{z)92m,v{z) ={v+2m)g%n-i,u(z)-z-'''~'goA^)9i,>'+27n-i{z) ^(v+27n){g\,„^,,^{z)-z"-'^''] >0, providedthat v+2m ispositive andzisnegative. Therefore, inthecircum- stancespostulated,thequotient isadecreasing function, andthealternation ofthezeros isevident. Theexistence ofthesystemofequations §9'7(8)nowshews thattheset offunctions -92s-2,A^)> +92s-i,M, .-.,(-y9o,,(z) form asetofSturm's functions. Thesignsofthese functions when ^^is-ooare +,+,....+,+,-,+,...,(-)*, andthere are salternations ofsign.When ziszero, thesignsofthe functions are ±,±,...,±,+,-+,...,(-)*, theupper signs beingtaken when—2s>v>—2s— 1,andthelowersigns beingtakenwhen—2s— 1>v>—2s—2; there aresand s+1alternations ofsignintherespectivecases. Hence, when—2s>v> —2s—l,92m,v(z)has nonegativezero;butwhen—2s—1>^>—2s—2,g2m,vi^)hasonenegative zero.Thetheorem stated isthereforeproved. 9*73. Positive andcomplexzerosofg.^^t,{z)when v<—2. Asin§9"72, define thepositive integersbytheinequalities -2s>7/>-2s-2. Itwillnowbeshewn* thatwhen vliesbehueen —2sand—2s— 1,gim,v{z) hasm—2spositive zeros; butthat,when vliesbetween —2s—1and—2s—2, g<imv{z)hasm—2s—1positivezeros. Provided that, ineach case,missolarge thatm+Vispositive. *This proofisofamore elementarycharacter than theproof given byHurwitz;seethe papercited in§9"72, 9-73] ASSOCIATED POLYNOMIALS 307 Inthe firstplace,itfollows from Descartes' ruleofsigns that, ineach case, g2m,A^)cannot havemore than thespecified number ofpositivezeros. For, when Vliesbetween -25and—2s—1,thesignsofthecoefficients of 1,z,z\...,z--^z^^\ ^•-«+^^^+^ ..., ^"^ing^^^^z) are +,+,+,...,+,-,+,-,..., (-)"*; and since there are iti—25alternations ofsign,there cannot bemore than m—25positivezeros. When vliesbetween —2s—1and—25—2thecoiTe- spondingsetofsignsis _____+_ /_V"• andsince there are tn—25—1alternations ofsignthere cannot bemore thanm—25—1positivezeros. Next,weshallprove byinduction from thesystemofequations §9"7(8) that there areasmanyasthespecifiednumber ofpositivezeros. When Vliesbetween -25and—25—1,thecoefficients ing^s,v{z)haveno alternations ofsign(beingall+)and sothisfunction hasnopositivezeros. Ontheother hand 'gAs+^A^) >^'gis+2,A+ X)=-X, andsog4s+2,v(2)hasonepositive zero, Ojisay;and,byreasoning already given, ithasnootherpositivezeros. Next, take^4.5+4,^(2^);from§9'7(8)itfollows that itssignsat0,«]_!,+ocare+,—,+;hence ithastwopositive zeros, andby thereasoning already givenithasnoothers. Theprocessofinduction (whereby weprovethat thezeros ofeach functionseparatethose ofthesucceeding function)isnowevident, andweinfer thatg-2m,v{z)hasw—25positive zeros, andnomore. Again, when vliesbetween -25—1and—25—2,the coefficients in gis+%v{z)havenoalternations insign(beingall— ),andsothisfunction has nopositivezeros. Ontheother hand andso5^4^+4, ^(2')hasonepositive zero,andbythereasoning already givenit hasnootherpositivezero. Byappropriatemodifications ofthepreceding reasoning weproveinsuc- cession thatgis+6,v{z), g4,+s,A^), •have 2,3,...jjositive zeros, andingeneral thatg<ijn,i>{2)hasm—25—1positivezeros. Bycombiningthese results with theresult of§972,weobtain Hurwitz' theorem, that, rvhen v<-2, andmissolargethat in+vispositive, g2m,f{^) has2scomplex zeros, where sistheintegersuch that -2s>v>-2s- 2. •20—2 CHAPTER X FUNCTIONS ASSOCIATED WITH BESSEL FUNCTIONS 10"1. Thefunctions Ji,{z) andE^ (2^)investigated byAnger andH.F. Weber. Inthischapterweshallexamine thepropertiesofvarious functions whose definitions aresuggested bycertainrepresentationsofBessel functions. We shall firstinvestigatefunctions definedbyintegrals resemblingBessel's inte- graland Poisson'sintegral, and, afterdiscussingthepropertiesofseveral functions connected with !'„(z)weshallstudyaclass offunctions, firstdefined byLommel, ofwhich Bessel functions areaparticularcase. The firstfunction tobeexamined, J^(z),issuggested byBessel'sintegral. Itisdefined bytheequation (1) J,(^)=-1'cos(i/^-^sin^)f/^. This functionobviouslyreduces toJn(z) when uhastheintegralvalue n. Itfollows from§6"2(4)that,when visnotaninteger,thetwofunctions are distinct. Afunction ofthesametypeasJ>(^)wasstudiedbyAnger*,but hetook theupperlimit oftheintegraltobe27r;andthefunction J„(^)is convenientlydescribed asAnger'sfunction ofargumentzandorder v. Asimilar function wasdiscussed laterbyH.F.Weberf, andhealso investigatedthefunction E^(e) defined bytheequation (2) E,(z)=i I"sin(v0-zsin0)dO. Inconnexion with thisfunction reference should alsobemade toresearches byLommel, Math. Ann. xvi. (1880), pp.183-208. 1/'2'rItmaybenoted thatthefunction—1cos(i*^-2.sin6)d6which wasactuallydis- cussed byAngeriseasily expressibleinterms ofJ^(2)andE„ (2) ;for,ifwereplace 6by ^TT—Bintheright-handhalfoftherangeofintegration, weget 1/"Sn- 1/"" 1 /""^ —- /co&{v6-ziim6)dd=z-Icob(vd-zsin 6)dd+--Icos(21/77- 1/^+2sin^)0?^ 27r_/o 27r_/o'2iTJ =cos2vn .Jt,(z)+sinvncosvtt .E^ (z). *Neueste Schriften derNaturf. Ges,. inDanzig,v.(1855), pp.1—"29.Itwasshewn byPoisson that V„Icos(j/^-2siu^)<Z^=(2- f)siuI'TT, Additions alaConn, desTemps, 1836, p.15(cf.§10-12), butashedidnomore itseems reasonable togiveAnger's name tothefunction. tZurichI'iirteljahrsschriJ't, xxiv.(1879), pp.33—76.Weber omits thefactorl/iriuhisdefi- nition ofHv(z). 10-1] ASSOCIATED FUNCTIONS 309 Toexpand J„(z) andE^(^)inascending powersofz,write^tt+(}ifur8 intheintegralsandproceedthus : sin™ sinv6d0— Icos"* sin{^vir+vcf)) dcf)J—hw =2sin^VTTcos"'(f)cosv(f)d(j) . TT .m !sin^j/tt 2'"r(im-|i/ +l)r(im +ii.+ l)' byaformula due toCauchy*. Inlikemanner, rnn aja'^ •'" !COSJpttsm'"dcos/'^cf^= z,—=-- ;-—^i- ;. But, evidently, I00/\m,^linfn 1^(_\tn^2m+l fn J,(^)=- 2\i ,, sin^"*^cosi/^(^^+- :i'^V Tvrsin^™+' ^sin?/^fZ6', sothat (3) J^{z)=cos^vrrX{-y-(^zr =0r(m- |j.+1)i'(m+^2/+1) +sinii'7r S andsimilarly (4)E,(^)=sinii^TTS=0r(m-ij/+1)r(/w+1;/+1)' (-)-(l^)^ or(m-iz/+l)r(m +ij.+l) —cos|z/7r1(-r(W^^' ,„=or(m-1^^+1)r(m+ii/+f) These results maybewritten inthealternative forms (5) J^{z)=smvir VTT sinvirs^ 22_i;2(•22_^2)(4-2_^2) {2?-v^)(^^-^- j.2)(G^- j,^)+... +- TTLl'-^' (l'-I^-)(S--Z^2) (12_j,2)('32_j,2)(52_y2) (6)E.(^)=1—COSvir VTT2- Z*1-- T.+,-^9i_J/-(2--i;^)(42-i;-) 1+COS I'TT IT+^' _r--V- (l^-I/^XS^-V^) (12_j,2)(32_j,2)(52_j,2)•• Resultsequivalenttothese weregiven byAngerandWeber. Theformula correspondingto(5)wasgiven byAnger (before thepublicationofhis memoir)inaletter toCauchy which wascommunicated totheFrench AcademyonJuly 17,1854; seeComptes Rendus, xxxix.(1854), pp.128—135. Mem. surlesintegrales definiea (Paris, 1825), p.40.Cf.Modern Analysis, p.263. 310 THEORY OFBESSEL FUNCTIONS[CHAP. X Forareason which willbeapparent subsequently (§10'7),itisconvenient towrite z z^ z^ (7) Sq^v\Z)- -.^ 2/'12_ ,.2\/Q2_,.2\"*"I2_y2 (p_j;2-)(32_j,2^ (1^-7^0(3— i/2^(52-I/-^)•••' _1 ^- z^ (8) s_,^,{z) =- -,+ ^,,^22_^2)- ^2(22_^2-)(42_ ^2-)+ •••' and,with thisnotation, wehave /n\ T/\sini/TT , j^sini"7r (9) J,{z)= 5o,,(z)-s_i ^(^'XTT TT /iA\ -ci /\l-4-cosy7ri;(l-coSi'Tr) (10) E,(z)= So,.(z)--^ ^ 5_i, ,(^).TT 7r Itiseasytodeduce thefollowingformulae from these results : (11)Icosv6 .cos(zsin6)dd=—vsin i^tt .s_i ^(2),Jo (12) sinvd .cos(2^sind)dd=—v(1—cosvrr).s_i j,(z),Jo (13)Isinvd .sin(^sin6)dd=sin i'tt .«„^(z),Jo (14)Icosvd .sin(2;sin6)dd=(I+cosi/tt).*«/(2^), J riff (15) Icosi'(^.cos(^^cos<^)rf(^=—1/sin-|-i/7r.s_j ^(^'X (16) cosv(f>.sin(zcos0)cZ(^=cos^vir.^o^(z).Jo Integrals somewhat resemblingtheintegrals discussed inthissection, namely f COS lesin* .(n6-cosd)dd, Jsin havebeenexaminedbyUnferdinger, WienerSitzutigsberichte,LVii.(2),(1868), pp.611—620. Also, Hardy, Messenger, xxxv. (1906), pp.158—166,hasinvestigatedtheintegral Ism(v^-2sm 6)-3-,Jo o andhasproved that,when visreal,itisequaltohn2?;„Jn(2),where)j„is1,or-1 accordingasv—aispositive, zero, ornegative. 10*11. Webe7-'s formulae connectinghisfunctions withAnger's functions. Itisevident from theformulae§10*1(9),(10), (15)and(16)that /I\ -r/\ -.• /\4COSwI/TTfi"^ ,. ,,, , (1) Jt,{z)+J^t,{z)= — ICOSvd)COS{zCOSd>)d(p,TT Jo /,\ -r/\» /X4sinAi/7r r*'' ,•/ ,.7, {2) J^{z)—J_^{z)='—cosv(f>sin{zcos(b)d(p,TT. 10-11, 10-12] ASSOCIATED FUNCTIONS 311 (3) E^(z)+E_^(2)= ^—cos1/0sin(^cosA)d^, (4) E^(z)-E_^ (z)=^^ I' cosv<bcos(zcos0)dxf). Itfollows onaddition that J,(z)=1cotIvTT[E^(z)-E_^ (s)}-Itan|i/7r{E^(^)+E_^ (z)\, sothat (5)sin I'TT .J„(z)=cos i'tt .E^(z)-E_^ {z), andsimilarly (6)sinvir .E^(2:)=J_^(z)—cos i^tt .J^(z). Theformulae(5)and(6)areduetoWeber. 10'12. Recurrence fornmlae forJ^(^) andE„(^). Therecurrence formulae which aresatisfied bythefunctions ofAngerand Weber havebeendetermined byWeber. Itisevident from thedefiniteintegralsthat J„_i (z)+J>.+i (2)^ J^(2)=- (cos^-- )cos{v6-zsin6^)dd Z TTjV Zj -—f":^{sin(i;^-^sin^)}rf(9 _2sinvir TTZ and E,_, (z)+E,+, {z)--R^{z)=~V fcos 6'-")sin{vO-zsin6)dd =^\^^.{cos{vO-^sind)\deTTZ f'C' 2(1—coSfTr) TTZ Itisalsovery easytoprovethat j._, (^)-j.,+, (s)-2j;{z)=0, 1e,_,(^)-e.+,(^)-2e;(^)=o. From these results wededuce theeightformulae 2i/ ,^2smTTT (1) J._,(^) +J.+:(^)=-J.(^)--^^, (2) j,_,(^)-j,+,(^)-2j;(^), (3) (^+I/)J,{z)=2J,_i (2)+(sin i;7r)/7r, (4) ("^-v)J,(z)=-zJ^+, {z)-(sin j;7r)/7r, 312 THEORY OFBESSEL FUNCTIONS[CHAP. X (6) E,_,(^)-E,+i(^)=2E;(^), (7) (^+z.)E,(2)=^E,_i (^)+(1-cosi/7r)/7r, (8) (^-v)E,{z)=-0E,+, {z)-(1-cosi^ttVtf, where ^,asusual, stands forz(djdz). Nextweconstruct thedifferentialequations;itisevident that (^2_y2)j^(^)=(^_ J,)[^j^_j (z)+(sin 7/7r)/7r} =£•(^+1— z--)J„_i (z)—{vsinv'Tr)l'Tr =—z'^Jy, (z)+(zsinv7r)/Tr-(vsinz/7r)/7r, sothat T7T/\(^- I')sin 7>7r (9) V^J^(^)=^ . TT Wealsohave C^'-v')E,(z)={'^-v) {zE,_, (z)+(1-cosi/7r)/7r} =^(^+1- i^)E^_i (^)-2^(1-cosi^7r)/7r =—^^E^{z)—z(l+cosi/7r)/7r—y(1—cosv'ir)l'ir, sothat (10) V,E,(^)= vr TT Formulaeequivalentto(9)and(10)were obtained byAnger, NeuesteSchriftender Naturf.Ges. inDanzig,v.(1855), p.17andbyWeber, ZurichVierteljahrsschrift,xxiv. (1879), p.47,respectively; formula(9)hadbeen discovered earlier byPoisson(cf. §101). 10"13.Integrals expressibleintermsofthefunctions ofAnger and H.F.Weber. Itisevident from thedefinitions that (1) J^(z)±i-E,{z)=- rex^{±i(vd-zsme)}de. Bymeans ofthis result, combined with formulae obtained in§§6"2—622, it ispossibletoexpressnumerous definiteintegralsinterms ofthefunctions of Bessel, AngerandWeber. Thus, from§6*2(4)wehave (2) re-''<-"inh« dt=^-{J,(z)-J,(z)], whenlarg^ <|7r; the result isvalid when|arg^|=|7r,providedthat R{v)>0.^ Again, wehave (3) re-'^-^-i'' dt=-^I/_,(z)-J., (z)}, J sin I'TT 10-13, 10-14] ASSOCIATED FUNCTIONS 313 SOthat,whenwecombine(2)and(8), (4) f" e-2«i"h« cosh vtdt=h7rtan^vtt{J,(z)-J,(z)]-\iv|E,{z)+F,(z)}, . (5) Ie-"'"'''sinh vtdt^hir cot|i/tt{J,(z)-J,(^)j-itt[E,(^)+F,(z)].Jo Theintegi'alIe~^^°'^^^ cosh ptdthasah-eady been evaluated(§6'3); but Jo r^ .' doesnotappeartobeexpressibleinasimple form; itsexpansioninascending powersofzcanbeobtained from theformula of§6'2'2(4), 2 /"'^ 2sinVTT f"^ /_^(z)+/.(z)=-e^<=°*^ cos;'^f^^+ g-^^^o^'^* sinh vtdt, but, since T" .-. /I7,1(—)*"sin i^TT„/ I'+7/i,v+m ,\ i„<'°>'"'"-^"-^""''"=W(ir+^•=^'(''"•^'^-- 2-;-V• theintegralunder consideration cannot beevaluated inanysimpleform *. Theformulae(2)—(5)arenugatory when visaninteger,butfrom§§6'21, 9"33wehave (6) f'e"f—i»h'dt=l{.%(z)-ttE, (z)-TTY,(z)},Jo (7)g-nl-z.uAU ^^=i(_y.+i {,S;^(^)+^E, iz)+TTF„(^)}. io Theassociated integrals f'^COS /^COJS e-"' .'U'sinhrf<, e""^ .(.fcosh«)c/i! ./«in'Josui havebeen noticed byCoates, Quarterly Journal., xx.(1885), p.260. Various integralsofthesetypesoccur inresearches ondiftVaction byaprism ;see, e.g. Whipple,Proc.London Math. Soc. (2)xvi.(1917), p.106. 10-14. Asj/mptotic expansions ofAnger-Weberfunctions oflarge argument. Itfollows from§1013(2)that, inorder toobtain theasymptotic expansion ofJ±A^) whenl^-jislargeand |arg^'|<^-7r,itissufficient toobtain the asymptotic expansionoftheintegrals IgTvt—zsmh t^Jf Jo Tocarryout thisinvestigationweshall firstexpandcosh vt/coshtand sinhi^f/coshtinaseries ofascending powersofsinh t. *SeeAnding, Sechsstelline Taft'lnderBe-fttelschen Ftinktionen imanhiriren Arfiumeuts (Leipzig, 1911) [Jahrbuchiiber dieFurtschritk' derMatli. 1911, pp.•l'J3— 494],andTakeuehi, Tohoku Moth. Journal, xviii.(1920), pp.295—296. 314 THEORY OFBESSEL FUNCTIONS [CHAP. X Ife^^=u,wehave, after themanner of§7"4, If(u+.llu+) f^hl/uh ]W*"+II-*"=^—• ^'"-^ \y+v\t \d^, SOthat coshz.^ 1r(«+.i/«+.i+)^i>'-^{^-l)d^ cosht 27riJ {^-lf--i^smhH 10-14] ASSOCIATED FUNCTIONS 315 sothat sinh vt_1 sinh 2t 27ri{»+,!/«+,1+)?"<«? 2-171}^i"(^-l)2-4^sinh^« 2^^Psinh^i' « L,„=o (r-ir"*+^• (r-lF{(^-l)^-4rsmh^«} whence itfollows that,ifwetakepsolargethatR{p-\-l±\v)> 0,thendK, sinh t-^sin\vir cosh^ TT (2m+1)!^ '^w= +^.3(-)-r(p +i^iz.)r(;.-M-^.) ^23inh0--(2p+])! Onintegratingthese results, itfollows that rsinh....-».-^.^^^i(-)-r(m+i-fHr(m +i-^^^ Jo 27r „,=o {\zf^+' If t-isrealand zispositive,theseasymptotic expansions possessthe propertythattheremainder afterj9terras isofthesamesign as,and isnumeri- callylessthan, the(p+l)thtermwhenpissolargethatR{p+1±\i>)^0. Itfollows from§§lOlS(2)and(3)combined with§lOll(6)that (1) J,(^)-./,(^) +smvir 772 sinVTT1 1_^ LI i . (2)E.(^)--r,(^)TTZ 1+cosvir TTZV_i;(2--V^)V{21'-v^(4--v") z z^ z^ 1—cosvir TTZz- Z*' V(2-- v'')V(2--V-)(4--v^) These results were stated withoutproof byAVeber, ZdrichVierteljahrsschnft,xxiv. (1879), p.48andbyLommel, Math. Ann. xvi.(1880), pp.186—188. Theywereprovedas specia>ncasesofmuch moregeneralformulae byNielsen, Handbuch derTheorie der Gylinderfunkiionen (Leipzig, 1904), p.228.Theproofofthissection does notseem tohave been given previously'. Since theonly singularitiesofcoshi/^/cosh«andsinht-i/cosh^, quafunctions ofsinht,areatsinh t=±i,itispossibletochangethecontours ofintegration intocurves inthe^planeonwhicharg(sinh t)isapositiveornegativeacute angle;andthenwededuce intheusualmanner (cf§6-1)thattheformulae (1)and(2)arevalid over thesector jarg2 |<tt. 316 THEORY OFBESSEL FUNCTIONS [CHAP. X 10*15. Asymptotic expansions ofAnger- Weberfunctions oflargeorderand argument. We shallnow obtainasymptotic expansions,ofatypesimilar tothe expansions investigatedinChapter viii,whichrepresent J^{z)andE„{z) when iV \and |z \arebothlarge. Inview oftheresults obtained in§1013, itwillbeadequatetoobtain asymptotic expansionsofthetwointegrals -fTJo AsinChapter viii,wewrite v=zcosh(a+t/3)=zcosh7, where ^/3:$ttand7isnotnearly equal*toiri. (I)We firstconsider theintegral /g—1/«—zsinh«-7/ =_ jg—z(<coshy+sinht)^/ ttJo '"Jo inwhich itissupposed temporarilythatvizispositive. When cosh7ispositive, tcosh7+sinh tsteadilyincreases from to00as^increases from to00;we shall take thisfunction of^asanew variable t. Itiseasytoshew that <isamonogenicfunction oft,except possibly when T=(2%+1)Trtcosh-y+sinh-y+ycoshy, where nisaninteger; and,when coshyispositive, none ofthese vakies oftisareal positive number;for,whenyisreal, (2?i+1)Tr/cosh ydoesnotvanish, and,whenyisapure imaginary (=2/3),thesingularitiesareontheimaginaryaxisandtheoriginisnotoneof them sinceyisnotequaltoiri. Theexpansionofdtfdrinascending powersofris ^=VaT-<-m' .;(o+) I dt^1 /"(+») dtwhere a„,=r^— .—rrrr,.T-dT'"liriJ T--"«+i dr 2iri j t^'"+»' and soa,„ isthecoefficient ofl/tintheexpansionoft"^*"-! inascending powersof t.Inparticular wehave _1_1_^~cosh7 1+cosh7''2(1+cosh7)^''24(1+ cosh7)7' 225-54cosh7+cosh-7a'3720(1 +cosh7)10 From thegeneral theorem of§83,wearenow inapositiontowritedown theexpansion (1)-fV''*-^«*"h«(^< ~-ilML?i« . Expansions valid neary=Triareobtained attheendofthissection. 10-15] ASSOCIATED FUNCTIONS 317 Thisexpansionisvalidwhenvjzispositive;ithas, sofar,been established onthehypothesisthat jarg^-j<^tt,but,byaprocessofswinging round the contour inther-plane,therangeofvalidity mayboextended tocover the domain inwhich |argz\<it. Next, weconsider themodifications causedbyabandoningthehypothesis thatcosh7isreal. Ifwewrite t=u+iv, thecurve onwhich risrealhas foritsequation usinh asiny8+Vcosh acos^-fcoshusinv=0. Theshapeofthiscurve hastobeexaminedbymethodsresemblingthose of§8'61. Forbrevity wewrite Itsinhasin/3+vcosh acos^+cosh usinw=cf>(u,v). Since <1>(ti,v)isunaffectedbyachangeofsignofboth aando,wefirst studythecurve inwhich a>0.Itisevident thatthecurve hastheoriginas itscentre. Since 9<I>{u,v)/du=sinhasin/3+sinh usin v, itfollows that,when vhasanyassigned value, d(i>/bu vanishes foronlyone value ofu,andsotheequationinu has, atmost, two real roots;andoneofthese isinfinite whenever visa multipleoftt. When >v>—TT,wehave <I>(—00 ,t')=—X,(J)(+00 ,y)=—CO ; and,when v=/B~ir,themaximum value of^{u, v),quafunction ofu,isat u—a,thevalue of<I>(u,v)thenbeing —cosh asin/5{1—atanh a+(tt—/3)cot/S}. Ifthis isnegative,theequation ^{u,/3—tt)=hasnoreal root,andsothe contour does notmeet the line v=/3—iror(bysymmetry)the line V=TT—fB. Henceprovidedthat thepoint (a,/3)liesinoneofthedomains num- bered 1,2,3inFig.21of§8-61, thecontour <1>(a,v)=0 liesasinFig. 25, thecontinuous curveindicatingtheshapeofthecontour when aispositive Fig. 25. andthebroken curve theshape when a.isnegative;thedirection inwhich r increases ismarkedbyanarrow. 318 THEORY OFBESSEL FUNCTIONS[CHAP. X Itfollows that theexpansion (1)isvalidwhen(a,^)liesinanyofthe domains 1,2,3. Next, wehave toconsider theasymptotic expansion when(a,/3)doesnot lieinanyofthese domains. Toeffect ourpurpose wehave todetermine the destinations ofthebranch ofthecurve <t»(w,v)=whichpasses throughthe origin. Consider firstthecase inwhich aispositive and/Sisacute. Thefunction '^(a,v)hasmaxima atv={2n+1)tt—/3andminima atv=(2??+1)tt+/3, eachminimumbeing greaterthanthepreceding; andsince(f)(a, fi— tt)isnow positive,itfollows that <^(a,v)ispositive when visgreater than—tt. Hence thecurve cannot cross the linew=aabove thepointatwhich y=—TT,andsimilarlyitcannot cross thelineu=—abelow thepointatwhich V=TT.Thebranch whichgoesdownwards attheoriginistherefore confined tothestrip—a<w<auntil itgetsbelow thelinev=—^Ktt+w— /3,where Kisthesmallestintegerforwhich 1-atanh a+{(2K+1)tt- /?}cot/9>0. Thecurve cannot cross theline v=—(2K+1)ir+^,and soitcrosses the lineu=aandgoesofftoinfinityinthedirection ofthelinev=—^Kir. Hence,ifaispositiveand /3isacute, weget (2)^r'^^' ,-.t-.sinh.^^^ L5(ML?^ , while, ifaisnegativeand/3isacute, weget (3) l[^^^^%-.-»i..h.^,^lV(2-)!a. TTJo 7r»«=oz"^"-' Bycombiningthese results with those obtained in§8'61,weobtain the asymptotic expansionsforthedomains 6aand7a. If,however, y3isobtuse andaispositive,thebranch whichgoesbelow the axisofwattheorigincannot cross thelineu=abelow(a,tt—13)and itdoesnot cross thei<-axisagain,soitmustgoto—xalongtheline v=—(2L+1)tt, whereListhesmallest integerforwhich 1-atanha-{(2Z+ 1)tt+/Sjcoty8>0. Hence,ifaispositiveand/3isobtuse, weget (4)- Ie-"^--"'^^dt^- S^ l-^''', while, ifaisnegative and^isobtuse, weget 1/•-x+(2i+i)7r;1 00/9mV a (5)l\.-.«-"i..^'*~i S<--|i^". 10-15] ASSOCIATED FUNCTIONS 319 Bycombiningthese results with those obtained in§8"61,weobtain the asymptotic expansionsforthedomains 4,5,6band 76. Since formula (1)istheonlyonewhich isofpractical importance, weshall notgivetheotherexpansionsingreaterdetail. Anapproximateformula fora,„whenmislargeandyiszero,namely a^-^-^'"^(^) wasobtained byCauchy, Comptes Rendus, xxxviii.(1854), p.1106. (II)Next consider theintegral 1 i"^-' 1r*_Qvt-zsm\it ^^—._/g-zi-<coshy+sinh«) (^^ TTJo TtJo Theonlydifference between thisandtheprevious integralisthechangein thesignofcosh7;and so,when 7liesinanyoftheregions numbered 1,4, 5inFig.21of§861,wehave where a,/ isderived from a„,bychangingthesignofcosh7,sothat ,_1, 2 9+cosh7 ^°- 1_cosh7'^'~ (1-cosh7)^'^'"24(1 -cosh7)' Thisexpansionfails tobesignificant when 7issmall, justasthepreviousex- pansion (1)failedwhen 7wasnearly equaltoiri. Todealwith thiscasewewrite v—z{\—e),T=^—sinh^, after themethod of§8-42. Itisthusfound that vrjo TTJooC^ 'Sir'm=0 andhence (7)-e"'"">'^'rfi~^S .i^a,m+i,• Aresult equivalenttothishasbeen given byAirey,Proc.Roi/alSoc.xciv. A,(1918), p.313. 320 THEORY OFBESSEL FUNCTIONS[chap. X 10*2.Hardysgeneralisations ofAirys integral. Theintegralconsidered byAiryandStokes(§6'3)hasbeengeneralised byHardy*inthefollowing manner: Ifs=sinh<^,then {2cosh20=46'-+2 2sinh3</>=85=*+65 2cosh4(^=16s*+165^+2 I,2sinhh<\)=32s«+405=*+10s,1 andgenerally 2^?'jj nc^={2s)\F, (-ir.,i-In;1-n;-l/s%SI thecosh orsinhbeingtakenaccordingasniseven orodd. Now write Tn(t,a)=P.si^,(-in,1-17j;1-n;-4a/«0. sothat r,ao)=f^+2a T.,{t,a)=t^+f]at T,(t,a)=P+iaf+'2o? Then thefollowingthreeintegralsaregeneralisations fofAiry's integral: (1) Gin{a)=i^cosTnit,a)dt, Jo (2) Sin(a)=rsinTnit, a)dt, Jo (3) Bin(«)=[" exp{-Tn(t,a)}dt. Itmaybeshewn;[that the firsttwointegralsareconvergent when ais real(whether positiveornegative)ifn=2,3,4 Butthethirdintegral convergeswhen aiscomplex;and itisindeedfairlyobvious thatEin(a)is anintegralfunction ofa. When nisaneveninteger,thethree functions areexpressibleinterms ofBessel functions;butwhen nisodd,the firstonlyissoexpressible,the other twoinvolvingthefunction ofH.F.Weber. Beforeevaluatingtheintegrals, weobserve thatintegralfunctions exist which reduce toCin(a)andSin(«)when aisreal;fortakethecombination Gin(a)+iSin (a)=exp {iTn{t,a)}dt. Jo *Quarterly Journal, xli.(1910), pp.226—240. tThe sine-iutegral inthecasen—Swasexamined byStokes, Camb. Phil. Trans, ix.(1856), pp.168—182. [3Iath. andPhys. Papers,ii.(1883), pp.332—349.] XHardy,loc. cit., p.228. 10-2, 10-21] ASSOCIATED FUNCTIONS 321 ByJordan's lemma, theintegral, when taken round anarcofacircle of radiusRwith centre attheorigin (the arcbeingterminatedbythepoints withcomplexcoordinates R,Re^'"'"'), tends tozero asR^^oc . And therefore rooexp(j7r?7«) Cin(cc)+iSin (a)= | exp {i1\{t,a)}dt =ei-^'/nexp{-T„(t,oLe-"'"'^)] dr, J where t=te"^"''";andthe lastintegralisanintegralfunction ofa.The combination Cin (cc)—iSin(a)maybetreated inasimilar manner, andthe result isthen evident. 10'21. 'TheevaluationofAiry-Hardy integrals ofeven order. Toevaluate thethreeintegrals Cin{oL), Sin{a),Ein{a) when niseven,we suppose temporarilythataispositive, andthen,makingthesubstitution t=2a*sinh{ujn) intheintegrals, wefind that,by§6'21(10), 2a* r°° Cin(a)+iSin (a)=—exp(2a*"icoshu)cosh{ujn)dunJ =iriai n-'e*'''>iTi/^^^' (2a*"), that istosav Cin(a)+iSi, (a)=^^^ {e*"^'/"J-vn (2a*")-e-*-V»J,^^(2a*«)}. Ifweequaterealandimaginary parts, wehave Inasimilar manner, 2a*T" £"4(a)=—exp(—2a*"coshu)cosh {u/n) du, "J sothat,by§6-22(5), (3) ^V„(a)=(2a*/n)iiri;.„(2a*"). Tljese results havebeenobtained onthehypothesisthataispositive;and theexpressions ontherightaretheintegralfunctions ofawhich reduce to Cin(a),Sin(a)andEin(a)when aisreal,whetherpositiveornegative. Hence, when aisnegativetheequations (1), (2),(3)arestill valid, sothat, forexample, wehave i whether abepositiveornegative^*"^"^" 2wsin(j7r/r0 {,,?ovilT(,n +l-lln)"" J,,pilr{>u+l+Vn)\' W.B.F. 2T 322 ‘THEORY OFBESSEL FUNCTIONS [onar. x Hence,replacingaby—8,weseethat,when£ispositiveandniseven,then , Oo) CigB=9,nya tnCBM)+Sin22), ' 8 ; . 6)Sin(-B)= 9,conha1229")~Sun28"), . 7 on -G) Big(BY5injn)ve2A)+Lun(28) 1followsfrom§431(9)that,whenmiseven,thefunetionsCi,(a)and Si,(a)areannihilated bytheoperator a 2ght, enter, andthatEi,(a)isannihilated bytheoperator a ntenters In the case ofthe first two fanctions itisdifficult toobtain this result* directlyfromthedefinitions, beeausetheintegrals obtained bydifferentiating twice under theintegral sign arenotconvergent. 10-22. Theevaluation ofAiry-Hardy integrals ofoddorder. ‘Toevaluate Ci,(a)when nisodd, wesuppose temporarily that ais positive, andthen, by§622(13), Cig(a)=[cos(2a!sinhu)cosh(un)du =2cLI) (ty ‘Thatistosay, . @ Cig(a)=228%GI)5c,(2a) at (2a)=Fyn(20 BrainpapafmGe) Usingthedevice explained in$1021, weseethat, when Aispositive, @ CB)aisin(ejayIe 2B")+Sn28H) . Itfollows that theequation §10-21 (4)istrue whether »beeven oroddand,whether nbeevenorodd,Ci,(@)isannihilated bytheoperator « i"nta@t—2,Get ea, for all real valuesofa. integral" 10-22] ASSOCIATED FUNCTIONS 323 Nextweevaluate Ein(a)when aispositive ;makingtheusual substitution, wefind that,by§1013(4), 2«iToo Ein{oi)=—exp(—2ai"sinhu)cosh{ujin)du ='^{tan(K/«) J./n(2a^")-E,/„ (2a*»)} Hence theseries whichrepresents Ei^(a)when wisoddandamayhave anyvalue is (3)m,(a)=-^^°L^^ 2^ ^"^ ?icos(^Tr/ti) m=or(??i+I-^/w)r(w+1+!/?«) ?isin'(7r/ri) 1^=0t^i !r(^'i+1—l//i) m=o^'*'T(m+1+l/n)f' andhence itfollows that (4)\j^^+n-a'^-4 Ein(«)=nai^''-'K Next consider Ci„(a)+iSin(a),where aistemporarilyassumed tobe positive. From§1013 (4)wededuce that 2a*r^ Gin(a)+iSin(a)=—exp(2a*'*isinh m)cosh(uln)du =—{tan(iTT/n) J„n(-2a*"0-E,/,(-20*'^-)} ft +•'^"\ ,{/-vu(-2a*'^0-/:/.(-2a*'H')}?isni(7r/;()^ '^ ?lcos(^tt/'/O ^.tor(m+1-i/n)V(m+f+^/n) +-^--,-. {e*-/'* /-i/n (2a*»)-e-*-''"/,/„(2a4'0},wsm(7r/n)^ andtherefore 7ra*<"+^' <«^ a"'" (5) ;5i,(a)=- -^-™-^— ^^^^^r(m+f-i/n)r(m +f+iM) 2??cos(^7r/w) whence itfollows that,when /3>0, .r/3*+.7, ,,{/-:/n (2/3*")-/,/.(2/3*")}. 2?icos(|7r/?i)^ L'l—2 324 THEORY OFBESSEL FUNCTIONS[CHAP. X andhence, forallrealvalues ofa, (7)17"2~^' "*'~i'^^'^(«)=-^'«*'""'* • Thisequationwasgiven byStokes inthecasen=3. Itshould benoticed that (8) Si,(a)+(-)4-+^) Ein(a)= ^^^^^^{sin{^M +(-l)-^<'^+'' } =w\ X{sin(Uhi) +(-1)*'"+''} nsm{Tr/n)' v- // \ j X{/_,/« (2/3i«)- J,/,,(2y3i«)l where^=—a,andaand /3arereal. Theformulae oftheprecedingthree sections areduetoHardy, though hismethods ofobtaining them were different andhegavesome ofthemonly inthespecialcasen=3. 10'3. Cauchysnumbers. Inconnexion with ageneralisationofBessel'sintegi'alwhich wasdefined byBourget,andsubsequentlystudiedbyGiuliani(see §10"31),itisconvenient toinvestigateaclass offunctions known asCauchy's numbers. Thetypical number, N_nic,m,isdefinedbyCauchy*asthecoefficient of thetermindependentofiintheexpansionof inascending powersoft.Itissupposedthat n,k,andmareintegersofwhich thelasttwoarenotnegative. Itfollows fromCauchy's theorem that 1 ,"(0+)/1\^"/ 1\»» (1) ^-..M.= 2;„.J t-'-'lt+^){t~-)dt 27r =I[e-'"^+{-y e»^*}cos*dsin'"BdS Orn+k r-rr= cos(^niTT—nd)cos*6sin'"Odd. TTJo Itisevident from thedefinition thatiV^_n,fc,miszero if—n-\-k +misoddor ifitisanegative integer. *Comptes Rendu><, xi.(1840), pp.473— 475,510—511; xii,(1841), pp.92—93;xiii.(1841), pp.682—687, 850—854. 10-3] ASSOCIATED FUNCTIONS 325 From(1)itisseen that (2) N_nX,n.=i-)^NnX^a=(-)""* N^Xm- These results, togetherwith recurrence formulae from which successive numbers maybecalculated, weregiven byBourget*. Therecurrence formulae are (3) N_n, k.in=^^-n-rl, k-1,m+^-n-i, k-l,)ii> ('*) -''—», t,m^^-^'—n+i,i-,m—1~-^'—n—i,fc,»(-i> andtheyareimmediateconsequencesoftheidentities r»{t+i/tf (t-1/0'"=t'-''(t+ijtf-' (t-i/t)""+r"-i{t+i/tf-' (t-1/0'^ r"(t+ijtf {t-1/0"'=«'"''(t+1/0^' {t-1/0™~'-1'""'^(^+1/0* (^-I/O'""'- Bymeans ofthese formulaeanyCauchy's number isultimately expressiblein terms ofnumbers ofthetypes H^n.k.O' ^-n,o,m- Adifferent class ofrecurrence formulae, alsodue toBourget,owes its existence totheequation df l\ 1 Itfollows that h^)\ {'-7) s{'-"('+7)}* 27ri(1 byapartial integration. Onperformingthedifferentiation weseethat (5) {m+1)N_n^ k,m=nN-n, k-l,m+i- (A'-1)i\^_„, k-2.m+2, andsimilarly (6 ) (^"+1)-^^-n, /t,m=nN_n. k+i,m~i-(m-1)iV_„, ^.+0„,_2. DevelopmentsduetoChessiu, Annals ofMath. x.(1895—6),pp.1—2,are s r=0 .s (8)-^-n, k,?rt=2(— )'gC^.iV_„ +3_2,-, fr,))i-8' Thesemaybededuced byinduction from(3)and(4). Another formula duetoChessin is (9) A^-n,k,.n=2i-YkCp-r-mOr, (•=0 wherep=hU-+m —n).This isproved byselectingthecoefficient oft"intheproduct {t+iiifxit-i/ty". *Journal deMath.(2)vi.(1861), pp.33—54. 326 THEORY OFBESSEL FUNCTIONS [CHAP. X 10"31. Thefunctions ofBourgetandGiuliani. ThefunctionJn,k{z)isdefinedbythegeneralisationofBessel'sintegral (1) JnA^)=2^.j'"^'r"-(^+ ^)'expji^(^-^)|rft where nisaninteger,andkisapositive integer. Itfollows that 1 f'" Jn jfc(2')=^r- exp{—i{nO—zsind)\.(2cosQfdd,ZttJ-„ andtherefore (2) J-,,^ (,-)=-r(2cos6'/-cos(?i6'-^sin6*)fZa TTJo Thefunction«/„,^(2)hasbeen studied byBourget, Journal deMath.(2)vi.(1861), pp.42—55,forthesakeofvarious astronomical applications ;while Giuliani, Giornale diMat. XXVI.(1888), pp.151—171,hasconstructed alinear differential equationofthefourth order satisfied bythefunction. [Note. Anearlierpaper byGiuliani, Giornale diMat. xxv.(1887), pp.198—202, containspropertiesofanother generalisationofBessel'sintegral, namely 1M- /cos(«(9-2PsinP^)(:/^, butpartsoftheanalysisinthispaper seem tobeincorrect.] Ifweexpandtheintegrandof(1)inpowersof2^,wededuce from§lO'Bthat and itisevident from(1)that (4) J„,o(^) =J„(4 Againfrom§lO'S(2)and(3)itisevident that (5) J_,,,(2)=(-)-^-J„,(^), (6) Jn,k {z)=Jn-xk-i {2)+Jn+i,k-i (z)', and, ifwetake ^'=1inthisformula, (7) J„,,iz)=^J,,(z). These results were obtainedbyBourget;andthereader should have no difficultyinprovingthat (8) 2/Va^)=Jn-,,ic{z)-Jn+,A'). Other recurrence formulae (duetoBourgetandGiulianirespectively)are (9) J,,,+, {z)=^j:,,,^, (z)-^J^±}1 {^,^_, ,(^)_/,^^,_, (^)}, (10) 4/'Vt_, (z)=/„., (z)-4/„,,_3 {z). 10-31] ASSOCIATED FUNCTIONS 327 The differentialequationismostsimply constructed bythemethod used byGiuliani;thus 1f'"d^nJn,k (^)=- ;Tn{-(n+zcos6)sin{jid-zsin6)](2cosdfdd ITjdd 2kr IT.in+zcos6)sin{nd-z^\u6)(2cosOf-'sinOdO =-2kzJ'n^k {z)+^[\jQcos{nd-zsin6)1(2cosdf-^ sinj^f/^ =-IkzJ'n^j, (z)-— cos{nd-zsin6');^{(2cos^)^-^sind]dd, andso V,Jn,k {Z)=-2kzJ'n„ {Z)-kK/,,_k (Z)+4.k(k-1)/,,,_, (z). d^ Operatingonthisequation by;7^,+1,andusing (10),itfollows that (^,+ l)IV,./.,, (z)+2kzJ'„^k (z)+k^Jn,k {z)\=k(k-1)/,,, (z), andhencewehave Giuliani'sequation (11)2^J'\, (z)+{2k+5)^J"'«,, (z)+{'2z'+(/^-+2)^- n^}J"n^, {z) +(2^•+5)zJ'n^k {z)+{z'+k+'2-n')J^,^. {z)=0. Itwasalsoobserved byGiuliani that (12) e'-«i"«(2cos6?)^-= Se,„,Ln,k{z)cos2nd rt= +iSe.n+iJ2n+i,k {z)sin{2n+l)d; this isverified byapplyingFourier's rule(cf. §2'2)tothefunction onthe right. Asomewhat similar function J{z ;v,Jc)hasbeen studied byBruhns, Astr. Nach. civ. (1883),col. 1—8.This function isdefined bytheseries Themost important propertyofthisfunction isthat (14) J{z; V,k)-J{z; V,J^+l)= j^^^^^f^:^^ff^f:^y whence itfollows that n^N // z^_%"2vJ^^.2m{z) ^^^^ •^^'' ''''^^-,„?,(v +2».-2)(. +2m+2)' 328 THEORY OFBESSEL FUNCTIONS [CHAP. X 10*4. Thedefinition ofStruve's function 'H.^{z). Now thatwehavecompletely examined thefunctions defined byintegrals resemblingBessel'sintegral,itisnatural toinvestigateafunction defined by anintegral resemblingPoisson'sintegral.This function iscalled Struve's function, althoughStruveinvestigated* onlythesjDecialfunctions ofthis typeoforders zeroandunity. Thepropertiesofthegeneralfunction have beenexamined atsomelength bySiemonf andbyJ.Walker;):. Struve's function H^(z),oforderv,isdefinedbytheequations sm(zcosO)sin''' Odd,2(izy r(i;+i)r(|).io providedthatR(v)>-h. Byanalysissimilar tothat of§3'3,wehave (1^)"^(-)"*227ft+».m! r(i),„=o(2m +l)!r(^ +m+f)' sothat (2) H,(^)= S Thefunction H^(z)isdefined bythisequationforallvalues ofv,whether R(v) exceeds —|ornot. Itisevident thatH^(z)isanintegralfunction ofv and, ifthefactor(^z)"besuppressed,theresulting expressionisalsoanin- tegralfunction ofz. Itiseasytosee[cf§§2-11(5),3-121(1)]that where (4)^'^«'W=IWPT?Ti)<'^'*' and I^0+1 1isthesmallest ofthenumbers|i/+f|, li'+fl, li'+li,— *Mem. deVAcad. Imp. denSci.deStPetersbourg, (7)xxx.(1882),no.8;Ann. derPhysik, (3)XVII. (1882), pp.1008—1016. SeealsoLommel, Archiv derMath, undFhijs.xxxvi.(1861), p.399. tPrograinm, Luisenschule,Berlin, 1890. [Jahrbuchilber dieFortschritte derMath. 1890, pp.340—342.] XTheAnalytical Theory ofLight (Cambridge, 1904), pp.892—895.Theresults contained in thissection, with theexception of(3),(4),(10)and (11), arethere given. 10-4] ASSOCIATED FUNCTIONS 329 Wecanobtain recurrence formulae thus : andsimilarly ^b-H (5V= ^ (-)-(2m +l)^- c?^^"^^^ ^^to 2''+-^-+i r(m+f)r(v+m^) «/_y«+i22m+2 ^,«"_i2'^+^+-^ 1^(m+I)r(i;+m+1) 1 ~2''r(z. +l)r(i)"^"H.+iC^ Oncomparingthese results, wefindthat (5) H,_. (.).H,„ (.)=.^H.(.)+ J^llV-^^^. (6) H._.(.)-H.„(.) = 2H/,.)-p^(i|]l^^. (7) (a+WH,(«)=2H,^,(2), Inparticular wehave (9) ^{m, (^)]=zU, (z),I{Ho (^)}=I-H,(^). Again, from(7)and(8),wehave^ C^-^- V-')H,(2)=(^-v)|^H,_i (z)} =z('^-v +l)II,_,{z) r(i.+|)r(i)^•"•'*^^^' sothatH^(z)satisfies thedifferentialequation (10) V,H,(^)= ^(^'+i)^(|)• Thefunction L^(z)which bears thesame relation toStruve's function as/^(z)bears toJ^(s) hasbeen studied (inthecase v=0)by*Nicholson, Qxiarterly Journal^ XLn. (1911), p.218. This function isdefinedbytheequation (11) L.(2)= 2 *^^^ ' ^^„,=or(w +#)r(:.+m +f) r(,.+A)r(1)/'^^"^^'''*^'*^-^'^"'''' ^^^' theintegral formulabeing valid onlywhenR{%>)>—\. Thereader should have nodifficultyinobtaining thefundamental propertiesofthis function. *SeealsoGubler, ZurichI'ierteljahrsschrijt,xlvii.(1902), p.421. 330 THEORY OFBESSEL FUNCTIONS [CHAP. X 10*41. Theloop-integral forH^(^). Itwasnoticed in§10*4that theintegraldefinition ofH^{z)failswhen R{v)<:—^,because theintegraldoes notconvergeattheupperlimit.We canavoid thisdisability byconsideringaloop-integralinplaceofthedefinite integral. Letustake (f-ly-^-sinzt .dt, Jo where thephaseoff^—lvanishes atthepointontherightof^=1atwhich thecontour crosses thereal axis,andthecontour does notenclose thepoint t=-l. IfwesupposethatTi{v)>—\,wemaydeform thecontour intotheseg- ment(0,1)ofthereal axis,taken twice, andwefindthat r(i+) n {t--ly-isinzt .dt=2icosVTT(1-t-y-^ sinzt .dt, Jo Jo where thephaseof1—^^iszero. Hence, whenR(v)>—^,wehave (1) H.{z)=^^ ;VilI <f'-^y-"-«i«'i 'dt-in1(2)•' Both sides ofthisequationareanalyticfunctions ofvforall*values ofv ; andso,bythegeneral theoryofanalytic continuation, equation (1)holds for allvalues ofv. From this result, combined with§6"1(6),wededuce that (2) j^{z)+ iH,{z)=^^^.;;Vi\-'^e-'{t^-1)'-dt. Totransform this result, let«beanyacuteangle (positiveornegative), and letthephaseofzliebetween —^tt-f<«and|-7r+w.Wethen deform thecontour into thatshewn inFig. 26,inwhich thefourparallellines make anangle—wwith theimaginaryaxis. Itisevident that, asthelines paralleltotherealaxismove offtoinfinity,theintegrals alongthem tend to zero.Theintegral alongthepathwhich starts fromandreturns to1+soie~''" isequaltoi/^w {z) ;andonthelinesthroughtheorigin wewrite t—iu,so thatonthem (f--1)"-*=eT{>'-h)W(1+^c^y-h, Itfollows that /,(z)+in,{z)=i/.a) (^)+jT^^y^p ^^^j^e-'(1+u^-^ du, *Theisolated values|,#,|,...areexcepted, because theexpression ontherightisthenan undetermined form. 10-41] ASSOCIATED FUNCTIONS 331 where thephaseof1+u^has itsprincipalvalue;andhence (3)9('l^V /•«exp(-itu) This result, which istrue forunrestricted values ofp,and foranyvalue of 2forwhich -tt<argz<7r, willbeapplied immediatelytoobtain theasym- ptotic expansionofH^(z)when |^ jislarge. Fig. 26. Aresultequivalentto(2)wasobtainedbyJ.Walker*, whoassumed thatR{v)>—^,B,(z)>0,sothat o)mightbetaken tobezero. Inthecase v=0,theresult hadpreviouslybeen obtained byRayleighfwith theaidof themethod ofLipschitz (§7'21). If,asin§6"12, wereplacecoby-drgz—^,itisevident that(3)maybe written intheform (4) n,{z)=Y,(z) +(h^y-'ccexp /(3 du, where—|7r</3<^ttand-^ir+/3<argz<i7r -\-^. Thisequation givesarepresentationofH^(z)wheniargi;!<tt.Toobtain arepresentationvalid nearthenegativehalfofthereal axis,wcdetinc Hi,(^) forunrestricted values ofargzbytheequation ^,, (5) H,(2e""^0=e'"^''+i)'^'H,(2),^_,/ anduse(4)with zreplaced byze^^K "TlieAnalytical Theory ofLight (Cambridge, 1904), pp.394—395. tProc.London Math. Soc. xix. (1889), pp.504— 507. [Scieiitijic Papers,in.(1902), pp.44—46.] 332 THEORY OFBESSEL FUNCTIONS [chap. X Ifwewrite z=ixin(3),where xispositive, weseethat,whenR{v)<\, and,byconsidering imaginary parts, wededuce that (6) lty{x)=-I_^{x)-2{W r(i/+*)r(i) josin{xu).(1+i{2)''-2 du aresult given byNicholson, Quarterly Journal, XLii.(1911), p.219,inthespecialcase in which V=0. 10*42. Tlieasyni])totic expansion ofH^{z)when jz \islarge. Weshallnowobtain anasymptotic expansionwhichmaybeused fortabu- latingStruve's function when theargumentzislarge,theorder vbeingfixed. Since thecorresponding asymptotic expansionofV^,(z)hasbeencompletely investigatedinChapter Vil, itfollows from|10"41(4)that itissufficient to determine theasymptotic expansionof Asin^7'2,wehave (-1)^2sv-JP^^i-y^'.ik-vXr.Uam w= mlz'' +(-)^.a-^)pw^ r (p-f^5^//-')-^?)uHy-p-^ dt. WetakepsolargethatR(v—p—h)^0, andtake Stobeanypositive angle forwhich |/3|-$^7r— S, Iargi:— yS|$Jtt—S, sothat zisconfined tothesector oftheplaneforwhich —TT+2S^argz^7r—28. Wethenhave sothatill\/t\,^. 5V1+—~ 1 !^sin 6, arg{I±ill\Jt <'rr. 1+~Y^* I^e^-i/WI(sin8)2i2W-2p-i=A^, say,whereA^isindependentofz. Itfollows onintegrationthat /J \-^/ 7)1= mlz^ where I^pi^-^"1^-- =(^-2^).aoexpip e-" it'^Pc?w 10-42, 10-43] ASSOCIATED FUNCTIONS 333 Wededuce that,when {argz\<Tr and |^islarge, providedthatR{p—v-[-\)^0; but, asin§7'2,this last restriction maybe removed. Thisasymptotic expansion mayalsobewritten intheformLo<- . 1/'-I V(m4-^\' (2) H,(^)=F,(^)+ Sp,^1Tm v^n.-.+i+Q(^'^~^M- ^' ^7r„i=ol (^+1—w)(^2-y"*"+1 Itmaybeprovedwithoutdifficulty that, ifvisrealandzispositive,the remainder after'pterms intheasymptotic expansionisofthesamesign as,and numericallylessthan the first termneglected, providedthat 'R{'p-\-\—v)'^^.Thismaybeestablishedbythemethod used in§7-32. Theasymptotic expausion* wasgiven byKayleigh, Proc.London Math. Soc.xix.(1888), p.504inthecase i'=0,byStruve, Mem. deVAcad.Imp.desSci.deStPetershourg, (7) XXX. (1882),no.8, -p.101,andAnn. derPhys.undChemiey (.3)xvii.(1882), p.1012 inthe case v=\;theresult forgeneralvalues ofvwasgiven byJ.Walker, TheAnalytical Theory ofLight (Cambridge, 1904), pp.394—395. IfVhasanyofthevalues^,%,...,then(1+u-/^-)""-isexpressible asa terminatingseries andY^{z)isalsoexpressibleinafinite form. Itfollows that,when vishalf ofanoddpositive integer, H^{z)isexjDressibleinterms ofelementaryfunctions. Inparticular [Hj (^)=(-)"(!-cos^), ^^^ l-r,r.X /^^^^2\/2\V• COS^ 10"43. Theasymptotic expansion ofSti'uve'sfunctions oflai-ge order. We shallnow obtainasymptotic expansions,ofatypesimilar tothe expansions investigatedinChapter viil,whichrepresentStruve's function H^{z)when jv \and |z \arebothlarge. Asusual, weshall write V=zcosh(a+t/3)=zcosh7 and,x^r simplicity, weshall confine theinvestigationtothespecialcase in which cosh7isrealandpositive.Themoregeneralcase inwhich cosh7is complex maybeinvestigated bythemethods used in§8'6and§10'15,but itis ofnogreat practical importance and itinvolves some rather intricateanalysis. *Foranasymptotic expansionoftheassociated integral seeKayleigh,Phil.Mag. (6)viii.(1904), pp.481—487.[Scientific Papers,v.(1912), pp.20G—211.] 334 THEORY OFBESSEL FUNCTIONS[CHAP. X Themethod ofsteepestdescents hastobeappliedtoanintegralofPoisson's t3"pe,andnot,asintheprevious investigations,tooneofBessel'stype. Inview oftheformula of§10"41(3),weconsider theintegral dw L-^z(l +u'-y which wewrite intheform div f'—ZT where t=w-cosh7.log(1+tv^). Itisevident that r,quafunction ofw,hasstationary points wherew=e-y, sothat, since7isequaleither toaortolyS,twocases have tobeconsidered, whichgiverisetothestationary points (I)e±«, (II)e±»^. Accordingly weconsiderseparatelythecases(I)inwhichzjvislessthan1,and (II)inwhichzjvisgreaterthan 1. (I)When 7isarealpositive number a,tisrealwhenwisreal,and,asiv increases from tooc,tfirst increases from toe~"—cosho.log(i -\-e~^), then decreases to e"^—cosh a .log(1+e^")andfinallyincreases to+ao . Inorder toobtain acontouralongwhich tcontinually increases, wesuppose thatwfirstmovesalongthereal axisfrom theorigintothepoint e~'^,and then starts moving alongacertain curve, which leaves thereal axis atright angles,onwhich tispositiveandincreasing. Tofindtheultimate destination ofthiscurve, itisconvenient tomake a changeofvariables bywriting w=sinh^,C= 1^+^'7> 6""=sinh^0, where^,rjand^0arereal. Thecurve inthe^-plane,onwhich tisreal,hasforitsequation cosh^sin?;=2coshaarctan(tanh ^tan1;), and ithasadoublepoint*at^o- Wenowwrite „ J,2arctan(tanh ^tan77) cosh^sin-t] andexamine thevalues ofi^(f, r})as^traces outtherectangle whose corners 0,A,B,Chavecomplexcoordinates 0,arcsinh 1,arcsinh 1+|tti, ^iri. As^goesfrom toA,F{^, t])\s,equalto2sinh|/cosh'- f,andthissteadily increases from to1. *Except when a=0,inwhich case ithasatriple point. 10-43] ASSOCIATED FUNCTIONS When ^isonAB,F(^, ??)isequalto \/2.arctan(—7^^ ).cosec77,335 V\/2 andthissteadilyincreases from 1to7r/\/2asrjincreases from to^tt. Note. Toestablish this result, write tanr]=tJ2andobserve that dU(l+2fi) ^\ 1 (t+2fi,I dt1—r~^'''^''7^ t^Tiu^) \t+7^-"'"''"7^*"' t+2t^ . . . . . 2fi(2+fi)because ^j5--arctant,which vanishes witht,hasthepositive derivate ^. When ^isonBG,F(^, 77)isequaltoirsech^,andthisincreasessteadily from irls^'I tottas^goesfromBtoC;andfinally when ^isonCO,F{^, -q) iszero. Hence thecurve, onwhichF{^, 77)isequaltosech a,cannotemergefrom therectangle OABC, exceptatthedoublepointonthesideOA ;andsothe partofthecurve inside therectanglemustpassfrom thisdoublepointtothe singular pointC. Thecontours inthe2t'-planeforwhich ahasthevalues0,iareshewn inFig.27by broken andcontinuous curvesrespectively. Fig. 27. Cdnsequentlyacontour inthetf-plane,onwhich risreal, consists ofthe partoftherealaxisjoiningtheorigintoe""andacurve from thispointto thesingular point i;and, aswtraces out this contour, rincreases ft'om to+00 . Itfollows that,iftheexpansionofd^/chinpowersoftis ^i= -hr'"ar „i=o 336 THEORY OFBESSEL FUNCTIONS[CHAP. X then o«j w=* andhence, by10-4(1),wehave /-.x XT /\•r/\ . 2(*^)'' V''^•^ (1) H,(^)~-i/,(^)+p— -^--^^—,S Itiseasytoprovethat bo= I, 6i=2cosh7, 62=6cosh-7—i,63=20cosh^7—4cosh7, (II)When 7isapure imaginary (=i^),tisrealandincreasessteadily from toCOaswtravelsalongtherealaxisfrom to00;andso •;.-(1+».)'-»dn,= /;.-"[^^^ *"|4r. Hence, from§1041(3)itfollows that (2) H.(.)~F.(^)+ ^//^/>;,,, i^'^- providedthat jargz \<^tt.This result canbeextended toasomewhat wider domain ofvalues ofarg z,after themanner of§8"42. From thecorrespondingresults inthetheoryofBessel functions, itistobe expectedthat these results arevalid forsuitable domains ofcomplexvalues ofthearguments. Inpcirticular, wecanprove that, inthecaseoffunctions ofpurely imaginary argument, (3) Ii„(vx)~I^(vx) when 11- 1islarge, |argv |<^tt,xisfixed, andtheerror isoftheorder ofmagnitudeof ]ri+j(i+x^).^..,^ .,,,7^• [— 2— ^''i' '~^^ ''"•^"^* times theexpression ontheright. [Note.Ifin(I)wehadtaken thecontour fromw—0 to ?<;=e~"andthence tow=— i, weshould have obtained theformulacontaining iJy(2)inplaceof-iJy (2).This indicates thatwegetacase ofStokes' phenomenon asycrosses theline/3=0.] 10"44. Therelation between H,i(z)andE„(z). When theorder ?;isapositive integer (orzero),Avecandeduce from §lO'l(4)that E,i(z)differs from—H„(z)byapolynomialinz;andwhen n isanegative integer,thetwofunctions differ byapolynomialiu1/z. 10-44, 10-45] ASSOCIATED FUNCTIONS 337 For,when nisapositive integerorzero,wehave ~ m=-nr{hu+\)T{\m +n+\)' and COp—\>mri(l„\n+m "^^^^«=or{\m+1)r(i//i+n+1)' andtherefore, since Jn(z)=Jn(z), wehave ,„=i1(1-pn)r(?i+1-|m) that istosay (1) E,.{s)=l% ii^»±^Ml£L^_H,.(.). Inlikemanner, when—??isanegative integer, (2) E_. (.)=L-)"ft^"'-"-^Vtf""'"'-H-»(4 10'45. T/iesi^nofStrlivesfunction. Weshallnowprovetheinterestingresult thatlrl^(x)ispositive when a;is positiveand vhasanypositivevaluegreaterthan orequalto^.This result, which waspointedoutbyStruve* inthecase i^=1,isderivable from a definiteintegral (whichwillbeestablished in§13'47) which isofcon- siderableimportanceintheTheoryofDiffraction. Toobtain theresultbyanelementajy method, weintegrate §10'4(1) by partsandthenweseethat, forvalues ofvexceeding |, iw" V{v+h)V{\) \icos(.a;cos6)sin-""^ 6 -(2z;-1) I' COS{xcosd)sin-"--^ cosddd"^ /=h/^^"^'fxn/1X11-(2^'-1)['"cos {xcosd)sin-"--^ cosOdd]1(^+2)^(i) I hJ =4^—tvWttsin^''--'6' cos^11-cos{xcos6)}dO ^0, since theintegrandispositive. *Mem. deVAcad. Imp. desSci.deStPetersbourg, (7)xxx.(188'2),no.8,pp.100—101. The proof given here isthenatural extension ofStruve's proof. w.B.F. 22 338 THEORY OFBESSEL FUNCTIONS [CHAP. X When pislessthanh,thepartial integration cannot beperformed ;and,when v=|,we have Hi(.^•)= (^.)*(l-cos.^')^0, andthetheorem iscompletely established. Acomparisonoftheasj'mptotic expansion which wasprovedin§10"42 with that of Y^(,v)givenin§7"21shews that, tc/ienxissufficientiji largeandpositive, H^{x)ispositive if 1/>3andthatHk (.r)isnotone-signed when j'<^;forthedominant term ofthe asymptotic expansionofH^{x)is or(— I^sin(x—Ivn-i-n) \lTXj' -r(.'+*)r(i) according as i/>ior i^<i.Thetheorem ofthissectionprovesthemore extended result that Struve's function ispositiveforallpositive values ofxwhen i/>iandnotmerely forsufficiently largevalues. Thetheorem indicates anessential difference between Struve's function andBessel functions;fortheasymptotic expansionsofChapterviishew that, forsufficiently large values ofx,J^{x)andY^(x)arenotofconstantsign. 10"46. Theisinger's integral. Ifwetaketheequation /"in- 1JUi'piQ J-hn^1-te'e r./"* fA-/ 1\"1 ,l+izdz7r2 IZz' andchoose thecontour tobetheimaginary axis,indented attheorigin* andthen write z=.+itau1^,wefindthat TT 4 andso-pT di> {h(^)-I»oix))=/cos{xcot^)logtan(iir+*0)-^^, (1) /o(^)-Lo i-v)=—ir" cos{xtan</>)logcot{h<i>)^^, aformula given byTheisinger, Monatshefte fUrMath, undPhys.xxiv. (1913), p.341. Ifwereplace xby.vsiu6,multiply bysin6,andintegrate, wefind,onchangingthe order oftheintegrationsintheabsolutely convergent integral ontheright, IEl{xtan<^)logcot(|0)-^=|f'" (^(^«^°^)'^o (-^sin6)}sin6de sothat \-e-' (2)/'" El{xtan0)logcot(*0)-^=fJ cos(p^ cos(f)2'X' onexpandingtheintegrand ontherightinpowersofx.This curious result isalsodueto Theisinger. *Thepresence ofthelogarithmic factor ensures theconvergence oftheintegral round tlie indentation. 10-46, 10-5] ASSOCIATED FUNCTIONS 339 10"5.Wliittaker'sintegral. Theintegral which isasolution ofBessel'sequation onlywhen 2visanoddinteger,has been studiedbyWhittaker*. Itfollows from§6"17 that, forallvalues ofv, (1 )VJ^He'^«P,_j (t)dt\=-lim[^ie'^« (1-f')P',-i(t)] =—cosvir .z^e'"'^. TT Ifweexpandtheintegrand (multiplied bye'~)inascending powersofzand integrate term-by-term i*itisfound that Theformula of§3'32suggeststhatwewrite andthen itiseasytoverifythefollowingrecurrence formulae, eitherbyusing theseries(2),orbyusingrecurrence formulae forLegendrefunctions : (3) W,_i (z)+W,+, (z)= (4)2v zI'• 'v(27r)r(f-^)r(fi-.) 2,1^^" z"-e~^^ W._, (z)-W.^. (z)=2W;(z)- ^J^^"p(^^_^^Y(l +v)' (5) {^+v)W,{z)=zW,_,{z)+*^''^'^^ V(27r).r(f-i/)r(i-t-i;)' Anasymptotic expansionofW^(z)forlargevalues of |zmaybeobtained by deformjflgthepathofintegrationafter themanner ofLipschitz (§7"21). *Proc. London Math. Soc.xxxv. (1903), pp.198—206. tByauseofLegendre's equation therecurrence formula /"I 2 may beverified; andwecanprove that/P,(t)dt==rT:, rrrTV^ \^yexpi^"<Ji"t? /_j''"- r(f+»')r(f- v) ^'(i~">4+'';1;5-2*)iiascending powersof1-f,andintegrating term-by-terni. 222 340 THEORY OFBESSEL FUNCTIONS [CHAP. X Thefunction isthusseen tobeequalto =--uUr L ^•""".^-'<-'>*- -vw) /,'"'"""^'-'^'^'" Now itisknown that*, neart=l, _/cos i^TryIr(m-i/ +|)r(m +i/+i)/I- ty X jlog(^^)- 2-«/r(m+1)+A/r(m-i;+1)+Vr(m+r+1)1, andsince •'-"\,-.n-./i-^r^^^^''^-^r(/.+i) 2/ii^M+l i:^",->..-„fL^M'<,,J'-'r(M+i) 1 V2/ •2'^^'^+i weobtain theasymptotic expansion (7)W,(^)~iiT,"' (^) +GC TTV(27r2^) +yfr(m+^+v)—yfr{m+J)-\og2z— ^iri] Some functions whichsatisfy equationsofthesamegeneral typeas(1)have been noticed byNagaoka, JournaloftheColl.ofSci.Imp.Univ. Japan,iv.(1891), p.310. 10'6. Thefunctions composing Yn{z). Thereader willremember thattheBessel function ofthesecond kind, of integral order,maybewritten intheform(§3"52) IX .,'"-^(n-m-\)\ ,, ,' +.!„ Jid'+l)!I^'°g(i^)-V'(''>+ !)->/.(. +»,.+1)1. The series ontherightmaybeexpressedasthesumoffourfunctions, each of which hasfairly simplerecurrenceproperties,thus *Cf.Barnes, Quarterly Journal, xxxix.(1908), p.111. 10-6] where (2)ASSOCIATED FUNCTIONS 341 and(of.§3-.582)m ! (3) Un{z)= 2(-rd^f^""" =0?u!(n+»i)!{i/r(/|+w +l)-->/r(l)} ThefunctionsT'n(2)^i-ndUn{z) have been studied bySchlafli, Math. Ann. ill.(1871), pp.142—147,though heused thesHghtlydifferent notation indicated bytheequations more recent investigationsareduetoOtti* andtoGi*afandGublert. Thefunction Tn{z)ismostsimply represented bythedefiniteintegral (4) Tn{z)=-["(iTT-d) sin(zsind-nO)d9. Toestablish this result, observe that Tn{z)=(-y»(|2)"+-»* de„i>_j„_j r{m+1+e)r(?i+m+1- e)Je=,-r^y^ 1 27rtLae,„>_j,,_j (n+2m)! j_i V^+'^^e=0 1(-)»" (i^y^+--"^/'«+>(1+tY+'''^ log^ 27n\^>.';„_i (/i+2m)! ./_if'^^^ 2TT Xe«'9(-i> sin^f+"-'".(^-Itt) ,. = ;I 2, ;;—CW, '7^^Jom>-irt-i (n+27H)! where thasbeenreplaced byeC-^'-'^ii. Itfollows that<^^ Now andsoI(-izsin^)»+2'«_jcosh(-izsin^) (neven) ,^>_j«_j (h+2?/i)!~ [sinh(-izsin6) (nodd) TTlJ *BernMittheilungen, 1898, pp.1—56. tEinleitimg indieTheorie derBesseVscken Funktionen, 11.(Bern, 1900), pp.42—09.Loramel's treatise, pp.77—87,should alsobeconsulted. 342 THEORY OFBESSEL FUNCTIONS [chap. X If6isreplaced bytt—^intheintegralobtained byconsidering onlythe second ofthetwoexponentials,theformula(4),which isdue toSchlafli,is obtained atonce. Thecorresponding integralforUn(z)isobtainedbyobservingthat Un{Z)=-~d_I(-)'^ (1^)"+^"^ r(1+6) _9e,„=om !r(n+7/i+1+e) €=0 aeKi^)-^r(i +e)j-,+,(^)} €=0 and so,from§6*2(4),wededuce that (o)p-„w=fiog(i^)-t(i)K»W -r+-r^sin(n(9-^sin^)rf^+(-)"[^e-««-^smhf ^^, TTJo y 10*61. Recurr-ence formulae forTn{z) and(Jn(z). From§10"6 (4)weseethat Tn-^{z)+Tn+^{z)-(27l/z)Tn(z) =-(Itt- (9)sin(zsin^-n^).{2cos6-2n/z]dO =-—r(hir-6)^ {cos(zsind-ne)}deTTZJQ"civ 4 4=-COS^4??7r Jn(z),Z^z^ onintegrating bypartsandusingBessel'sintegral. Thus (1) Tn-. {z)+7^,,+, {z)=(2n/2) T^{z)+4{cos^Invr-J„{z)]\z. 22/'^ Again T„'(^)=— I(Itt—^)sin^cos{zsin^—nd)(^5*, andso (2) From these formulae itfollows that (3) (SV+n)Tn{Z)=zTn-, (Z)-2COS^1IITT+2J„(z), (4) (^-»)Tn{z)=-zTn+, {z)+2cos^*nTT-2/„ (^), andhence(cf.§1012) wefindthat (5) V,jTn{z)=2[zsin21?i7r+ncos^Iwtt}-4hJn(z). 10-61, 10-62] ASSOCIATED FUNCTIONS 343 With theaidofthese formulae combined with thecorrespondingformulae forJn{z),Y„{z)andSn[z),wededuce from§10"6(1)that (6) Un-, {z)+Un^, {z)={2n/z) Un(z)-(2/z) J,,(z), (7) f^„_: (Z)-Un+, {Z)=2U,:(Z)-{21Z)J,{z), (8) (^+n)Un{z)=zUn-. [z)+2J, {z), (9)^ (^-^0 U,(z)=-zUn^,(z), [of.§§3-58(1), 3-58(2)] (10) V,,Un(z)=-2zJ,^,(z). Thereader may verifythesedirectlyfrom thedefinition, §10'6(3). Itisconvenient todefine thefunctionT_n(z),ofnegative order, bythe equivalentof§10"6(4).Ifwereplace^bytt—^intheintegralwefindthat T_n(z)=- f"(17^-^)sin(zsin6+nO)cW =(ivr—^)sin(^sin6—nO }•inr)dd, andso (11) T_,,{z)=(-r+^TA^). Wenowdefine U^niz) bysupposhig §10"6(1)tohold forallvalues ofn; itisthenfound that (12) t7_,{Z)=i-Y {Un(Z)-Tu(Z)+Sn(z)]. 10-62. SeriesforTn{z) andUn{z). Weshallnowshewhow toderive theexpansion (1) Tn{z)= t-{Ju+2nAz)-Jn-,,niz)]m=l"'' from§10-6(4).Themethod which weshall use istosubstitute ^sin2m0 TT^---0=^ s intheintegralforTn(z),andthenintegrate term-by-term.Thisprocedure needsjustification,since theFourier series does notconverge uniformlynear ^=^d6=iT,and, infact,theequation justquotedisuntrue forthese two values of0. Tojustify theprocess*,letSand ebearbitrarilysmallpositivenumbex's. Since the seriesconverges uniformly when S^^^tt—S,wecanfindaninteger mosuch that ^^sm2m(9 {\7T-6)-2 m=im<f, *Theanalysis immediately followingisduetoD.Jackson, Palermo Rendiconti, xxxii. (1911), pp.257— -262.Thevalue oftheconstant Ais1-8519... . 344 THEORY OFBESSEL FUNCTIONS [CHAP. X throughouttherange8^6^ir—8,forallvalues ofMexceeding Wq-Again,forallvalues of6between andtt,wehave ,-M^sin2m0 m=l »l=I' {l+2cos2^ +2cos4^+...+2cos2i¥i!}o?^ _|''i'rsin(2i/+l)«t J(dt g t SUl t ,[h^s\n('2,M+\)t-,f(M+h)^ sin 07, =iTT;^—dt= JttI ao;- ]^t' ./(23f+l).f. ^ forsome value ofbetween 6and^tt,bythesecond mean-value theorem, since tlsintis amonotonic(increasing)function. Bydrawing thegraphof.r~isinjp itiseasytoseethat thelastexpressioncannot /"^sniT exceed Jrr I— '-dxinabsolute value;ifthisbecalled hnA.wehave-jo^ Tn{z) 2 I sm(2Sin^-«^)rf^^m=iJ"* /Tf ^sin2?u^l J(i:r-^)- 2-^^[sin (zsine- nd)dd n['m=l «iJ sin(2sin6—n6)\d6^U y5 r JTT-Sj11' m=l™J <-{7r.48 +(7r-2S)f}^, where5istheupper bound of |sin{zsin6-n6) |. Since (2J.84-e)5isarbitrarily small, itfollows from thedefinition ofan infinite series that* Tn{z)=- Ir'i'^-?^sin(zsind-nd)cW ~^I3rWn+2m (-3")—Jn-2m \^)h andtheresult isestablished. Itwillberemembered thatUniz) hasalreadybeen defined(§3'581) asa series ofBessel coefiicientsbytheequation andthat, in§3-582, thisdefinition wasidentified with thedefinition ofUn(z) asapowerseriesgivenin§10Q(3). 10"63.Graf's expansion ofT^{z+t)asaseriesofBesselcoefficients. Itiseasytoobtain theexpansion (1) Tn(z+t)=1Tn-,,{t)J^\z),m=—00 *Thisexpansion wasdiscovered bySchlafli, Math. Aitn. iii.(1871), p.146. 10-63, 10-7] ASSOCIATED FUNCTIONS 345 for,from§10-6(4),itisevident that T;^(^+^)=^["(1TT-6)sin{tsine-nd-\-z sin0)dd TTl. j/i=-cc =^["(1,^-^)I ./,„(^)sin(^sin^-(71-m)l9)r/^, TTj m=-CO byusing §2"1 ;since theseries under theintegral signisuniformlycon- vergent,theorder ofsummation andintegration maybechanged,andthe result isevident. Theproofoftheformula given byGraf, Math. Ann. xliii. (1893), p.141,ismorecom- plicated;itdepends ontheuseoftheseries of§10-62combined with§2-4. There seems tobenoequally simple expressionfor£/"„{z+t). 10"7. Thegenesis ofLommeV sfunctions S^^y(z) ands^^t,{z). Afunction, which includes asspecialcases thepolynomials zOn{z)and 8n{z)ofNeumann andSchlafli, wasderived byLoramel, Math. Ann. ix.(1876), pp.425—444, asaparticular integraloftheequation (1) V,y=kz>-+\ where kand/*areconstants. Itiseasytoshew that aparticular integralof thisequation, proceedinginascending powersofzbeginningwith z'^^'^, is (2) y=k + (^+1)._,r-[{f,+\r-- v^-]{(^+3)^- .^j =kz^-^ i(-)-(iy»+--^ ^, ,_iI(-)>n(l^)em+2r(l;.-l^ +l)r(l^ +|.+|) ,„to V{l^Ji-\v +m+^,)^{\^l-^\v +m+^) Forbrevitytheexpressionsontherightarewritten intheform Thefunctions^^^{z)isevidentlyundefined when either ofthenumbers /x+fisanoddnegative integer*. Apartfrom this restriction thegeneral solution of(1)isevidently (3) 2/='^,(2r)+A-S,,,(0). Inlikemanner thegeneralsolution of IS (5) y-z-'-^''-'Hr^,{z)+ks,,A^))- *Thesolution oftheequationforsuch values of/xand visdiscussed in§10-71. 346 THEORY OFBESSEL FUNCTIONS [chap. X Next letusconsider thesolution of(1)bythemethod of"variation of parameters." Weassume asasolution* whereA(z)andB{z)arefunctions ofzdeterminedbytheequations J,{z)A'(z) +J_A^)B'(z) =0, J',(z)A'(z)+J'_, {z)B'{z)=kz^-\ Onusing §3"12(2),weseethat A(^)=.lTT zsm v-TTz'^J_^{z)dz,B{z)=-;7r 2smvTTz>^Jy{£)dz. Hence asolution fof(1)is (6) i/=72sinvKJ^{z)z^J_^{z)dz-J_^{z)z^J^{z)dz where thelower limits oftheintegralsarearbitrary. Similarlyasolution of(1)which isvalid for allvalues ofv,whether integersornot, is (7) y=i^'T^ F,{z)z^J^{z)dz-./,{z)z^Y^(s)dz Itiseasytoseethat, ifboth ofthenumbers/x+i^+1havepositivereal parts,thelower limits in(6)and(7)maybetaken tobezero. Ifweexpand theintegrandsinascending powersofz,Aveseethat theexpressiononthe rightin(6)isexpressibleasapowerseriescontainingnopowersofzother than z^^^, z^'^'^, z^'^^,—Hence, from(3),itfollows that, since neitherofthe numbers/m±visanoddnegative integer, wemust have (8)V..(^)=TT 2sinVTTJ^{z)z''/.^(z)dz-J. Jozi^Jv{z)dz Inobtainingthisresult itwassupposedthat visnotaninteger;but if weintroduce functions ofthesecond kind,wefindthat (9) s^^Az)=l7rY^{z)z^J^{z)dz-J^{z)z^Y^(^)dz andinthisformula wemayproceedtothelimit inmakingvaninteger. Itshould beobserved that, inPochhammer's notation(§4"4), (10)V^(^)-(^_^ +l)(^^^_^l) x,2^2(l; \ii-\v^\,\ii^\v +\\-\z'). *Cf.Forsyth, Treatise onDifferential Equations (1914), §66;itissupposed temporaiily that Visnotaninteger. fThegeneraUsation ofthis result, obtained byreplacingzt^+'^ in(1)byanarbitrary function of2,wasgiven byChessin, Comptes Rendus, cxxxv. (1902), pp.678—679; and itwasapplied by him,Comptes Eendus, cxxxvi.(1903), pp.1124—1126, tosolve asequence ofequations resembling Bessel's equation. 10-71] ASSOCIATED FUNCTIONS 347 Theassociated function S^„(z)isderived from aconsideration ofasolution of(1)intheform ofadescendingseries.Wenowproceedtoconstruct this solution andinvestigateitsproperties. 10*71.Theconstructionofthefunction S^„(z). Aparticular integraloftheequation 110"7(1),proceedingindescending powersofz,beginningwith z'^~^, is (1) t/=kz>^-' ^(^-ly-V^ ^{(f,-If- v-^]K^- 3)-^-v^ z" This series, however, doesnotconvergeunless itterminates;but ifitterminates, itisasolution of§10-7(1),and itwillbecalledkS^^^(z). The series terminates if/u,—z^isanoddpositive integer,oriffj.+visan oddpositive integer,and innoother case. Intheformer casewewrite/jl=v+2p+1,andthenwehave ^^^ ^to r(m+i)r{v+»i +i) =(-}p2^-'r{if^-iv +l)r(i/j, +iv+i)J.(2) +s^^,(z) ^ - - ,^/I 1 INcosA(u+I^)TTr/X ,\=-2>^-^r(i^-1.+1)r(lya+11.+1)l-^"^^^-L{z)+V^(^)- Whenfx—V=2p+\,thefunction vanishes, and so,when a-z/isanoddpositive integer, wehave X[cosl{^-v)7r. ./_^{z)—cos|(/i+i')tt .J^{z)]. Since both sides ofthisequationareeven functions ofv,theequationis true alsowhen/a+1^isanoddpositive integer,sothat itholds inallcases in whichS^^^(z) has,asyet,been defined. Weadoptitasthegeneraldefinition ofSf,^^(z), except that,when visaninteger,wehave tousetheequivalentform (3) S,^,(z)=s,^,(z)+2^-^r(lyL.-ii.4-1)r(Iac+|i/+i) X[sinh(/Jb-v)TT .J^(z)—cos^(fJb—v)TT .Y^(z)]. Itwillbeshewn in§1073 that^^^v{z)hasalimitwhenfji+vorfx-v isanoddnegative integer,i.e.whens^_p{z)isundefined;and so,ofLommel's twofunctionss^^^{z)andS^^^(z),itisfrequentlymore convenient tousethe latter. 348 THEORY OFBESSEL FFNCTIONS[CHAP. X Itwillappearin§10"75 thattheseries(1),bymeans ofwhichS^^„{z)is defined when either ofthenumbers/x+yisanoddpositive integer,isstill ofsignificance when thenumbersfx±varenotoddpositive integers.It yields,infact,anasymptotic expansionofS^^y{z)valid forlargevalues of thevariable z. 10*72. Recurrence formulae satisfied hyLommeVs functions. Itisevident from§10-7(2)that that istosay Again,itiseasytoverifythat d^[z"Sm,V{z)\={^l+v-\)z'' S^_i, ,_i{z), sothat (2) s'^ ^{z)+{vjz) s^^y(z)=(/x+v-1)s^_,, ^_i(z), andsimilarly (3) sV.„(z)-(viz) .9^,^(z)=(fl-V-l) S^i, ^+1(z). Onsubtractingandaddingthese results weobtain theformulae (4) {2v/z) s^^^(z)=(^+v-l) .9^_,, ,_,(z)-{fx-v-l) s^,, ,+1(z), (o) 2sV,. (^)= {fJ'+V-l) S^_,,^.i {z)+{fM-v- 1)S^_i.^+i (z). Thereader will find iteasytodeduce from§10"71(2)that thefunctions of thetype s^„(z) maybereplaced throughoutthese formulaebyfunctions of thetype >S^_^(z) ;sothat (6) S,^,,.(z)=z'^^^-{(^+iy- „^|,SV..(z), 0) S',,. (Z)+{viz) S,,,(Z)= {fji+V-1)>^^_,._, (z), (8) S'^^^z)-(vjz) S^^,(z)=(fM-v-l) 5f^,.,+, (z), (9) {2vlz)S,^,(z)^(fi+v-l)S^,,^_,{z)-(fi-v-l)S^.,^,^,{z), (10) 2S',,,(Z)=(,M+V-1) ^,_,,,., (z)+ifjL-V- 1)>^^_,.+x (Z). These formulae maybetransformed invarious waysbyusing (1)and(6).Theyare duetoLommel, Math. Ann. ix.(1876), pp.429—432,buthismethods ofproving them were notinallcasescompletely satisfactory. 10'73.LommeV sfunctions S^^y(z) whenfi±visanoddnegative integer. Theformula§10"71(2)assumes anundetermined formwhen/j,—vorfi+v isanoddnegative integer*. Wecaneasilydefine'S^„_2p_i,v(^)interms of 'S„_i,^(^) byarepeateduseof§10'72(6)whichgives m 9.,x "^^ (.yn,.-.p^.n (-)PS^_,,^{Z) (1) ^.-.p-^,Az)- _-^2.^^(_^^^^^ (^_^)^^^+ 2..^, (1_^)^• *SinceS^,v(z)isaneven function ofv,itissufficient toconsider thecase inwhich fi-v isan oddnegative integer. 10-72, 10-73] ASSOCIATED FUNCTIONS 349 Wenext defineS^^i„(z) bythelimitingform of§1072(6),namely (2) ^„_i,„{2)=lini ^^^_l 1_(/L6—y+1)(/U,+1/+1)j' Thenumerator (whichisananalyticfunction of/xnear/x=v—l)vanishes whenfx=v— 1,and so,byL'Hospital's theorem* S^_,^^(z)=^Z,VJfl IJ.=P—1 Now itiseasytoverifythat Also d.=.-1' ^\,Zo{m +iy.r(v+m +2) X(2logz+^|r{\)+y}r(v+l)-^|r {m+2)- yjr(v+m+2)}. ^{2'^+ir(l/.-l^+f)r(l^ +ii^+|)cosl(^ +z07r}y^i _M=i'-i and 'd_ d/jb andhence itfollows that=2"r(i.+1)sinvir{log2+|-f(1)+l^|r(v+l)+^7rcot vir}, {2'^+!r(l/x-ii.+I)rCl/i+ii;+1)cos1 (/^-^)77} (-r(i^)""jn=i'—12''-^7rr{v +l), (3) >Sf,_,,(^)=i^T(z.) S i=o^>i! r{v+'in+ 1) X{2\og hz- y}r{v+m+1)-f(m+1)}-2"--^ ttT(v)Y,(z), and thisformula, whichapjjearstobenugatory whenever visanegative integer, is,ineffect, nugatory onlywhen i'= ;forwhen v=—7i(where nis apositive integer) wedefine thefunctionbytheformula >S^_«_i,_n (z)='S-w-i, n(2), inwhich thefunction ontherightisdefinedbyequation §1073(1). Todiscuss thecase inwhich i'=0,wetakethefm-uiula V, (z)- whichgives 'S-i,o(2)= ,^,{s''^'-^%^2,0(^)} M=-l SinceS,,,,,{z)=^z--^{V {^+f)}-^2 ,/'^J, +2»+1{r(l/L.+i)}2{cos AyxTT.J'o(z)-Sini/xTT.Jo(--)), itfollows, onreduction, that *Cf.Bromwich, Theory ofInfinite Series, §152. 350 THEORY OFBESSEL FUNCTIONS [CHAP. X 10*74. Functionsexpi-essibleintermsofLomtneVs functions. From thedescendingseriesgivenin§10'7l(1)itisevident thatNeumann's poljmomial On{z)isexpressibleinterms ofLomrael's functions bytheequations (1) 0.,,n{Z)=(If2)S,^^n.(Z), 0^,^+1 {z)={(2//i+1)/^} /S„,,,n+x {z), and Schlafli'spolynomial Sn{z)issimilarly expressible bytheequations (2) S^(z)=4m>Sf_i, 2,„(z), S^,^^ (z)= 26'o,o,„+, (z). Itisalsopossibletoexpresstheimportant integrals Izi^J^{z)dz,Iz<^Y^{z)dz interms ofLommel's functions;thuswehave ^{0"J,(z).^1-"^^_i, ,_;(z)]=zJ,., {z)8^_,^ ,_i(z)+(^-v-l) zJ,{z)^^_2. ,{z), d^[z'-'/,_i {z).Z--S^^,(z)]=-zJ,(z)S^_,{z) +{fi+v-l) zJ,_, (z) ,Sf^_,, ,_i(z). Oneliminating /S^_j^_i(2:)from therightoftheseequations, andusing §10'72(6),wefindbyintegratingthat (8)['z^J,{z)dz={fi+v-l) zJ^(z)>SV_3. ._,(z)-zJ,^, (z) /Sf^.,(z), andproofsofthesame nature shew that (4)j%:^Y^iz)dz={,^+v-l)zY,(z)S,_,,,_,(z)-zY,.,(z)S^,^(z), and,moregenerally, (5)j^z'^K{z)dz={^+v-l)z^, (z)S^_, ,_,(z)-z%\^, (z)^V.(z). Specialcases ofthese formulae areobtainedbychoosing /u,and i>sothat thefunctions ontherightreduce toNeumann's orSchlafli'spolynomials, thus (6)fz^,,, (z)dz=z^ I-J^ ^%„ (z)0,,_, (z)-<^^,_, (z)0,„,(^)l, 0) r''^2,0+, (Z)dz=^Z {'(^,,ri+, (Z)S^„(Z)-%rn(z)^2m+i (z)]. Oftheseresults, (1), (3),(4)and(6)arecontained inLommel'spaper, Math. Ann. ix. (1876), pp.425—444; (6)and(7)were given byNielsen, Handbuch derTheorie der Cylinderfunktionen (Leipzig, 1904), p.100,buthisformuljie contain somemisprints. Itshould benoticed thatLommel's function, inthose easeswhen itis 10-74, 10-75] ASSOCIATED FUNCTIONS 351 expressibleinfinite terms, isequivalenttoGegenbauer's polynomialof§9"2. Theformulaeconnectingthefunctions are* / (^)2''r{v+m)V+2m lit i z A (X-2''+^r(^^+m +l)i/+2w+l^., Itfollows thatthemostgeneralcase inwhich theintegral (5)isexpressible interms ofelementaryfunctions andcylinderfunctions isgiven bytheformula (•' ) j^ (^.+.. (^)dz- ^.p(^^^,^^ L,.+2/M-1 V+2m Thefunction defined bytheseries (-)'"2''+2'»4:Z's )((.lor(i/+2??i+i) r(..+i)*''+7,i^^'* hasbeenstudied ingreat detail byW.H.Voung+ ;thisfunctionpossesses many properties analogoustothose ofBesselfunctions, buttheincrease ofsimplicityoverLommel's more general function seems insufficient tojustify anaccount ofthem here. Tiieintegralv j ,/ .-sdthasbeen studied (whenvisaninteger) byH.A.Wel)b, Messenger,xxxiii.(1904), p.58;andhestated that,when v=n,itsvalue is0„(z).This is incorrect(aswaspointed outbyKapteyn);andthevalue forgeneral values ofvisJ {>'^'uA-2)-^'%Ku{-Z)}Iz, when Ji(v)>and |arg{-z')\ <7r. 10'75. Theasymptotic expansion ofS^„(z). We shallnowshewbyBarnes'method§ that,when/x±varenotodd positive integers,thenS^^^{2) admits oftheasymptotic expansion when i^' jislargeand |argz\<7r. Letustaketheintegral _^p.-Phini-lfi +^v-sjTi^-lfM-^v-s) TTJ-Uy-'ds /Stti j_co/_^,+j r(^— ^/j,+^v)r(I— h/J'—^f)' sinsTT Thecontour istobedrawnbytaking ^jtobeanintegersolargethatthe onlypolesoftheintegrand onthe leftofthecontour arepolesofcosec sir,the polesoftheGamma functionsbeingontherightofthecontour. *Oegenbauer, WienerSilzungsberichte, lxxiv.(2),(1877), p.126. +Quarterly Journal, xliii.(1911), pp.161—177. ICf.Gubler, ZurichVierteljahrsschrift,xlvii.(1902), pp.422—428. §Proc. London Math. Soc.(2)v.(1907), pp.59—118;cf.§§6-o, 7-5,7-51.•^(;.-1)^-^.^[(f,-If- v^-\{(^- 3)-^-V- z"- z* 352 THEORY OFBESSEL FUNCTIONS [CHAP. X Theintegralisconvergent when [arg^ |<tt,and itmaybeseenwithout difficultythat itis(zf^'-P). Itmaybeshewn from theasymptotic expansionoftheGamma function thatthesameintegrand, whenintegratedround asemicircle, ofradiusRwith centre at—p—f,ontherightofthecontour, tends tozero fisR-^oo,provided thatRtends toinfinityinsuch amanner thatthesemicircle neverpasses through anyofthepolesoftheintegrand. Itfollows that theexpression givenabove isequaltothesum ofthe residues of ^'^' r(^-ifi +iv)r{i-ifi-iv)• sinsTT atthepointss 0,-1,-2,. ..,-(^-1), 1,2,3,..., 1 1,,_li, 3_1,, _1,, 5_1,, _!,. 2—2A* 2^'' 22/^2*'' ^2/* 2'^---) 22/^'2^> 22A''+2'> ^2^^2*^' When wecalculate these residues wefindthat _j^^1(-)'"r{l-ifi +^v+711)r(I-l/i-1 1;+m) ,_,^(-rr(i +ifjL-iv)T{i +^f.+iv)qz)^ r(|-|yu. +|i^)sini'7r^=oWi! T{l—v-\-m) 2>-^7rra+ia-i.)-^(-rd^r-- ^^ sothat sini/TT X[cosl(/jL-v)7r. /_,(z)-cosl(fi+i')'7r. J,(z)]=(z'^-'P), and so,by§1071(2),wehave theformula andthis isequivalenttotheasymptotic expansionstated in(1). 10-8] ASSOCIATED FUNCTIONS 353 10'8.Henii-cylindrical functions. Functions Sn{z) whichsatisfythesinglerecurrence formula (1) S„_, (^)-S,+, (^)=2S,/ (^) combined with (2) S,{z)=-S:{z) have been studied ingreatdetailbySonine*. Theywillbecalled hemi- cylindricalfunctions. Itisevident that S^i{z)isexpressibleintheform S,{z)=/„{D).So(z), whereD=d/dzandfn(D)isapolynomialinDofdegreen ;andthepolynomial fn(^)satisfies therecurrence formula combined with /o(^)=l,/(^)=-e Itfollows byinduction(cf.§9"14) that /n(^)=H{-^ +v(r+i)i"+{-i-v(r+i)}'a andtherefore (3) S„{z)=^[{-D +^{D-^+l)]-+{-D-,/{D-^ +l)\n].s,{z). Ifitissupposedthat(1)holds fornegativevalues ofn,itiseasytoseethat (4) S_n{z)={-rSn{z). Toobtain analternativeexpressionto(3),put^=sinht,andthenf J''^^'{-Qmhnt (wodd) '2l^4lr+ ••.{neven) "1 n^n{n'- 1-)^^ [-1,^-3;'r----- (^odd) Hence (5) S„(^)=a- n-(n^—2^) So{z)+^So"(z)+—^^ So'^{z)+.,.(neven)2|--"\-y41 PSo{z)~So{z}-....{nodd) Itistobenoticed thatOn{z),T,^(z)andEn(z) arehemi-cylindrical functions, butSn(z), Un{z)andH„{z)arenothemi-cylindrical functions. Itshould beremarked thatthesinglerecurrence formula S,_,(^) +X.+,(^)=^S.(^) givesrisetofunctions ofnogi-eaterintrinsic interest thanLommel'spolynomials. *Math. Ann. xvi.(1880), pp.1—0,71—80. tSeee.g.Hobson, Fhme Tiigonometry (1918), §264. w.B.P. 23 354 THEORY OFBESSEL FUNCTIONS [CHAP. X 10*81. TJieaddition theoremforhemi-cylindrical functions. Weshallnow establish Sonine'simportant expansion* (1) S„,(^+0=iJn{t)S,^-n{z); n=—00 theexpansionisvalidwhen z-\-tliesinside thelargest circle, whose centre is atthepoint z,which doesnotcontainanysingularityofthehemi-cylindrical function under consideration. Take ascontour acircleCwith centre zsuch that So(^)hasnosingularity inside oronthe circle. Then Ziri jc^—z—t The seriesconverges uniformlyonthecontour, andsowehave iTTl ,i=o JC =^.l.nJn{t)fni-^)\ ^-^d^ 2in ,j=o VdzjJc^-z "^ fd\=S€nJn{t)fn[-J-^j^m{z)dz =J^.,./.(«/„(-|)/.,.(j'^)s.(.)d.. But itiseasytoverifythat 2/n(-^)/« (^)=/r«-« (f)+(-)"/<«+« (f). sothat S^(0+O=^o(OSm(^)+ iJn{t)[S,n-n{t) +{-T^,n^n{t)], M=l whence Sonine's formula isobvious. Itshould benoticed that, ifS„{z)denotes afunction ofamoregeneral typethan ahemi-cylindrical function, namelyonewhichmerelysatisfies the equation withoutsatisfyingtheequation S-^{z)=—So'(^), westillhave fn(-^)S„,{Z)=S,n-n (z)+(-)"S„,+„ (z), andsotheformula(1)isstill valid.Wethushaveanalternativeproofof theformulae of§§5-3,91,934and 10-63. *Math. Ann. xvi.(1880), pp.4—8. SeealsoKonig, Math. Ann. v.(1872), pp.310—340;ibid. XVII.(1880), pp.85—86. 10-81, 10-82] ASSOCIATED FUNCTIONS 355 10*82.Nielsens functional equations. Thepairofsimultaneousequations |(1) ^.-1U)-^;+i (^)-2F; (2)=2/,{z)/z, 1(2) F^,, (z)+F^^, (z)-{2v/z) F^(z)=2g^{z)/z, where/„(^)andg^iz)aregiven arbitraryfunctions ofthevariables vand z, formanobviousgeneralisationofthepairoffunctionalequations whereby cylinderfunctions aredefined. IthasbeenshewnbyNielsen* that the functionsf^(z)andg^,{z)mustsatisfytherelation /;_! (z)+/,+! (z)-(2v/z)f, (z)-(/,_!(z)-g,+, (z)-2gJ{z) ; and ithasbeenproved fthat, ifthis relation issatisfied, thesystemcanbe reduced toapairofsoluble differenceequationsofthe first order. Forbrevitywrite f.{z)+g,(z)=a,{z), /,(z)-g^{z)=^,(z), andthegiven systemofequationsisequivalenttom (^+P)F,(z)=zF,^, (z)-a,{z), 1(4) (^-v)F^(z)=-zF,^, {z)-13,{z). Itisnowevident that (^^- v^-)F,{z)={'^~v) [zF,_, {z)-a,{z)] =-z'F, {z)-0^,_i (z)-(^-^)a.{z), sothat V,F, (z)=-z/3,.^ (z)-(^-v)a,(z). Again (^^- v"-)F,(z)=(^+.)[-zF,^, (z)-13,(z)] =-z-'F, (z)+za,^, (z)-(^+7.)^,(z). Wearethus ledtotheequation (5) V,F, (z)=z^, (z), where jiQ)znr,(z)=-z^,_, {z)-(:^-v) a,(z), 1(7) z^,(z)=+za,^, (z)-C^+v)/3,(z). On/Comparingthese values ofto^(^), weareatonce ledtoNielsen's condition (8) /,_! (z)+/;+! (z)-(2u/z )/:(z)=g,., (z)-g,+, (z)-'2gJ (z). Itnowhastobeshewn that Nielsen's condition issufficient fortheexist- ence ofasolution ofthegiven system.Toprove this,weassume(8)tohv *Ann. diMat. (3)vi.(1901), pp.51—59. tWatson, Messenger,xlviii.(1919), pp.49—53. 23—2 356 THEORY OFBESSEL FUNCTIONS [chap. X given, and, afterdefining ct^(z)by(6)and(7),wesolve(5)bythemethod ofvariation ofparameters.Thesolution is (9) F,(z)=J,(z)\c,-l7r Y,{t)^,(t)dt +Y,(z)\d,+iTT J,(0^,(0dt\, where aand harearbitrary constants; andCt,anddt,maybetaken tobe independentofz,though they will, ingeneral, dependonv. Itremains tobeshewn that c^and c?„canbechosen sothat thevalue of F,{z) given by(9)satisfies(1)and(2),or(what comes tothesamething) that itsatisfies (3)and(4).If(3)issatisfied, then zJy-i (^) ]c.-^TT IY,(t)m^(t)dt ]-^irzJy(z)Y^{z)^^{z) +ZF^_i {Z)\d^ -k-^TT{J^(t)-ST^(t)dt[+^TTZF„(Z)J^(Z) Tff^{z) zJ^_^ {z) j(_1,•v—\ 2'TTY,_,{t)7n^_,{t)dt\ that istosay, ^J"^_i {z) +^F,_i(2)+^F^-i {z) \d„-i+l-TT J,_i(0^^-1 (t)dt\-a^ {z), --1-l'^\\Y"(0^.(0-Y^_, (t)t^,_,(0} Jadt d^-(/,_!+Itt {J,(t)^At)- J,-, (t)CT,_i(Ojdt+ofp(z)=0. But itiseasytoverifythat d dz._,(z)/3,_i {z)-^,{z)«,{z)]=isr,{z)^,(z)-t^,_i (z)'^,_,{z\ since(6)and(7)aresatisfied; andso(3)issatisfied if zJ^_^ {z)\c„-c^_i-Itt F,_i {z)/S._, {z)-F,{z)a,(z) +^F^_i {z) \d^-fZ^_, 4-Itt /,_! {z)/3,_i (^)-J^{z)a,(z)+a,(z)=0, andthiscondition, by§3*63(12), reduces to zJ^_i (z){c^-c„_i+Itt [F^_i(a)/3^_i(a)-F^(a)a^(a)]} +^F,_, (^){o5,-d,_,-Itt[J,_, (b)/3._, (6)--/.(b)a,(b)]}=0. Consequently,sofaras(3)isconcerned, itissufiicient tochoose c„andd^ tosatisfythediiferenceequations |(10)c^-c^_i=-^TT {F^_i(a)yS^_i(a)-F^(a)«^(a)), i(ll) d,-d,_,=Itt{J,_, (6)/3,_, (6)-/,(6)«,(6)} ; 10-82] ASSOCIATED FUNCTIONS 357 andthereader willhave nodifficultyinverifying that, ifthese same two differenceequations (withvreplaced byi/+1throughout)aresatisfied, then thevalue ofF^(z)given by(9)isasolution of(4). These differenceequationsareofatypewhose solutions mayberegarded asknown*; and sothecondition(8)isasufficient, aswell asanecessary, condition fortheexistence ofasolution ofthegiven pairoffunctionalequa- tions(1)and(2). If,as ^^-^00 , f^(z)=0(z^-% g^{z)=0{zi-^), where S>0,thenwemaymake a-*-oo,h^~ zc,andwehave Cy=c„—i, cti,^rt„_i , sothatthegeneralsolution maybewritten (12) F,(z)=/,{z)L,(v)+Itt["F,(0^.(t)dtl +F,(Z)ItT, (l/)-ItT["J,(t)7Z,(t)dtl, where7ri(i/) and773(1/) arearbitrary periodicfunctions ofvwithperiod unity. Note. Someinteresting propertiesoffunctions whichsatisfy equation (2) 07ili/areto befound inNielsen's earlierpaper,A)in. diMat.(3)v.(1901), pp.17—31.Thus, from a setofformulae ofthetype F„_J{z)+F, ^,{z)- (2vlz) F(z)=2g,{z)lz, itiseasytodeduce that (13) F,,,,{z)=F,{z)R„,,{z)-F,_,{z)R,,.,,,^,{z) n-l +(2/2)2gv^-miz)Rn-m-hv^m +\{z); thefirsttwoterms ontherightarethecomplementaryfunction ofthedifference equation, andtheseries istheparticular integral. *Anaccount ofvarious memoirs dealing withsuchequationsisgiven byBarnes, Proc.London Math. Soc.(2)11.(1904), pp.438—469. CHAPTER XI ADDITION THEOEEMS 11*1. Thegeneralnature ofaddition theorems. Ithasbeenproved (§4"73) thatBessel functions arenotalgebraic functions, and itisfairlyobvious fromtheasymptotic expansionsobtained inChaptervii thattheyarenotsimply periodic functions, and,afortiori,thattheyarenot doubly periodicfunctions.Consequently,inaccordance with atheorem due toWeierstrass*, itisnotpossibletoexpress Jv{Z \-z)asanalgebraicfunction ofJv(Z)andJy(z).That istosay,that Bessel functions donotpossess addition theorems inthestrict sense oftheterm. There are,however, twoclasses offormulae which arecommonlydescribed asaddition theorems. Inthecase offunctions oforder zerothetwo classes coincide;andtheformula forfunctions ofthe firstkind is 00 Jo{'\/{Z'^+Z-—2Zzcos^)}=Se^nJm {Z)Jmi^)COSm(f>, 7)1= which hasalreadybeen indicated in§4"82. Thesimplest rigorous proofofthisformula, which isdue toNeumannf, dependsonatransformation ofParseval'sintegral;anotherproofisdue to Heine|,whoobtained theformula asaconfluent form oftheaddition theorem forLegendrefunctions. 11*2.Neumanns additiontheoi'eni^. Weshallnow establish theresult 00 (1) ^0M=S^mJ'm {Z)J,n{z)COS7/10, where, forbrevity, wewrite CT=sJiZ^+2^-2Zzcos0), and allthevariables aresupposedtohavegeneral complexvalues. *Thetheorem wasstated in§§1—3ofSchwarz' edition ofWeierstrass' lectures (Berlin, 1893) ;seePhragmen, ActaMath. vii.(1885), pp.33—42, andForsyth, Theory ofFunctions(1918), Ch.xniforproofs ofthetheorem. tTheorie derBesseVschen Functionen (Leipzig, 1867), pp.-lO—70. XHandbuch derKugelfunctionen,i.(Berlin, 1878), pp.340—343;cf.§5-71andModern Analysis, §15'7. §Inaddition toNeumann's treatise cited in§11-1, seeBeltrami, AttidellaR.Accad. diTorino, XVI.(1880—1881), pp.201—202. 11-1-11-3] ADDITION THEOREMS 359 Wetake theformula(Parse val'sintegral) If'' 1 /"'" ^TTJ-n ^TTJ_^ which isvalid forall(complex)values ofsranda,theintegrand beingaperiodic analyticfunction of6withperiod27r.Wenextsupposethat aisdefinedby theequations tssina=Z—2cos(f),-urcos a.=zsin<^, and itisthenapparentthat \[" . ,/„(ct)=X— Iexp [i(Z—^^cos^)sin^+i^sin cf)cos^]c?^ 2^S/,„(2) }H=-Xgj/it9+(zsin((^-6)f^0 ^TT )».=—: HI=-X X=t./„,(^)J..(^)e-'^ »l=—X theinterchangeoftheorder ofsummation andintegration followingfrom theuniformity ofconvergenceoftheseries, andthenextstepfollowingfrom theperiodicityoftheintegrand. Ifwegrouptheterms forwhich thevalues ofmdifferonlyinsign,we immediatelyobtain Neumann's formula. Thecorresponding formulae forBessel functions oforder±^were obtained byClebsch, Journal filrMath. LXi.(1863), pp.224—227,fouryearsbefore thepublicationofNeumann's formula; seeJ511'4. 11*3. Graf's generalisation ofNeumannsform'tda. Neumann's addition theorem hasbeenextended tofunctions ofarbitrary order vintwo differentways.Theextension which seems tobeofmore immediate importanceinphysical applicationsisdue toGraf*, whose formuJ^ is iZ—ze~^)^^°° (1) J^{^).\^^\=SJ,^,„(^)/,„(^)e-*, (^ze^) III=-00 andthisformula isvalidprovidedthatboth ofthenumbers\ze^''^\areless thanj^l. *Math. Aim. xliii.(1893), pp.142—144andVerhandlungen derSchiceiz. Natitrf. Gen. 1896, pp.59—61.Aspecial caseoftheresult hasalsobeenobtained byNielsen, Math. Ann. lii.(1899), p.241. 360 THEORY OFBESSEL FUNCTIONS [CHAP. XI Graf'sproofisbased onthetheoryofcontourintegration, but,twoyears after itwas pubhshed, anindependent proofwasgivenbyG.T.Walker, Messenger, xxv.(1896), pp.76— 80; thisproofisapplicabletofunctions ofintegralorderonly, and itmaybeobtained from Graf's proofbyreplacingthecontourintegrals bydefiniteintegrals. Toprovethegeneral formula, observe thattheseries ontherightin(1) isconvergentinthecircumstancespostulated,and so,ifargZ—a, wehave m=—'x =s-7 2exp\lZ(t--)\r"---'J,,(z)e-^'Ut ^T!"*w=-oo J-a)exp(-«a) (, \ f/J there isnospecial difficultyininterchangingtheorder ofsummation and integration*. Now write {Z-ze-''^)t=-uTU,{Z- ze''l')/t=^ju, where, asusual,-sr=\/{Z'^+z^—2Zzcos(f>), and itissupposed nowthat thatvalueofthesquareroot istaken which makes -07-^+Zwhen z^^0. For alladmissible values ofz,thephaseof•ay/Zisnowanacuteangle, positiveornegative.Thisdetermination of-nrrenders itpossibletotake the w-contour tostart fromandendat—ooexp(— 1/9),wherey3=argzsr. Wethenhave 2J.„ (Z)J,(.).»-=i-.l^^^)'r exph.(u- ')]^m=-oo 27rl V CT /J-ooexp{-ip) [ Vuj }v''+' (Z-ze-'^\^^ by§6-2(2);andthis isGraf's result. Ifwedefine theangle i/rbytheequations Z—zcos=OTcos-v/r,zsin(f>—nrsinyjr, whereyjr^^asz-^{sothat, forrealvalues ofthevariables, weobtain the relation indicatedbyFig. 28),then Graf's formula maybewritten (2)e"^^/.(tsr)=i/.+,.(^)/,.(^)e'»'*m=—00 and,onchangingthesignsof^andyjr,wehave (3) e-^'*M^)= tJ.^m{Z)Jm{z)e--^'\ Wl=—00 *Cf.Bromwicb, Theory ofInfinite Series, §176. 11-3] whence itfollows that (4)ADDITION THEOREMS 361 olll »»=—00 oilljn=—00 Fig.28. If,inthisformula, wechangethesignsofvand in,wereadilydeduce from 8'54that (5) andso (6)sin^,„-_«^ ^ ' 'sm"^ sm )»=—Qo sill Theformula(5)wasgiven byNeumann inhistreatise inthespecialcase ^=0;see alsoSommerfeld, Math. Ann. XLV.(1894), p.276; ibid. XLVii.(1896), p.356.Some physical applicationsoftheformulae areduetoSchwarzschild, Math. Ann. lv.(1902), pj).177—247. Ifwereplace Z,zandwin,theseequations byiZ,izand i-^respectively, itisapparentthat 0) (8)sin 1M=—rr, Olll)»=—00 00 bill iij=— cr !5J11)H=—X Oftheseresults, (7)wasstated byBeltrami, Atti della R.Accad. diTorino,xvi.(1880— 1881), pp.201—202. Thefollowing special results, obtainedbytaking (^=|7r,should benoticed : (9) (10)9^,{m)(,o%v^= S{-y^9^.+,,n{Z)J,„^{^\ '^,(B7)sini;r/r= S{-rK+,m+i{Z) J,m+, {z), ?Ji=—X where Z=^cos•<^,z=-stsinyfrand\z\<\Z\. Forthephysical interpretationofthese formulae thereader isreferred to thepapers byG.T.Walker andSchwarzschild;itshould beobserved that, in thespecialcase inwhich visanintegerandtheonlyfunctions involved are ofthe firstkind, theinequalities|2^e***\<\Z\need notbeinforce. 362 THEORY OFBESSEL FUNCTIONS [CHAP. XI 11'4.Gegenhauer^'saddition theorem. ThesecondtypeofgeneralisationofNeumann's addition theorem was obtainedbyGegenbaiier* nearly twenty yearsbefore thepublicationof Graf'spaper. IfNeumann's formula of§111 isdifferentiated ntimes withrespectto cos<^,wefindthat Jn(«r)"Jm+n {Z)J,n+n (^)d""COS(m+?i). (1) -^^=2 e. tsr'^ ,>,'Zo'^^'' Z"" z'' d(cos <f>y' Thisformula wasextendedbyGegenbauertofunctions ofnon-integralorder bymeans ofthetheoryofpartialdifferentialequations (see §ir42); but Soninef gaveaproof byadirect transformation ofseries, and thisproofwe shallnowreproduce;itistobenoted that, in(1),^isnotrestricted (asin §11'3)with reference toZ. WetakeLommel'sexpansionof§5'22,namely j.wa+h)] ^^(-hh)pj.^A^) andreplace fandhbyZ'^+z^and—2Zzcos(}) respectively;ifwewriteO inplaceofJ^(otj/ct"forbrevity,itisfound that -(Zzcos<t>)P J.+pW(Z^+2') } ^« I(-yzP+^COSP<f> J^p^giZ) ^ro,to 2^.p\qlZ'^^' byafurtherapplicationofLommel'sexpansionwith ^andhreplaced byZ^ and2l But,by§5-21, J,+p+g{Z)_9qlv+p+2k r(p+p+k)T^- {:,k\{q-k)\29 Tiv+p-^q+k +iy"^^^'^^^^' andso ^_^^4(-)g {v+v^2k)r(v+P+k)zP^'i cos^J,+j,+^Zl ^ro^ro^ro 2'^plkl{q-k)ir{,>-\-p +q+k+l)Z" thetripleseries ontheright being absolutely convergent, bycomparison with 00 00qi S2S /)=0g-0 A:-0r(r+p+A;)zP^-^'iZ'^^^ 2P+-2q+2kj)i/.\^^_ ^.)!r(j,+p^2k)r{v+p +q+k+i) *Wiener Sitzungsberichte, lxx. (2),(1875), pp.6—16. tMath. Ann. xvi.(1880), pp.22—23. 11-4, 11-41] ADDITION THEOREMS 363 But, foranabsolutely convergent series, q=k=i) k=M=() andso *^I(-)fe+» (jj+p+2k)r(v +p-\-k)zP+^+'^n cQs^^J^^p+,k {Z)- ,;:,,ro„=o2^*+'^«p\k\n\T{v +p+^k+n+l) Z" ^^^(-)^2-^^(^+p+2^-)r(i^+/J+A;)cos^ </>J.^^^{Z) J^,_;+,^) ^I:^(-)fc2-+"'--^-(t^+m)V{v+m-k)cos^'^-^'^ffi /,+m(Z)J^+M -fc=ow =2fr (m-2k)lkl Z" z" »<hn(_)A;2>'+>n-2A; (j,+^^)p(V+??i- /i;)COS"*-"*^</)J,+,„,(Z) ./,+,^ (^) m=0^-=0 (m-2^')!yt! Zr z" (2)^^^=2"r(.)2(.^m)-'^^^" ^±'^C,,"(cos <^),„ <J.'"(-)^-2^»--^r(i/+?/i-A;)cos'"--*^d) ^,Now 2^^' ^\i.,i,J..—^=C,/(cos0), where, asin§3"32,C^"(cos <^)denotes thecoefficient of«"*intheexpansion of(1—2acos<p+ct")"" inascending powersofo.Wehave therefore obtained theexpansion which isvalid forallvalues ofZ,z,and0,and for allvalues ofvwith the exceptionof0,—1,—2,— Inthespecialcase inwhich i/=|,wehave (3)?i2."=^I(»+1)^-^^-.'-'^P,„.(cos ,^). This formula isdue toClebsch, JournalftlrMath. Lxi. (1863), p.227; itisalso given byHeine, JournalfilrMath. LXix. (1868), p.133,andNeumann, Leipziger Berichte., 1886, pp.75—82. Theformula inwhich 2i/isapositive integer hasbeen obtainedby Hobson, Proc.London Math. Soc.xxv. (1894), pp.60—61,from aconsideration ofsolutions ofLaplace's equationforspaceof2v+2dimensions. Anextension oftheexpansion (2)hasbeen given byWendt, Monatshefte furMath,xmd Phys.XI.(1900), pp.125—131;theeftect ofhergeneralisationistoexpress icr-''-Psin2P0j'^, +p(rar) asaseries ofBessel functions inwhich the coefficients aresomewhat complicated determinants. 11*41. Themodified form ofGegenbauer'saddition theorem. Theformula (1)—4^^=2"r(.)2(-)- (.+m)-^^^^^^ ^-^^a/(cos <^) maybeestablished inthesamemanner astheGegenbauer-Sonineformula of §11*4. Thisformula does notseem tohavebeengiven previously explicitly, 364 THEORY OFBESSEL FUNCTIONS [CHAP. XI thoughitisusedimplicitlyinobtaining some oftheresultsgiven subsequently inthissection. Unlike theformulae of§11"4, theformula istrueonlywhenl^'lisso small thatboththeinequalities |ze^^^\<\^\aresatisfied; but,inprovingthe formula, itisconvenient first tosupposethatthefurtherinequalities \2Zzcos(f>\<\Z^ +z''\,\z\<\Z\ aresatisfied. Wethen useLommel'sexpansionof§5'22(2)intheform i)=0P! which isvalidwhen |A |< |^|. Itisthenfound bymaking slightalterations intheanalysisof111"4that _^^I(-)y+i {v^-2i+2k)V{-V-p-q-k)zP+-'9 cos^ <f>J-,-p-^ {Z) "3oA/t=o 2-^'iplk\(q-k)ir{l-v-p-k)Z" ^^^Z(-)P+' (i^+P+ '2k)r(-v-p-2k- n)^P-H*+^" cos^(t>J-,.p-,k jZ)- .;i,C<>X 2^^-^-p\k\n\Y{\-v-p-k)Z^ _IIi-y^^ 2"^{v+p+2k)T{v +p+k)cosy^J_,_p_^(Z) J,+p+,k (z) piok% p\k\ Z^ z^ I<4»^(-)m-fc2''+"'-2^-{v+m)r(i/+m-k)cos"'"^^ </>J-^-m (Z)J^+m (z) ~,.=oA=o {m-2k)lk\ Z" z" =2-'r(.)i(-r (^+m)^-7/^^%^G^"(COS ,/,), SOtherequiredresult isestablished under theconditions \2Zzcos<f>\<\Z' +z^\,\z\<\Z\. Now thelastexpressionisananalyticfunction ofzwhen zliesinside the circle ofconvergenceoftheseries* I(y+7n)Z-'"-"^ z'^Cm''(cos (b) ,„=or(l-v-m)T{\-\-v^-m)' and this circle isthecircle ofconvergenceoftheseries X (^Ij0^"(cos <^). Hence thegivenseriesconvergesandrepresentsananalyticfunction ofz provided onlythat j2re*»* j< |^ |;and,when thispairofinequalitiesissatisfied, J-^ (ra-)y''5r''isalsoananalyticfunction ofz. *Cf.§5-22. 11-41] ADDITION THEOREMS 365 Hence, bythetheoryofanalytic continuation, (1)isvalidthroughthe whole ofthedomain ofvalues ofzforwhich I^e*** I< i^ I• Ifin(1)wereplaceyby—i^wefindthat Again,ifwecombine(1)with§ir4(2),weseethat, forthedomain of values ofznowunder consideration, (3) ^^-P=2"r(.)2(.+m)%^>%MC,"(cos cf>), and so,generally. (4)^)=2^r(.) i(.+^)%i^)^%^c',/(cos<^). Ifin(8)wemake v^-0andusetheformulae Co"(cos0)=1,lim[r{v)(v+m)C,,,"(cos0)}=2cosmcj), (mi^0) wefindthat 00 (5) Fo(t3-)= S€,nY,n{Z) J,n.(z) cosm(f).m=0 Theformulae(1)and(2)have notbeengiven previously; but(3)isduetoGegenbauer, and(5)wasgiven byNeumann inhistreatise(save thatthefunctions ¥,„wererej^laced bythefunctionsFt"*)). Theformula(3)with vequaltoanintegerhasalsobeenexamined byHeine, Handhuch derKugelfunctionen,i.(Berlin, 1878), pp.463—464.Somedevelop- ments of(4)areduetoIgnatowsky, Archiv derMath, undPhys. (3)xviir.(1911), pp.322— 327. Ifwereplace Z,zand otbyiZ,izandmintheformulae of§11'4and thissection wefindthat (6)^-#=2^r(.)S(-)'" (.+m)%^^%^^ C^"(cos </>), tJT-,„=o Z" Z- T (7) J^_-Ap=2''r{v) i(-)-(.+,n)^^^^>%^6;/(cosc/,), (8)^>=2^r(.)i(.+m)^^^^^^C.,r(cos <^). Ofthese formulae, (8)isduetoMacdonald, Proc.London Math. Soc. xxxii. (1900), pp.156—157; while(6)and(7)were givenbyNeumann inthespecialease v=h 366 THEORY OFBESSEL FUNCTIONS[CHAP. XI Theformulae of§11"4andofthissection areofspecial physical importance inthecase v=^.Ifwechangethenotation bywriting kxt,krand forZ. 2and (bweseethattheformulae become sinkvX/^+a-—2arcos6) .^^V(/-'+fl"'-2arcos^) cos Z;v'(?^+«"-2arcos 6?) ^^^ V(r^+a--2ar cos (9) fi.=0 V« V'* exp{—AVC?*^+0-—2orcos^)} v(^+a^—2arcos^) =2(2,„+l)^^iii±iA^^'Ii±i<^)p,„(oos«). 171=0 \« \r These formulae areofimportanceinproblemsinwhichpulsations emanate from a point ontheaxisofharmonics atdistance afrom theorigin,inpresenceofasphere whose centre isattheorigin. Cf.Carslaw, JJatk. Ann. lxxv. (1914), p.141 etseq. Thefollowing specialcases of(4)werepointedoutbyGegenbauer,and areworthrecording: If <f)=TT,wehave If=|7r,wehave UZ=z.6=0, and "if^istaken tobe/„, aformulaalreadyobtained(§o'o)byadifferent method; inthisconnexion thereader should consultGegenbauer,Wienei'Sitzungsherichte, lxxv.(2), (1877), p.221. Moregenerally, taking Z=z, ^=^0,'fe'Y=/»,,wehave Gegenbauer,loc. cit.givesalsospecialcases ofthisformula, obtained bytaking (^=^TT.(b=TT. 11-42]ADDITION THEOREMS 367 Again,itcanbeshewn that*, ifB{v)>—^, Isin-"(^Cm"(cos (/))0/(cos <^)c^0<_itT{'Iv-\-m) _0 [~22''-J(^ +m).m!{r(i/)py'^^'V) and so,providedthatR{v)>—\, ^ .'o (^^+2:2-2^^008 0)*"^^ V2/V2/^^ ^z, and,moregenerally, /-,Kx T"'^".'fV(-^'+^'-2^^cos4>)]^, ,X• .,.^.^ ='^r(2>/ +7H)'^Um(^)>/.+., (^) 2"-^m!r(i/) Z" 2" Asimple proofofthisformulaf,inthespecial case inwhichm=andthecylinder functions arefunctions ofthe first kind, wasgiven bySonine, Math. Ann. xvx.(1880), p.37,Another directproofforfunctions ofthe firstkind isduetoKluyver, Proc. Section ofScL,/{.Acad, van Wet. teAmsterdam, xi.(1909), pp.749—^755.Anindirectproof, dependingon§12'13(l't,isduetoGegenbauer, WienerSitzungsherichte, Lxxxv.(2),(1882), pp.491—502. [Note. Aninteresting consequenceof(4),which wasnoticed byGegenbauer, Wiener Sitzuiiffsberickte,Lxxiv.(2),(1877), p.127. isthat,ifjse***\<\^\ throughout thecontour ofintegration, then fcf.§9'2) (18)— -—-^ A,„,(z)&=2"r(.) (.+m)^ g,/(cos cj,). Specialcases ofthisformula, resembling theresults of§9*2,areobtainable bytaking equaltoortt.] 1142. O'egenbauer's investigation oftheaddition theorem. Themethod usedbyGegenbauer, WienerSitzungsberichte, LXX.(2),(1875), pp.6—16, toobtain theaddition theorem of§11"4 isnotquitesoeasytojustifyasSonine's transformation. Itconsists inprovingthatQisasolution ofthej)artial differential equation d^a. 2i/+l 3q 1d2j22;/cot(^9Q dz^ z dz z^d<p^z^d(f)~ ' andassuming that i2canbeexpandedintheform 0=1^,„.C,/(cos</)),m=0 whpre-iJj,!isindependent of0,andC'„/(cos(^)isapolynomialofdegreemincos</);it follows that {|,+ 2.cotc^l}c,/(cos0) *Gegenbauer, Wiener Sitzungsberichte, lxx.(2),(1875), pp.433—443,andBateman, Proc. London Math. Soc.(2)iv.(1906), p.472;cf.alsoBarnes, Quarterly Journal, xxxix.(1908), p.189; Modern Anabjsi.% §15-51 and Proc-. London. Math. Soc.(2)xvii. (1919), pp.241—246. tFormula(16)hasbeengiveninthespecial casei/^ObyHeaviside, Electromagnetic Theory,in. (London, 1912), p.267, inasomewhat disguised form. 368 THEORY OFBESSEL FUNCTIONS[CHAP. XI isaconstantmultipleofC^(cos^),andsoC^"(cos0)maybetaken tobethecoefficient ofa'"intheexpansionof(1—2acos+a^)~ ".Andthen5^,quafunction of2,satisfies the diflferential equation sothati?misamultipleofz""./^^™ (2),theother solution ofthis differential equation not being analytic neartheorigin. From considerations ofsymmetry Gegenbauer inferred that5„i,quafunction ofZ,is amultipleolZ~'"Jy^yf^{Z\sothat i2=2Om rr;——VrrCK^O&(p),m=0•^ -^ where6,„isafunction ofvandmonly ;andh^isdetermined bycomparingcoefficients of z^Z"^cos'^(f)inQandintheexpression ontheright. Asimilarprocess wasusedbyGegenbauertoestablish §11-41(3),buttheanalysis seems lessconvincing than inthecase offunctions ofthefirstkind. 11'5.Thedegenerate form oftheaddition theorem. Theformula (1)e^«>«*= (£)*^i^(2/1+1)t"Jn^ {Z)Pn(COS C^) wasdiscovered byBauer* asearlyas-1859;itwasgeneralised byGegenbauerf, whoobtained theexpansion (2)g«cos.*=2''r(i;) i{v+m)i^^"-^"^y^ Cj'(cos <^);m=0-2^" Bauer's result isobviouslythespecialcase of"thisexpansioninwhich t"=j. Inthelimitwhen v-^0, theexpansionbecomes thefundamentalexpansion of§2-1. Gegenbauer'sexpansionisdeducible from theexpansionof§11'41(4)by multiplying byZ""^*andmaking Z-*00;itisthenapparentfrom§11"41(9) and(10)that thephysical interpretationoftheexpansionisthat itgives theeffect due toatrain ofplanewaves comingfrominfinityontheaxis of harmonics inaform suitable forthediscussion ofthedisturbanceproduced bytheintroduction ofaspherewith centre attheorigin. Asimple analytical proofoftheexpansionconsists inexpandingz"e'^''^°^^ inpowersofzandsubstitutingforeachpowertheseries ofBessel functions supplied bytheformula of§5'2;wethus findthat „.^i"-cos" (b^cos1^__> z.2"'^nVgJZCOSC w=o w! _*t"cos" (^*2"+'*i^v+n+2^').r(z/+/I+^')r /X —-— j^ r-jJv-it-n-\-'ik\Z)- *JournalfilrMath. lvi.(1859), pp.104, 106. tWienerSitzuvgsberichte, lxviii.(2),(1874), pp.355—367; Lxxiv.(2),(1877), p.128;and Lxxv.(2),(1877), pp.904—905. 11-5] ADDITION THEOREMS 369 Ifwerearrangetherepeatedseries bywritingn=m—Ik,wededuce that w=oA-=o fci{m—Zh;)l =2"r(v)I(i^+»i) t'"-/,+„, (^)C',/(cos </)),m=0 andthis isGegenbauer'sresult. Modified forms ofthisexpansion,alsodue toGegenbauer, are _(3) gzcos*^2"r(v)I(i.+m)^-^^±1^ Cm"(cos </>),m= -^ (4)e-~^cos<^ 3=2"r{v)%(-r•('^+"0^-^^^^ G^n"(cos </>), (5) cos(^cos<^)=2''r(,0 5(-y^.{v +2m)'^^^±^!^C''^(cos(f>),m=0• •^ 'Ji'+2m+i \^) (6)sin(^cos</))=2''r(i;) S(-)'».(i.+2/h+1)"^^ ,^''' C'',„,+, (cos0),m=0•^ (7) 1=2" :£(i^+2m)m=o^ ^?i! (8)/;.>c»*,..>(e„s,).„=.,c/,= ?:i>^<^|p^i»%i!i. The last isageneralisationofPoisson'sintegral,which wasobtainedbya different method in§3'32. Itisvalidonlywhen R{i')> —^. These formulae aretobefound onpp.363—365ofthefirst ofGegenbauer's memoirs towhich reference hasjustbeenmade. Equation (1)wasobtained byHobson,Proc.London Math. Soc.xxv. (1894), p.59,bya consideration ofsohitions ofLaplace's equationinspaceof2i/+2dimensions,2i/+2being aninteger. Amoregeneralsetofformulae maybederived from(2)byreplacing cos (f)bycos (f)cos</>'-fsincf)sin(f)'cos-v/r,multiplying bysin^""^^jr,and inte- gratingwithrespecttoyjr.Theintegral* r C'^i"(cos (^cos </)'+sin<^sin cf)'cos-yjr)sin'-""^-»/rfZ-v/r ^=^"rir'i^l'^^'^-''(^^« '^)^™''(c°^ ^')> which isvalidwhenR(i')>0,shews that exp[iz(cos (^cos^'+sin^sin </>'cos-v/r)]sin^""^ -i/rrf\^ =2-- {r(.)}^2'71^9^Itr^'%^^'«"'^"^' '^>^'"'' ^"^^^ '^'^^ *Cf.Gegenbauer, Wiener Sitzungsbertchte, lxx.(2),(1874), p.433; cii.(2a), (1893), p.942. W.B.F. 24 370 THEORY OFBESSEL FUNCTIONS [CHAP. XI andso t/„_j {zsin^sin</>') {zsin<f)sin0')",„ , t/„_i12^Hill (DSillro )r., ,,, Theintegralused intheproof converges onlywhenK{v)>0,butthefinal result istrue forallvalues ofv,byanalyticcontinuation. This result wasgiven byBauer, MunchenerSitzungsheriehte,v.(1875), p.263inthe case v=\;thegeneral formula isduetoGegenbauer, Monatshefte furMath, tmdPhys.x. (1899), pp.189—192; seealsoBateman, Messenger,xxxiii.(1904), p.182andaletter from GegenbauertoKapteyn, Proc. Section ofSci.,K.Acad, vanWet. teAmstcrdam,lY. (1902) pp.584—588. Interesting specialcases oftheformula areobtained bytaking </>'equalto<^orto^ ; and,ifweput^'equalto^tt,multiply bye^^^°^^ sin^"^andintegrate, wefindthat (10) "TTi {''Jv-\(^sin0)e'^cos.^ sin'^"(^rfc/) z-J =2"/f27r) i (V»^^"^"*""'^•^^"^^"^^ '^'' •"^"^^^^ '^'^^^"^^^^ sothat theexpression onthe left isasymmetric function ofzandZ;thisformula also wasgiven byBauer inthecase v=\. 11'6.Bateman sexpansion. Weshallnow establish thegeneral expansion (1)^0J"^(2^008cos<J>)«/^ (2^sin^sin<I>) a)=CDS'"<^cos*^^sin"<^sin"^2(-)" (/i+y+2;?+1)/^+^+27i+i (^) ^ .!r(;.4-. +l){r(.+ l)r^.F.(-.,^ +.+n4-l;. +l;sm-c/>) Xo^i{-n,ix-^v +n+l\ 2'+1 ;sin-<l>), which isvalid forallvalues offxand vwith theexceptionofnegative integral values. Some oftheresults of§11*5arespecialcases ofthisexpansion^which was discovered byBateman* from aconsideration ofthetwotypesofnormal solutions ofthegeneralised equationofwave motions examined in§4'84. Weproceedtogiveaproofoftheexpansion byadirect transformation. *Messenger,xxxiii.(1904), pp.182—188;Proc.London Math. Soc.(2)in.(1905), pp.111—123. 11-6] ADDITION THEOREMS 371 Itiseasytodeduce from theexpansion (§5"21) ofaBessel function asa series ofBessel functions that ^zJ^(zcos(f)cos<J>)J^(zsin(/>sin^) «(-)"»(i^y+2"»+i(cos<f)Cos<I>y+-'" r/ J^^=i^—^'^\^,^^^—/,(^sm(/>sin4>) (-)"»cos-"*(^cos^'"<I>=cos'"(f)cos'" 4>sin"^sin" <J>S ?rt=0 X7/i !r(/A+/?i+1) H-ti'+2m+2n+l (^) .3^J'X2^1(- ??,yu,+ z^+2»;,+/^+1;z^+1 ;sin- sin-<35) (J cos'"(f)COS'" <l>sin"4>sin" *1>S )i='0(/i+ Z^+272-+1)J^+^+sn+i (^) »f(-)"^COS-'" (^cos^'»^.r(fi +P+n+m+l)X' ^(j'^o Im!(?i-7?t)!r(y +l)r(yu.+w+1) X.ii^i(m—n,/x+v +711+w-I-1;z--+1;sin'-^cf)sin'^<5») "(=cos*" (f)COS*"Osin" cf)sin" <J>S H=(»yn,+ z^+ -2/;+l)r(;u,+z/+n+l)^.. 7i!i(ii+ 1)i(z'+ 1) X^i(-)i,/M +v+n+l; fju+1,v+1;cos-</>cos-<&,sin^<^sin-<l>) whereJ[p4denotes thefourthtypeofAppell's* hypergeometricfunctions of twovariables, defined bytheequation Wenowhave totransfornif Appell'sfunction intoaproductofhyper- geometricfunctions inorder toobtainequation (1);ineffectingthetrans- formation weassume thatR(/x)>0,though obviouslythis restriction may ultimatelyberemovedbyusingthetheoryofanalyticcontinuation. Thetransformation isaconsequenceofthefollowing analysis,inwhich series arerearranged, andafreeuse ismade ofVandermonde's theorem : cos-'" *i> .jfi{— n, fjb+p+n+I;/j,+l,v+l]cos-cf)cos- 4>,sin- cj)sin-<P) ^IY(->')r+s{f^+v +n+lUsJ{-y sin''-^''4>2(-rsin^'+^"0 ^=0»<=o rl{v+ l)r t=otl(s- ty.„to ulif^+1%-u *Comptes Rendus, xc.(1880), pp.296, 731. tThis transformation liasnotbeen previouslynoticed toexist exceptinthespecial ease in which 4>=0,seeAppell, Journal deMath.(3)x.(1884), pp.407—428;some associated researches areduetoTisserand, Annales {Mimoires) deI'Observatoire(Paris),xviii.(1885), mem. C. 24—2 372 THEORY OFBESSEL FUNCTIONS[CHAP. XI ~^ro *=o^-r .«=,•^!{v+l)r(t-r)\(r+s-ty.{u- r)!(/a+\)r+B-^, ~ <=«=r=*=*^-.r * !(y+l)r{t-r)\{r+S-t)\{u-r)\{^l+ l)r+s-u ~ ^=0M=or=o r!(i/+1),.(^-r)\{u-r)\(^+1),,_« n 00^22(-«),(/^+..+«+1),^i^,.(.+.,+1)(-^)„_„^._^,„^ = {-Y^~^^^.,F,{-n,fi+v-\-n+\: v+\-sin=c/,) X2^1(-fi-n,v-vn-\-\; v+l\ sin-^) X2^1(- /I,/i+1/+n+1;t-+1;sin-4)). Hence weatonce obtain theresult ^zJ^{z cos COS<I>) cTj,{zsin(/>sin$) =cos'^4>cos'^^sm^ sm" <i>J^ n!T(/x+1)T(.+1)''^^^^^> x(-r^^^~j|"..i^i(-n,A*4-i/+^^+l;r+1;sin^<^) X2^1(- ??,/A+2^+w+1;j;+1;sin^$), fromwhich Bateman's form oftheexpansionisevident. CHAPTER XII DEFINITE INTEGRALS 12'1. Varioustypes ofdefinite integrals. Inthischapter weshallinvestigatevarious definiteintegralswhich contain either Bessel functions orfunctions ofasimilar character under theintegral sign,andwhich have finite limits. Themethods bywhich theintegralsare evaluated ai-e,forthemostpart,ofanobvious character; theonlynovel feature isthefairly systematicuseofamethod bywhich adoubleintegralisregarded asasurfaceintegralover aportionofaspherereferred tooneorother of twosystemsofpolarcoordinates. Themostinteresting integralsarethose discussed in§§12-2—12-21, which areduetoKapteynandBateman. These integrals,fornoveryobvious reason, seem tobeofamuch more recondite character than theotherintegralsdiscussed inthischapter;their realsig- nificance hasbecomeapparentfrom therecent workbyHardydescribed in §12-22. Thenumerous andimportant typesofintegrals,inwhich theupper limit ofintegrationisinfinite, aredeferred toChapterXlil. Thereader mayherebereminded oftheveryimportant integral,due to Sonine andGegenbauer,which hasalreadybeen established in§11*41, namely r^^.{V(^^ +.--2Z.coscA)}..,,,.x,i^.. .^. ^7r^(2^/ +m)'^.+,„ (^)/.+,» (^) "l^-KmlViy)Z" z" 12'11.Sonine'sfirst finite integral. Theformula (1) /,+.+, (z)=2TC+l) /r^^^''^"^^^'^^"^' ^°^'""'' ^^^' which isvalidwhen bothi^(/i) andR(v) exceed -1,expresses anyBessel function interms ofanintegral involvingaBessel function oflower order. Theformula wasstated inaslightlydifferent formbySonine*, Rutgers-f and Schafheitlin:[:, and itmaybeproved quite simply byexpandingtheinte- *Math. Ann. xvi.(1880), p.36;seealsoGegenbauer, Wiener Sitzungsberichte,lxxsviii. ("2), (1884), p.979. tNieinv Archief voor IViskunde, (2)vi.(1905), p.370. JDieTheoria derBesseV sclicn Funktioncn (Leipzig, 1908), p.31.Schafheitliu seems tohave beenunaware ofprevious researches onwhat hedescribes asanew integral. 374 THEORY OFBESSEL FUNCTIONS [CHAP.XII grandinpowersofzandintegratingterm-by-term,thus f'/^{zsin6)sin'^+^ ^0082"+! 6'c;^ 'Jo U2'^+''+-'«mir(/Lt +m-M)r (2/+1)2''r(i^+i) 00 /\m/i^V+''+2ni+i ^=0m!r(yu.-F2/-fwH-2)' andthetruth oftheformula isobvious. Itwillbeobserved thattheeffect ofthefactor sin'^+i 6intheintegrand istoeliminate thefactorsT(/u,+ni+l)inthedenominators. Ifwehadtaken sin'"*^ 6asthefactor, weshould haveremoved thefactorsm I.Hence, when R{v)>—1andfiisunrestricted, wehave (2) f*V^ (zsin6)sin^-'^ ^cos2''+^ Ode=^^^^^^j i>^ z .^---^^r(/.)- Inparticular, bytakingv=—\,wehave (3)(-)fV^(2sind)sin>-^ ^^61=H^_j (2). Aformula* which iseasilyobtained from (1)is ^ (4) J^{zsin^)/,{zcos^)tan'^+1ddd=Vfi ,1 .i!-^^(^X whenR(v)>R(/jl)>—1.Thismaybeproved byexpanding /^(zcos^)and integrating term-by-term,andfinally makinguseofLommel'sexpansion givenin§5"21. Thefunctional equation, obtained from(1)bysubstituting functions tobedetermined, F^andF^^^ +i,inplaceoftheBessel functions, hasbeenexamined bySonine, Math.Ann. LIX. (1904), pp.529—552. Somespecialcases oftheformulae ofthis section have been given byBeltrami, Istituto Lomhardo Rendiconti, (2)xiii. (1880), p.331,andRayleigh,Phil.Mag. (5)xil.(1881), p.92.{Scientific Papers,I.(1899), p.528.] Itwillbeobvious tothereader thatPoisson'sintegralisthespecialcaseof(1)obtained bytaking /x=-^. Forsomedeveloimientsoftheformulae ofthis section, thereader should consult two papers byRutgers,NieuivArchiefvoorWishmde, (2)vi.(1905), pp.368—373;(2)vii.(1907), pp.88—90. 12"12.Thegeometrical proof ofSonine' sfirst integral. Aninstructiveproofoftheformula oftheprecedingsection dependson thedevice(explainedin§3'33)ofintegratingover aportionofthesurface of aunitspherewith various axes ofpolarcoordinates. If(I,m,n)arethedirection cosines ofthelinejoiningthecentre ofthe *Due toEutgers, Nieino Archief voorWiskunde, (2)vii.(1907), p.175. 12'12]DEFINITE INTEGRALS 375 spheretoanelement ofsurface dcowhoselongitudeandco-latitude are^and 0,itisevident fromanapplicationofPoisson'sintegralthat r(fi+l)r (i)(1zy+' f" J"^(^sinO)sln'^+i dcos^-'+i Odd J =(|2X+''+' I"IV^s'°<'cos0gin2f^+i^cos2''+i^sin2"<^rf(^()f^Jo.'o =(1^)"+"+'11e^^hn"!^ 71-"+' d(o ={lzY+''+'11 e''^» l'^7)1'"+' dco =(^zy*''+' jiV^cose gin-'^+2.<+2 (,og2^^sin^-'+i</)(/(/>c^^ Jo 2r(/.+i.+f) Jo andthetruth ofSonine's formula isobvious. Anintegral involvingtwoBessel functions which canbeevaluated bythe same device* is I^ J,(zsin26)/,(zcos26)sin^-'+i (9cos^-'+i 6dO, J inwhich, tosecureconvergence, Riy)>—I- Ifwewrite w^=sin^6^+cos^^-2sin-^ cos^^ cos<^=1-sin^26cos^(j>, anduse111*41(16),weseethattheintegralisequalto ^_1M!_^ [*''I"'lA^sin^^+i 6cos'"-^' esin^"6debdO (Izyf'^ri'^ J^{^v/(l-sin2^cos"(f))|. .^,/, .o,7^7^= 2^-^^r(:+i)r(l) JoJo- (l-sm-^^cos-^)^-'^"""^"^^"--^^^^-^-^^^ 2--r(. +i)r(i)jj,^o,.^o (1-^^)*" J,(2sin6)sin-'+i ^cos-" (/>cos-" dddcj)2^"+^r(z/+I)r(1)jj,^0,.>o (1-n')^" ^2-^^ro. +^)r(i)j-iJo sothatfinally, by|12-11 (1), (1) I*V, {zsm'0) J,{zcos'6)sin'^''+^ 6cos-=''+^ ^cZ^= \^^''+^r\v +l)z^' *This integral hasbeen evaluated byadifferent method byRutgers,Nietiw Archie/ voorWis- kunde, (2)vii.(1907), p.400;cf.also §12-22. 376 THEORY OFBESSEL FUNCTIONS[CHAP.XII Someintegrals which resemble this,butwhich aremuch more difficult toevaluate, havebeeu thesubjectofresearches byBateman, Kapteyu andRutgers;see§12•2. Asasimple exampleofaniutegi-al which maybeevaluated bythesame de\dce, the reader mayprove that,whenR{v)>—i, ^{x^- t^)h''cos t./,W(x^- 1^)}^^= 2^?r(l +a)' bywritingtheintegral onthe leftintheform 2''+ir(v+|)r(i) jojot)sm(pau(((ji. Thisformula wasgiven (withv=0)byB6cher, AnnalsofMath. viii.(1894), p.136. 12'13. Sonines secondfinite integral. Theformula (1)\y,{zsin6)JaZ cosd)sin'^+^ ^cos-'+^Odd='^^^^^^^^^^, which isvalidwhen bothR(fi)andR{v) exceed —1,isalsoduetoSonine*; and, infact,heobtained theformula of§12'11 from itbydividingboth sides oftheequation byZ"andthenmaking Z^0. Asimple method ofproving theformula istoexpandtheintegral inpowers ofzandZ andtoverifythattheterms ofdegree fi+v+2montheleftcombine toform (-)'-g^Z'-(Z2 +22)'» mlT(fjL+v+7n+2)' Theproof bythismethod islefttothereader. Woproceedtoestablish Sonine's formula byintegratingoverportionsof thesurface ofaunitsphere. Under thehypothesisthat R(fi) andR{v) exceed— |,weseethat, with thenotation of§12'12, wehave Tiw/iyw /M(^sin^)/,(^cos^)sin'^+'^cos''+^^(^^ ^\\Ie''^«"^^'=°«*+'^«=°«^«=°«'''sin2'^+i^cos2''+i^sin='^<f)sin-''-v/rf?(f)ff>/rc/^Jo.J ^izi+izncos^i ^y^iH- 11-"+^ sin^"-fdcody^ =IIfQizm+iZlcos^' „2M[iu+i ginS.-^dwdyjrJoJJn^O.l^O = I e^«'n«(^cos*+zsm</.cos.A)cos2M^sin2''+i0sin'''+-^sin=''A/rf/(f)f/^(^-JrJO.'O J-hw IIgisuie (zn+ZD^^2u cos-''6sm-''+-edQ)dd.'OJ .'m>0,n>0 iT| J.?)i>0 I/gisin(zl+Zm)j^2u(josSf^ sm'"'+^ddwdd JJn'^0 / / e»sinesiii.#,ucos^+zsmWcos2''<f)sin</)Cos=^^sin-''+-6'fZ-v/rf?0rf^. *Math. Ann. xvi. (1880), pp.35—36. 12-13] DEFINITE INTEGRALS 377 Nowthe'exponentialfunction involved here isaperiodic analyticfunction of yjrwithperiod 27r,and so,byCauchy's theorem, thelimits ofintegrationwith respecttoyjrmaybetaken tobeaand 27r+a,where aisdefinedbythe equations STcosa=z,-CTsina=Z, and OT=\/(^'+^')-Ifweadoptthese limits ofintegration, andthen write \/r+afor -v/r,thetriple integralbecomes q\^Sin6sni^cos A>coS""</)Sm(^COS'''QSm^"+-^c/-v/r f/</)f/^, andthisintegral mayalsobeobtained from itsprecedingformbyreplacing ^bytsrandZbyzero.Onretracingthestepsoftheanalysiswith these substitutions wereduce thetriple integralto iff/'TTfTT g/-Grsin0cos<f>sin"^'^+' ^C0S-''+i^sin-'^</)sin""l/rf/0fZx/rc/(9, J r(i-'+1) Jj„i^i),n^O _r(^+i)r(i)p r(i;+l) Jo.'o^ andweobtain Sonine's formula byacomparisonofthe initial and final expressions. Sonine's ownproofofthisformula wasbased ontheuseofinfinite dis- continuousintegrals,andtheprocessofmakingitrigorouswould belongand tedious. Theformula maybeextended tothedomains inwhich —h'^R(fi)>-l, and—i^i^(i/)>—1,byanalyticcontinuation. InSonine's formula, replace Zby^/{Z'+^-—2Zl^cos(f>),multiply by sin'''(/)/(^- +^--2Z^cos0)i^ andintegrate.Itfollows from§11-41(16)that (2>^[V^{zsind)J,(Zcos 0)J,(^cos 6)sin'^+i^ cos6(16 " _^^^-r r/M..^.{v(^^+^--+r-2irrcos(^)} 2''r(v+^)r{^j}o (^•^+Z^'+^—2Z?cos<^)i"^+''+^'^^^' providedthat R{^)>-1, R{v)>-h This result isalsoduetoSonine,ibid.p.45.Inconnexion with theformulae ofthis section thereader should consult Macdonald's memoir, Proc. London Math. Soc.xxxv (1903), pp.442, 443. 378 THEORY OFBESSEL FUNCTIONS [CHAP. XII 12*14.Gegenhauer's finite integral. Anintegralwhich somewhat resembles the first ofSonine'sintegrals, namely {zcos6cosy\r)J^_j {zsin6sin-v/r)0/(cos6)sm''+i6dd,'^cos sm hasbeen evaluatedbyGegenbaiier*; weshalladoptournormalprocedureof usingthemethod ofintegrationoveraunitsphere. Itisthusseen that rgzzcosflcos^j^_^ (2sindsini/r)C/(cos0)siw+iddd Jo (-^zsin'\lrY~^ C'^f'^=- r./N-n; 1X e''^*'°'^'""*+'*°**'^°"''•'°'*^C/(cos 6*)sin^-"6sin-"-' <idd.cZ^ i{v)i {^)JoJo^ iUsin i|r)_"-* (i^sinaI^)"-*A(J^)l (i)J.m>0 Qiz(?COS*+»nsin ^)(^^v(^)^j2,.-i ^^^ (^zsin\!ry-* ri'' r-''=HyxrTAxe^^sinecos(*-^)c^..(gin^cos0)cos-"-» 6sin^c^<^d^ A{v)1(2)JoJo =T^/NrT/ix e'^«>"^c°*"^C/{sin^cos(0 +-f)}cos2"->^sin(9rf(/)fZ^,A(^)••(^) JJ since thepenultimate integrandisaperiodic analyticfunction of<^with period27r. Ifweretrace thestepsoftheanalysis, usingthelastintegralinstead ofits immediatepredecessor, wefindthattheoriginal integralisequalto (^0smijr)"->r I^.^^^^^(ICOSyjr-msiny\r)n'"-' day A(^)i(2).-«>o =^^^f^^^}\ 11e'^»Or"(ncosf-lsinyjr)wi-"-idw ^(^^sm^r)" p;%ucosec^.(cos>/rcos^-sin>|rsin^cos<^)sin2"^sin-^"-i<^cZ(^fZ^.I(^)I(2) •'O-'o Now,bytheaddition theorem fforGegenhauer's function, 0/(cos yjrcos6—sinyjrsin^cos(jj) =,V. .Mo2 Vi/^— ^^(2v+2/j-1)sm^^ sm'^-v^ J Xr:;(cos^)C;;:;(cos ^|r)C^*(coS c^). *Wiener Sitzungsberichte, lxxv.(2),(1877), p.221andlxxxv.(2),(1882), pp.491—502.• tThiswasproved byGegenbauer, Wiener Sitzungsberichte,lxx.(2),(1874), p.433; cri.(2a), (1893), p.942. 12'14, 12-2] DEFINITE INTEGRALS 379 When this ismultiplied bysin-""^<^andintegrated,alltheterms oftheintegral ofthesumvanishexceptthe firstwhich is t!\^^^''\ 0/(cosd)C/(cos ylr)fsin-"-^4>d(b.Vi^v+r) Jo Wethus findthat g(>cosecos*j^_^ (^zsin6sina/t)C'/(cos6)sin^'+^ddO =^•!r(2i^)-(i^sinx/r)---^r -^.^^^^,^^. ^^.^^^^^^ andhence, by§3"32, (1) [e»^cosecos^,/^_j(^sin^sin-f)CV(cos^)sin''+*6d6 Jo — )i''sin"-*t/tC/(cos i/r)/^+,. (z). Ifweequaterealandimaginary parts,weobtainGegenbauer'sformulae (2)Icos(2cos^cos-»/r)/,_j(;2sin^sin-v/r)C/(cos^)sin''^i^f//9 J J(-)*'(~)sin"-*fC,y(cos -v/r),/,+, (^), (?•even) [o(?•odd) and (3) ["sin {zcosdcosi/r)/,_j (^sin6sini/r)C/(cos6)siw+^Odd J '0, (reven) I(_)i(/-:)f^\^ sin"->|rC,"(cos -v^)J.+ri^)- (rodd) 12*2. Integrals deduced fi^omBateman's expansion. InBateman'sexpansionof§11-6, write <I>^ (/>;andthen, notingJacobi's formula* 2rZlo/^i (-n, ^J.+v+n+l,v+l,sin^(^)]-cos^'^+i sin-^"+^<^c?<^ n!r(/. +»+l){r(i.+ l)p (yn+1/+2n+1)r(;ii+y+?i+1)r(i;+/i+1) wededuce that,when R{^l) andR{v) bothexceed -1, (1)^Jy,{zcos^<^)J^(^;sin-^)sin<pcos<j)d(fi= 2!(-)" J^+„+on+i (^), Jo "=0 *Journal filrMath. lvi.(1859), pp.149—175 [Werke,vi.(1891), pp.184-202]. 380 THEORY OFBESSEL FUNCTIONS [CHAP.XII that istosay (2) r~J^{t)J.{z-t)dt=2i(-rJ-^+,+,„+,(4 .' n=(i Animportantdeduction from thisresult isthat,whenR(/x)>andR{v)>—1, [z M[z JoC J sothat (3) fMt)JA^-t)^='^^^±^. Jo t/£ Thisformula isduetoBateman*; somespecialcases hadbeen obtained independently byKapteynf, whoconsideredintegralvalues of/xand vonly. Itwillbeobserved thatwecandeduce from(2),combined with§2-22(2), that (4)rj^(t)J.^(2-t)dt=sin2,jJ^(t)J,_^(z-t)dt=J,(z)-cosz, Jo Jo when—1<R(/x)<1,andwhen—1<R{/x)<2respectively. Byinterchanging /xwith vand twith 2-—^in(3),weseethat, ifR(fi) andR(v)arebothpositive,then ^Jot z—t\flvj z Itseemsunnecessarytogivethesomewhatcomplicatedinductions by which Kapteyndeduced(3)from thespecialcase inwhichix=v=\, otto describe thedisquisition byRutgers;):onthesubjectoftheformulaegenerally. 12'21. Kapteynstrigonometrical integrals^. Asimplerformula than thosejustconsidered is (1)1cos{z—t)Jo{t)dt=zJo{z).Jo Toprove this,weputtheleft-hand sideequaltou,andthen itiseasily verified that d^uJ-, , andtherefore w=2"JoC^)+-4cosz-\-Bsinz, whereAandBareconstants ofintegration. *Froc.London Math'. Soc. (2)ra.(1905), p.120.Some similar integrals occurringinthetheory ofintegral equations areexamined bythesapje writer, ibid.(2)iv.(190G), p.484. tProc. Section ofSci.,K.Akad. vanWet. teAmsterdam,vii.(1905), p.499;Niemv Archief voorWishunde, (2)vii.(1907), pp.20—25;Mim. delaSoc.E.desSci.deLiege, (3)vi.(1906),no.5. XNieuiv Archief voor Wislainde, (2)vii.(1907), pp.385—405. §3Iem. delaSoc.E.desSci.deLiege, (3)vi.(1906),no.5. 12-21]DEFINITE INTEGRALS 381 Now, Avhen zissmall, andsoJ-=5=0,andtheresult isestablished. Itfollows from(1)bydifferentiation that (2) \\\xx{z-t).J,{t)ilt=zJ^{z\ Jo and,byapartial integration, (3)Isin(z-t).Ji(t)dt=sinz-zJ^{z). Jo Theformula (4)sin{z- 1)^^dt=-^(-r J,^,n-,^ {z), Jo^ Mw= which isvalidwhenR(/i)>0,isofamore elaborate character, andtheresult oftheprecedingsection isrequiredtoproveit. Wewrite v= Jo(z-t)J^(t)dt,Jo andthenwehave Jo+v= {J"(z-t)+J,(z- t)\J^(t)dt+j;{z)dz- : =\^^'^^^JMdt+j;{z) Jo z—t =fMz-t)'^dt+j;(z) =fiJ^(z)/z,. by§12-2. Bythemethod ofvariation ofparameters (cf§7'38),wededuce that [^ J(t) v=Acosz-\-Bsinz +fx\sm(z -t)-^—dt, Jo* and, since v=^^^ _^+{z''+% when zissmall, itfollows that,whenR(/x)>0, A=B=0. Hence weobtain therequiredresult. Bydifferentiating (4)withrespecttozwefindthat (5) f'cos(Z-t)"^dt=- I(-)" enJ^+,n {z). 382 THEORY OFBESSEL FUNCTIONS[CHAP. XI [ 12"22.Hardy's method ofevaluating finite integrals. Asatypical exampleofaverypowerful method ofevaluatingfiniteintegrals*, weshall nowgiveaproofoftheformula(cf.§12'12) (1)f*" J^izsin26)J,{zcos26)sinV+16cos^"^^6d6= V^^^^i^^^~^l.'^''^r,l\^^^ , which isvalidwhenR(fi)>-iandR{v)> -h. Themethod ismore elaborate thananyothermethod described iuthischapter,because itinvolves theuseofinfiniteintegrals combined withanapplication ofLerch's theorem t onnull-functions. finLet IJ^{zr"'sin26)J,{zr^cos26)r^'*+2.'+3^{^2^.+1q^os^"^^6d6=f^(r), Bychangingfrompolar coordinates(r,6)toCartesian coordinates(x,y)andusing v^13'2(5)weseethat,whenever t>\I{z)\, then Iexp (-r2 •A(?•)c??-=exp (-x^t)J^{zx^)x"^"+^dx exp(-y^t) Jy.{zy^)yV+1dyJo' .'LI y = /exp(-j-2i;)./2(r)o?j-, andhence, byanobvious modification ofLerch's theorem, /i(r)isidentically equalto f^(r);andthisestablishes thetruth oftheformula. 12'3.Chessin'sintegral forY„ (2). AcuriousintegralforYji (2)hasbeen obtainedbyChessin, AmericanJournal., xvi. (1894), pp.186—187, from theformula 1 1 1 rii-^n+m ifwesubstitute thisresult inthecoefiicients oftheascendingseries forY,, (5),weobtain theformula inquestion, namely (1) Y„(2)=2(y+logi2) J„(2)- \'m-''-Kn-m-r)\ -/m=o in \ \-tdt. *Imustexpress mythanks toProfessor Hardyforcommunicating themethod tomebefore thepublication ofhisowudevelopmentsofit.Themethod wasusedbyRamanujan toevaluate many curious integrals ;andthereader mayuse ittoevaluate theintegrals examined earlier in thischapter. tActaMathematica, xxvii.(1903), pp.839—352.Theform ofthetheorem required here is that,if/(')isacontinuous function ofrwhen ?•>0,such that jbX'p(-rH).f(r)dr^O forallsufficiently large positive values oft,then/(r)isidenticallyzero. CHAPTER XIII INFINITE INTEGEALS 13*1. Varioustypes ofinfinite integrals. Thesubjectofthischapteristheinvestigationofvarious classes ofinfinite integralswhich contain either Bessel functions orfunctions ofasimilar character under theintegral sign. Themethods ofevaluatingsuchintegralsarenot verynumerous;they consist, forthemostpart,ofthefollowingdevices : (I)ExpandingtheBessel function inpowersofitsargumentand inte- grating term-by-term. (II)ReplacingtheBessel function byPoisson'sintegral, changingtheorder oftheintegrations,andthencarryingouttheintegrations. (III) ReplacingtheBessel function byoneofthegeneralisationsofBes.sel's integral, changingtheorder oftheintegrations,andthencarryingoutthe integrations;thisprocedurehasbeen carried outsystematically bySonine* inhisweightymemoir. (IV)When twoBessel functions ofthesame order occur asaproduct under theintegral sign, theymaybereplaced bytheintegralofasingle. Bessel function byGegenbauer'sformula (cf§12*1), andtheorder ofthein- tegrationsisthenchanged f. (V)When twofunctions ofdifferent orders butofthesameargument occur asaproductunder theintegral sign,theproduct maybereplaced by theintegralofasingleBessel function byNeumann's formula(§5"43), and theorder oftheintegrationsisthenchanged. (VI)TheBessel function under theintegral signmaybereplaced bythe contourintegralofBarnes'type (§6*5)involving Gamma functions, andthe order oftheintegrationsisthenchanged;thisverypowerful method hasnot previouslybeeninvestigatedinasystematicmanner. Infiniteintegrals involvingBessel functions under theintegral signare notonlyofgreatinterest tothePureMathematician, buttheyareofextreme importanceinmanybranches ofMathematicalPhysics. Andthevarioustypes aresooiumerous that itisnotpossibletogivemore than aselection ofthe mostimportant integrals,whose values willbeworked outbythemost suitable methods;carehasbeentaken toevaluate several examples byeach method. Inspiteoftheincompletenessofthischapter,itslength must becontrasted unfavourablywith thelengthofthechapteronfiniteintegrals. *Math. Ann. xvi.(1880), pp.33—60. tThisprocedure hasbeen carried outbyGegenbauevinanumber ofpapers publishedinthe Wiener Sitzungsberichtc. 384 THEORY OFBESSEL FUNCTIONS[CHAP. XIII 13*2. Theintegral ofLipschitz,withHankeVsgeneralisations. Itwasshewn byLipschitz*that (!) J ly.KU(U)dt=-^^. whereR(a)>0,and, inorder tosecureconvergenceattheupperlimit ofin- tegration,both thenumbers R{a±ih)arepositive. That value ofthesquare root istaken which makes |a+\/{a-+lf)\>\h\. Thesimplestmethod ofestablishingthis result istoreplacetheBessel coefficient byParseval'sintegral (§2-2)andthenchangetheorder ofthein- tegrations—aprocedurewhich maybejustified withoutdifficulty.Itisthus found that roc TT ./ ^ cie ITjQa—ihcos6 =1/V(a"+6"), andtheformula isproved. Now consider themoregeneral integral je-"W,{bt)t''-Ult. Thisintegralwas firstinvestigatedinallitsgenerality byHankelf, ina memoirpublished posthumouslyatabout thesame time astheappearance of twopapers byGegenbauer:[:.These writersproved that, ifR(/j,+v)> 0,to secureconvergenceattheorigin,andthepreviousconditionsconcerninga and bare satisfied, tosecureconvergenceatinfinity,then .theintegralis equalto a'^r(i.4-l) -A 2'/^+^+^, +l;_^^ (I.4-1)-^ '\2'2''—'aV Toestablish this result, firstsupposethat bisfurther restricted sothat I6 I< ja|.Ifweexpandtheintegrandinpowersofbandintegrate term-by- term,wefindthat JoIe-<'UAbt)t>^-^dt= XA-iir^^-^ rJt'^-"'^"-"'-' e-^'dt /o m=ow'I{v+m+l)Jo ~ r>i=oncTVlv +m+1)a'^+''+^'» *Journal furMath. lvi.(1859), pp.191—192. tMath. Ajin. viii.(1875), pp.467—468. 1Wiener Sitzuvgsherichte, lxx. (2),(1875), pp.433—443; ibid, lxxii.(2),(1876), pp.343—344. luthejormer, thespecial caseix—v-^\ wasinvestigated bytlieintegral givenin§3*32; inthe latter, Gegenbauer obtained thegeneral result bysubstituting Poisson's integral forJv(bt). 13-2]INFINITE INTEGRALS 385 The final series converges absolutely,since |6 [< |a|,and sotheprocessof term-by-termintegrationisjustified*. Hence (2) re-^^J.{bt)t^-'dt Jo _(^b/ayr{fi+v)(iM+vfi+v+i .6^ The result has, asyet,beenproved onlywhenR(a)>andI6 |< Ia |; but, solongasmerely R(a+ih)>andR(a—ih)>0, thenboth sides of(2)areanalyticfunctions ofh;and so,bytheprincipleof analytic continuation, (2)istrue forthismore extensiverangeofvalues ofh. Again, byusingtransformations ofthehypergeometric functions, (2)may bewritten inthefollowingforms : (3)I"e-«' /.(ht)V'-'dt Jo (Way r(^+,.)i^^ir^-^^^_/._-M±i _q^+1.,+1._ ^^ai^Tip+l) V a-J-'\ 2'2' ' a- (^byVifi +v) ^,ff^+vl-f^+i^ . , -,.b^- ~(a'+¥f^^^+"^r{v+l)- '\2'2'"' 'a'+bV Theformula(2)hasbeen usedbyGegenbauerfinexpressingtoroidal functions asinfiniteintegrals;specialcases of(2)arerequiredinvarious physical researches, ofwhich those byLambJmayberegardedastypical. Bycombining twoBesselfunctions,itiseasytodeduce that (4)[e-"tl\(bt)t''-^dt =cot vir J-r 2^1^^^ ) '^ v+ i;..,., —cosec VTT—, ; 2'^i ~^r~ > «— >^-";——rs , („2^.^,-2^i(M-.')r(i-v)' '\2'2 a^+6V provided R{ix)>\ Il{v)\ andR{a±ib)>0;specialcases ofthisformida areduetoHobson, Proc.London Math. Soc.xxv. (1892), p.75,andHeaviside, Electromagnetic Theory,iii. (London, 1912), p.85. Itisobvious thatinteresting specialcases oftheformulae sofardiscussed maybe *Cf.Bromwich, Theory ofInfinite Series, §176. tWiener Sitziaigsberichte,c.(2),(1891), pp.745—766; Gegeubauer also expressed series, whose general terms involve toroidal functions andBessel functions, asintegrals with Bessel functions under theintegral sign. XProc. London Math. Soc. xxxiv.(1902), pp.276—284; (2)vii.(1909), pp.122—141. See alsoMacdonald, Proc. London Math. Soc.xxxv. (1903), pp.428—443andBasset, Proc. Camh. Phil. Soc. V.(1886), pp.425—4.33. W.B.F. 25 386 THEORY OFBESSEL FUNCTIONS [CHAP.XIII obtained bychoosing fiand vsothat thehypergeometric functions reduce toelementary functions. Thus, bytaking /xequaltov+1orv+2,weobtain theresults (5) j\-a^JAbt)t^dt=^^^^l^i^,Jo^ . (a'+b'-y+i ^tt' ^jo^ ^ {a^+b-^y+^^TT' providedthatR{v)>—§,^(i')>—1respectively. These formulae wereobtained byGegenbauer,WierierSitzungsberichte,Lxx.(2),(1875), pp.433—443;theywere alsonoticed bySonine, Math. Ann. xvi. (1880), p.45;andHardy, Trans. Camb. Phil. Soc.xxi. (1912), p.12;while Beltrami, Atti della R.Accad. delle Sci. diTorino, xvi.(1880—1881), ^^.203,andBologna Alemorie, (4)ii.(1880), pp.461—505, has obtained variousspecial formulae bytaking ^=1and vtobeanyinteger. Otherspecial formulae are (7) f%-.....(60^'=W(«^+'''-'"' (8) e-''iJ^{bt)dtvb" W{a'+b^)-a}'' [Note.Itwasobserved byPincherle, Bologna Memoric, (4)viii. (1887), pp.125—143, thattheseintegralsarederivable from thegeneralised form ofBessel's integrals (§6'2)by Laplace's transformation(cf.§9"15). Thisaspectofthesubjecthasbeen studied by Macdonald, Proc.London Math. Soc.xxxv.(1903), pp.428—443,and Cailler,3le'ni. dela iSoc.dePhysiquedeGeneve., xxxiv. (1902—1905), pp.295—368.The dift'erential equations satisfied by(5)and(6),quafunctions ofa,have beenexamined byKapteyn,Archives Merlandaises, (2)vr.(1901), pp.103—116.] . .Jo(bt)tdtwasobtained byNeumann, Journal fiirHath. Lxxii. Sinn Trt (1863), p.46,asalimit ofaseries ofLegendre functions(cf.§14"64). Theintegral does notseem tobecapable ofbeingevaluated infinite terms, thoughitiseasytoobtain a series foritbyusing theexpansion cosech 7rt=22e-(2»+^)^t. n=0 Aseries whichconverges morerapidly (when bislarge)willbeobtained in§13-51. Someintegralsofthesamegeneral tyi3earegiven byWeber, JournalfiirMath. Lxxv. (1873), pp.92—102; andmorerecentlytheformula rJAbt)t''dt ^{2b)''r{u +^)I1 Jo e^i-l Jn n=i(7i2^2 +^,'y+i' which isvalidwhen Pt{v)>0 and |7(6) |<«-,hasbeen obtained byKapteyn, Mem. dela Soc.R.desSci.deLiege, (3)vi.(1906), no. 9. 13*21. TheLipschitz-Hankel integrals expressedasLegendre functions. ItwasnoticedbyHankel thatthehypergeometricfunctions which occur intheintegrals justdiscussed areofthespecial typeassociated withLegendre functions; subsequently Gegenbauer expressedtheintegralsinterms oftoroidal functions (whichareknown tobeexpressibleasLegendre functions), anda little laterHobson*gavetheformulae insome detail. *Proc.London Math. Soc. xxv.(1893), pp.49—75. 13-21] INFINITE INTEGRALS 387 Toobtain thefundamental formulae* ofthistype,weshallchangethe notation bywriting a=cosh a,h=isinh a, where aisacomplex number such that -|7r^/(a)^^7r; wethus obtain theformula (1)re-^cos\^'^ /,{tsinhOL)t^'dt=V{fi^v+\)P-" (cosh a), providedthatR{fji+ v)>—1. Thespecial caseofthisformula inwhich^=0hadbeen given byCallandreau, Bull, des Sci.Math.(2)xv.(1891), pp.121—124,twoyearsbefore Hobson pubhshedthegeneral formula. Itfollows atoncefrom(1)that (2) [""e-^cosha XUsinha)t''dt=.^^"^^Y(/u,-v+1)Q/ (cosh a), providedthatR(/jb+1)>\R(v)\. Themodification of(1)which hastobeusedwhen theargumentofthe Legendrefunction-f*ispositive and lessthan 1is (3) [""e-^cos^ j^(^sin/9)t^dt=V(fj.+v+1)P^-" (cos^),Jo andhence wefindthat (4)re-too^^YAtsinl3)t^dt=- .'^"^^" ,r(^-. +l) Jo Sm{/M+V)TT IT X[Q^"(cos /3+Oi)e*-"^'+Q/(cosy3-Oi)e-^""']. Somespecialcases ofthisformula havebeen given byHobson, loc. cit.p.75,andby Heaviside, Electromagnetic Theory,in.(London, 1912), p.85. Anapparentlydifferent formula, namely (5)re-.c.,>..^^(,)<|=ftzi(?5^. hasbeen studiedbySteinthal;]:.Thisformula isconnected with formulae of theprevious typebyWhipple's §transformation ofLegendre functions, which *Since, byachange ofvariable, theintegrals areexpressibleinterms oftheratio ofhtoa,no generalityislost.Thevarious expressions forLegendre functions ashypergeometricseries which arerequired inthisanalysis aregiven byBarnes, Quarterly Journal, xxxix.(1908), pp.97—204. fThereader willremember that itiscustomary togiveadifferent dehuition fortheLegendre function insuch circumstances; cf.Hobson, Phil.Tram, oftheRoyalSoc.clxsxvii. A,(1896), p.471;andModernAnalysts, §§15-5,1.5-6. XQuarterly Journal, xviii.(1882), pp.337—340. §Proc. London Math. Soc.(2)xvi.(1917), pp.301—314. 25—2 388 THEORY OFBESSEL FUNCTIONS [CHAP. XIII expressesafunction ofcoshainterms ofafunction ofcoth a.Themore generalformula ofthesametypeis cosVTTQlZl (cosh a) (6) f"e-^'^o«i^<^/^m^'^-irf^=^ 'o sin(/i+i^)TTV(2'^)-sinh'^-*a* Inthese formulae, R{fji1^v)>^andR(cosh a)>\., Onreplacing?/by—i/in(6),wefindthat r-^ , P*"^ (cosh a) (7) e-t^'''^'^K,{t)t^-'dt=^|{h^'^).T{^JL-v)T{^JL^v)~^^^-, J^Sinn'**a. andthisformula isvalidwhenR(/x)>\R(i/)\andR(cosh a.)>—1. Ifwetakecosha=0,wededuce that (8)/JKAt)t--^dt=2'^--rl^^)r('^y aresult given byHeaviside* inthecase v=0. Whenfi=l, (7)becomes ,„. [^ ,wT' /s7 TTsinhva JosinVTTsniha andhence,ifv=0, J„ ..,arcsinhv'Cot^~1)_^'"'^^"^V(l- <*^)_*rccosa "^^-^^^-^—^_________ =._^^__. Ifwereplace aby±zb,wefindthat andso,when \I{b)\<l, /n\ r°°•/7Nr^ /N7arcsinh 6 (11)j^sm{bt).Koit)dt = -j^^-^. Theformer ofthese isduetoBasset, Hydrodynamics^ii.(Cambridge, 1889), p.32. [Note.Various writers have studied theLipschitz-Hankel integrals from theaspect of potential theory ;totakethesimplest case,if(p,0,z)arecylindrical coordinates, wehave 1/: / Itissuggested that, since e~P'J,,{^t)isapotential function, theintegral ontheleft isa potentialfunction finite atallpointsofrealspace except theorigin andthatonttheplane 2=0 itisequalto1/p,andsoitisinferred that itmust bethepotentialofaunitchargeat theorigin. Butsuchanargument doesnotseem toprecludethepossibilityoftheintegral beingapotential function with acomplicatedessential singularityattheorigin, andso thisreasoning must beregardedassuggestiverather than convincing. *Electromagnetic Theory,in.(London, 1912), p.269. tOntheaxisofz,theintegralisequaltoaconstant divided by\z\. 13-22} INFINITE INTEGRALS 389 Forvarious researches onpotential theorywiththeaidoftheintegralsofthissection, thereader mayconsult Hafen, Math. Ann. lxix. (1910), pp.517—537. Forsomedevelop- ments based onthepotentialfunction seeBateman, Messenger,sli.(1912), p.94.] f m 13"22.Applications oftheaddition formulatotheLipschitz-Hankel integrals. Itiseasytodeduce from theresults oftheprecedingsections combined with§|11"41 (16) that, ifallfour ofthenumbers R(a±ib±ic) arepositive andR{}i+21/)>0,while sriswritten inplaceof^/(b'+c-—26ccoscfi),then (1)re-"' ./,(bt) ./,(ct)f^-'dt -V{v-\-h)r{h)]Jo ^^ ^^ ^^,F,r-~^-,'^— ^v+l;-~-]sm'''cf>d<t>.7ra>^+"'r{2v+l).^o'\2'2''' a-^ Thehypergeometricfunction reduces toanelementaryfunction if/x=1or2; andsowehave (2)/;.-.J,mJ,(ct)dt=^Q,.,("^i|^). Thecase^l=2maybederived from thisbydifferentiation withrespecttoa. These formulae, orspecialcases oftheiu, havebeenexamined bythefollowing writers: Beltrami, Bologna Memorie, (4)ii.(1880), pp.461—505;AtH della R.Accad. delle Sci.di Torino, xvi. (1880—1881), pp.201—205; Sommerfeld, Konigsherg Dissertation, 1891; Gegenbauer, Monatshefte fiirMath, undPhys.v.(1894), p.55;andMacdonald, Proc. London Moth. Sac.xxvr. (1895), pp.257—260. Bytaking /x=—1,v=lin(1),wefindthat ^-at'-'i l^;cu^1 ;^/(„2+2-2cos</))-a}(l+cos0)(^0,t- "IttJ,, sothattheintegral ontheleft,which wasencountered byRayleigh, Phil. Mag. (5)xlii. (1896), p.195[Scientific Papers,iv.(1904), p.260],isexpressibleasanelliptic integral./. Anintegralwhichmaybeassociated with(1)is i- de(S^[cosatI,{bt)K,(ct)dt= .'0 .0\/{a-+(b+c)--4>bcsin'd]' Thiswasdiscovered byKirchhoff* asearlyas18.53; thereader should have nodifficultyindeducingitfrom§13"21 (10)combined with§11"41(16);it isvalid ifallthenumbers R(c±b± ia) arepositive. *JournalfiirMath, xi.vin.(1854), p.364. 390 THEORY OFBESSEL FUNCTIONS[CHAP. XIH Asomewhat similar result, namely (4)Ie-<'U^-''J^ibt)J^{ct)dt r(v+^)r(^) J(«2+2iaccos<^-c2cos2</,+62)^+^' which isvalidwhenR{a±ib±ic)>0 and It(fi)> —h,isdue toGegenbauer, Wie?ier Sitzungsherichte,Lxxxviii.(2),(1884), p.995. Itismosteasily proved bysubstituting integralsofPoisson's typefortheBessel functions. Inthememoir citedGegenbauer has alsogiven alistofcases inwhich theintegral ontherightisexpressible byelementary functions(cf.§13-23). 13'23. Gegenbauersdeductions fromtheintegrals ofLipschitz andHankel. Aformula duetoGegenbauer, Monatshefte filrMath,undPhys.iv.(1893), pp.397—401, isobtained bycombiningtheresults of§13"2with theintegral formula of§5*43 fortheproductoftwoBessel functions;itisthuspossibleto expresscertainexponential integralswhich involve twoBessel functionsby means ofintegralsoftrigonometricalfunctions *.Thegeneralresult obtained byGegenbauerisdeduced bytakingtheformula 2r*''J ft.(bt)J^(bt)=—Jy,+v (^btcos(f>)cos{/jL—v)(b deb, ttJ multiplyingitbye'^"-^ f^^"andintegratingfrom tox ;itisthusfound that, ifR(a)>\I (b)\andR(ti+v)>-^,then re-''''^J„(bt)J^(bt)t>'-^''dt=- [j^e-'''^J^^^{2btcos4))t''+''cos(ix-v)(}).d(bdtJo "^JoJo =- f'"[ 6--"*/,.+. {2btcos(j>)i''+''cos(/ji-v)(f). dtd(f>TTJO Jo 2r^-(46cos0^+"Tifi+v +l) = ;;-Jo (4«-+46-cos-c{>y-^-i V^'''^^~'^'^•^^^- Theinversion oftheorder oftheintegrations presentsnogreattheoretical difficulties;hence (1)re-"-^'J^{bt)J,{bt)t>^+UltJo _T{fl-\-V +\)b^^" fi'COS>^+''(f)COS(fl-v) <f) TT^ Jo (a'+b'-cos'(f>y+''^i*^" Thisresult, inthespecial case inwhichfi=v=0,hadbeen obtainedpreviously by Beltrami, Attidelta R.Accad. delle Sci.diTorino,xvi.(18S0— 1881), p.204. Asparticular cases of(1)take/i=1and vequaltoandto-1.Itisfound that (2) re-^'''Ji{bt)Jo{bt)tdt= i^~^ Jo '2nbJ{a^-\-b^)' (3) rc-^'^^JHbt)dt(2«^+^'^)^-2(a^+6^)i: ^^jo'mat- ^62v'(a2 +6-^) *Seealsoanearlier notebyGegenbauer,ibid. pp.379—380. 13-23, 13-24] INFINITE INTEGRALS 391 where themodulus ofthecomplete elliptic integralsKandEis.bls'{a^+b^)-Beltrami's correspondingformula is (4)/>-"-'»^('")*=W{5t-6?- Replacingbby ib,wededuce from(2)that (5).f^-^"^A(bt)h{ht)tdt=^i5'"i''!Yy: 2Trb{a'-b'^)' where R{a)>\R{b)\,andthemodulus1:^oftheelliptic integralsisbja.Thefonnulae(3) and(4)maybemodified inasimilar manner. Itwasstated byGegeabauerthat theintegralsin(2), (3)and(5)areexpressible by means ofeUiptic integrals, buthedidnotgivetheresults indetail;some formulae deducible from theresults ofthis section weregiven byMeissel, KielPrograming1890. {Jahrbuchiiber dieFortschritte derMath. 1890, -^-g.521—522.] 13°24. Wehersinfinite integral, after Schafheitlin. Theformula j,(t)dt_ r(iji) (1)^,_^+i 2''-'^+i r(l/-iyLt +l)' inwhich <R{^-)<R{u)+^,wasobtained byWeber* forintegralvalues of V.Theresult wasextended togeneralvalues ofvbySoninef ;andthecom- pletely generalresult wasalsoproved bySchafheitlin:{:. Theformula isofamore recondite typethan theexponential integral formulaegiven in^13'2;itmaybeestablished asalimitingcaseofthese formulae, for,since theconditions§ ofconvergencearesatisfied, wehaveby§13"2(3) ''-^^limfe-atJ^(0dt f-i^+i '".F.fs,!^:^^';.^!;!2T(v+l)"'-^V2' whence theformula isatonce obtained. Adirect method ofevaluatingtheintegralistosubstitute Poisson'sintegralforthe Bessel function, andthenchangetheorder oftheintegrations;this isthemethod usedby Schafheitlin, buttheanalysisisintricate because theresult isestablished first fora limited rangeofvalues of/xand vandthen extended bytheuseofrecurrence formulae andpartial integrations. Analyticaldifficulties are, toalarge extent, avoided byusingcontour integralsinstead ofthedefiniteintegralsofSchafheitlin. Ifwesupposethat *Journal furMath. lxix.(1868), p.230.Thespecial case inwhich u=was setbyStokes as aSmith's Prize question,.Jan. 29,1867. [Math, andPhys. Papers,v.(1905), p.347.] tMath. Ann. xvi.(1880), p.39. +Math. Ann. xxx.(1887), pp.157—161. §Cf.Bromwich, Theory ofInfinite Series, §172. THEORY orBESSEL FUNCTIONS [CHAP. XIII R(fi)<andR(v)> —^,wethen have (the integrals being absolutelycon- vergent) ..(-0--^- 2-^r(.+i)r(i)J,.<-'^'~'jy^^''''^>''^''''^^^' =2-r(. +i)r(i) Jo^^^'^^^^^ ^^^ —2tsin/MTT.r(-|yLt) Bythetheoryofanalytic continuation, thisresult isvalidwhen/aand t-are subjectedtothesinglerestrictioni?(/u,)< R(v+%). WhenR(yu)>0,wedeform thecontour intothepositivehalf ofthereal axistaken twice, andweatonce obtain theWeber-Schafheitlin formula. Theintegral* (-ty-i^^' maybetreated inexactlythesame manner;theonlydifference inthe analysisisthat cos(tcos6)hastobereplaced by—sin{tcos6),and so,by Euler's formula(adaptedforcontourintegrals),thefactor cos|/i7rhastobe replaced by—sinI/att. Itisthusfound that rt"+)H. (- dt_2isinfx-rr.T(|/z)tanIfx-rr j+x {-ty->'^^" 2''-'^-^ir(i/-iyLi+l)' providedthatR{fi)< R{v+%)andR(/j,)^0. WhenR{fx)>—1,thecontour maybedeformed intothepositivehalfof therealaxistaken twice, sothat .,,. rtlAt)dt^r(| /.)tan(l/x7r) providedthat—\<R{ix)^0 andR{ii)<R(v)+1. Ifwetake/a=0,v=l,weseethat (3)Hi(t)dt_^ fxTT. Thisresult, combined with theasymptotic formula ir ^JXH.i{2t)dt2(\\\cos(2a;-|-i»r) wasusedbyStruve, Ann. derPhysik undChemie, (3)xvii.(1882), p.1014, totabulate i_r'A^mdt forboth smallandlarge values ofx.The lastintegralisofimportanceintheTheoryof Diffraction. *Generalisations obtained byreplacing Bessel functions byLommel's functions(§10-7)in theintegralsofthis section and inmany other integrals arediscussed byNielsen, K.Danske Videnskahernes SelskabsSkrifter, (7)v.(1910), pp.1—37. 13-3]INFINITE INTEGRALS 393 [Note. Bjdifi'erentiating (1)under theintegral signweobtain Weber's result (4) fJ^{t)\ogtdt=-y-\og%; thisformula hasalsobeen investigated bjLerch, Monatshefte fiirMath, undPhys.i. (1890), pp.105—112. Theformula forfunctions ofthesecond kind, correspondingto(1),is (5) Pl\{t)dt^Vjhfi)r(hn-p)cos(hfi-v)iT providedthatIR{v)\<R{}i- v)<'i. This result hasbeen given byHeaviside, Electro- magnetic Theory,in.(London, 1912), p.273,when i'=0.] 13'3, Weber'sfirst exponential integral and itsgeneralisations. Theintegralformula (1) l^i"Jo{at)expi-if-t').tdt= ^^exp(-^3) wasdeducedbyWeber* from hisdoubleintegralformula which willbe discussed in§14-2. Thisintegraldiffers from those considered earlier inthe chapter bycontainingthesquareofthevariable intheexponentialfunction. Itissupposedthat |argp|<jtttosecureconvergence,butaisanunrestricted complexnumber. Itisequally easytoprove Hankel'sf moregeneral formula, (2)rJ,(at)ex^{-p-t^).t'^-'dt =— 2p^r(.+ l) ^-^^(i^+^^-^+ l'-4p^j' byadirect method. Tosecureconvergenceattheorigin,itmustnowbe supposed that:|: R{lJi+v)>0. Toobtain theresult, weobserve that, since(by §7'23) [/],,da I• Iexp{-pH-) 1. 1i'^-i Idt Jo isconvergent,itispermissible^toevaluate thegiven integral byexpanding Jt,{atyi^ powersoftandintegrating term-by-term, *Journal fiirMath. lxis.(1868), p.227.Weber alsoevaluated(2)inthecase;U=v+2,vbeing aninteger. fMath. Ann. viii.(1875), p.469. See'alsoGegenbauer, Wiener Sitzungsherichte, lxxii.(2), (1876), p.346. %This restriction maybedisregardedifwereplacethedefinite integral/bythecontour .' /|('+) integral §Cf.Bromwich, Theory ofInfinite Series, §176. 394 THEORY OFBESSEL FUNCTIONS [CHAP.XIII Itisthusfound that J,(at)expi-pH').f^-'dt=2)^\^'^ '—-f+i^+^-i exp{-pH')dt andthis isequivalenttotheresult stated. IfweapplyKummer's firsttransformation(§4"42) tothefunction onthe rightin(2),wefindthat (3)rJ^{at)ex^{-p-f).t^-^dt andsotheintegralisexpressibleinfinite terms wheneveryu,—i/isaneven positive integer. Inparticular, wehave (4)JV.{at)expi-pH^).t^^^dt= (—^^^^P ("^)o/> providedthatR{v)>—1.Thisintegralisthebasis ofseveralinvestigations bySonine, Math. Ann. xvi. (1880), pp.35—38;some oftheseapplicationsare discussed in§13"47. Inorder that thehypergeometricfunction ontherightin(2)maybe susceptibletoKummer's second transformation(§4"42),wetakefi—\]and, ifwereplace vhy 2v,wethen findthat (5)|V,.(aOexp(-^^i^).rf«= ^exp{-^).L {^), aresultgiven byWeber inthecase v=^. Ifwereplacei/by— j;,itiseasytoseethat (6) \^Y.,,{at)ex^{-pH'')dt .' when\R(v)\<^; and, ifwemakep-*0,(abeingnowpositive), wefindthat r* -r-r / X7 tanVTT (7) Y,Aat)dt= , Jof^ when \R{v)\< ^,hy using §7*23;and, inparticular, (8) rY,{t)dt=0. J 13-31]INFINITE INTEGRALS 395 Formulae(5)and(6)were given (whenv=0)byHeaviside, Electromagnetic Theory^ill. (London, 1912), p.271. Another method ofevahiatingtheintegral onthe leftof(3)issuggested byBasset, Proc. Camb. Phil. Soc. viii.(1895), pp.122—128;theintegi'als have alsobeen evaluated withthehelpofLaplace'stransformation byMacdonald, Proc.London Math. Soc.xxxv. (1903), pp.428—443; seealsoCurzon, Proc.London Math. Soc.(2)xiii.(1914), pp.417— 440;andHardy,Trans. Camb. Phil. Soc. xxi.(1912), pp.10,27,forformulae obtained by making j9-apureimaginary. Forsomeapplicationsoftheintegralsofthis section totheTheoryofConduction of Heat, seeRayleigh,Phil.Mag. (6)xxii.(1911), pp.381—396[Scientijic Papers,wi.(1920), pp.51—64]. IS'Sl. Weber's secondexponential integral. The result ofapplyingtheformula§ll"41(lt))totheintegral justdis- cussed istomodifyitbyreplacingtheBessel function under theintegral signbyaproductoftwoBessel functions ofthesame order. If -S3-=\/{a-+lr— '2abcos(f>)and i{R(v)>—1,R{2v+/x)>0,\ argpI<^tt, wethusdeduce that rexp(-pT-)J,(at)J,[bt)f^-^dt J 27r^'-r(2i^+l) V; Jo^\^pV \ 2'' 4^j7^^ Thehypergeometricfunction reduces tounitywhen/x=2;sothat (lahlp'YIa?+hfx[' fnbcos<f,\... ,, Ifweexpandtheexponentialunder theintegral sign,wefindthat (1)[Jexp(-pH')J,(at) ./,(bt)tdt= ^^exp(-^^-')7. (1^,). Thi^ormula isvalid ifj^(v)>—1and |argp<^tt. Like theresult of§13-3, thisequationisduetoWeber, JournalfiirMath. LXix. (1868), p.228;Weber gaveadifferent proofofit,asalsodidHankel, Math. Ami. viii.(1875), pp.469—470.Theproof givenhere isduetoGegenbauer, WienerSitzungsberichte,lxxii. (2),(1876), 2J.347. Other investigationsareduetoSonine, Math. Ann. xvi, (1880), p.40; Sommerfeld, Konigsberg Dissertation, 1891;Macdonald, Proc.London Math. Soc.xxxv. (1903), p.438;andCailler, Mem. delaSoc.Phys.deGeneve, xxxiv. (1902—1905), p.331. Somephysical applicationsareduetoCarslaw,Ih-oe. J^ondon Math. Soc.(2),viii. (1910), pp.365—374. 396 THEORY OFBESSEL FUNCTIONS [CHAP. XHI 13*32. GeneraUsationsofWeber's secondexponential integral. When theBessel functions inintegralsofthetype justconsidered are notofthesame order, itisusually impossibletoexpresstheresult inany simpleform. Theonlymethod ofdealingwith themostgeneral integral I"^ ./„(at)J,(ht)exp(-jft-)t^-' (It . istosubstitute theseries of§11*6 fortheproductofBessel functions and integrate term-by-term, but itseemsunnecessarytogivetheresult here. Inthespecialcase inwhich X=v— fx,Macdonald* hasshewn thattheintegral isequalto (hay-"r^'^,„,/,• .^,/, 7-fabsin6\/b-+aHin-0^,, byatransformation based ontheresults of§§12'11, 13"7. Anexceptionalcaseoccurs when a=b:i^R(\+ /x+i')>0. wethenhave J^(at)J,(at)exp(-pH-)t^-'dt=-^-r-^-^^ -n— ," r ^{lx+v-\-l fi-\-v+1\+iJb+v ^ ^ ,a-\ "^^n^'2'2'^+1.^+1./^+"+!-• -p2J' byusingtheexpansionof§5"41.Somespecialcases ofthisformula have beeninvestigated byGegenbauer+. 13'33. Struve'sintegral involving products ofBesselfunctions. Itwillnowbeshewn that,whenR{^-^v)>0,then (1)/'JMt)JAt)^^^T(,M+v)r{^) ^M+v 2'^+"r(fL+V+^)r (fi+^)r (v+1) This result wasobtained byStruve, Jle'm. deVAcad.Imp.desSci.deStPe'tersbourg, (7) XXX.(1882), p.91,inthespecialcase/x=z/=l ;theexpression ontherightisthen equal to4/(37r). Inevaluatingtheintegralitisfirstconvenient tosupposethatR(/u.)and R{i')bothexceed h.Itthen follows from§3-3(7)that J^(t)JAt)^.^^(2/x-l) (2^.-1) ^+, 2'^-''-Vr(/x-hi)r(i;-F|) c=^rh-ch-Q[jx(ts,i'a.d)s,\n(tsv[\d>) .,.,/, ,„,•-a• •7/1jj ;* I^^^^— ^^^cos-''"- 6cos-""-<^sin^sinddd(pdt. .JJ t' *Proc.London Math. Soc.xxxv.(1903), p.440. tWiener Sitzungsberichte,lxxxviii.(1884), pp.999—1000. 13-32, 13-33] INFINITE INTEGRALS 397 Inview ofthefactthat t~"sin{tsin6)sin{tsin^)doesnotexceed numeri- callythesmaller of\jt-and sin6sin0,therepeated integral converges absolutely,andtheorder oftheintegrations maybechanged. Since* psin{tsin6)sin{tsin(/>)._i^irsin6, {6^(^) Jo^^"|i7rsin(/), (^^(^) wefindthatthetriple integralisequalto i-TTr /cos-'*-- 6cos-''-==</)sin^6sin(f)ddd(f> -I-^TTI ICOS-''-- ^cos-"--(/)sin6s\n^(f)d(f)d6. JJ But,byapartial integration,wehave (2v-l)['"cos^''-^(j)sin(^ jjcos-'*-^ ^sin^ (9g?^1dcf) =-cos-''-^ cos-'^-- (9sin^Odd + cos^-'-i(^.cos^'*-^(^sin-<^(/(/> L •'0 Jo^ _V{^i+v-l)T {^) •2r(^+ i/+*)• Theother integralisevaluated inthesame manner, andsowehave /<-JAt)•L{t)^^_V{^l+v-l)V{%){{2^,-l) +{^v-l)] whence theresult stated isevident. Theextension overtherangeofvalues of//.and Vforwhich merely R{fi+v)>0isobtained bythetheoryofanalytic continuation. Itmaybesliewn inasimilar manner that,when R{fi+v)ispositive, then also r-H^(OH.(0^^_r(;x+.)r(A) ^ ji> t^"^" 2'*+T(iu +i/+i)r(/i+i)r(i/ +i)' This result wasalsoobtained byStruve{ihid. p.104)inthecase \j.=v= 1. Byusing §10'45wefind that,whenR(/x)andRiy)exceed|, 'H^(0H.(0dt (2^-l)(2v-l) r*p'-[h^{l-cos{tsin6)\{1-cos{tsin0)} ~2'"+''-"^7rr(/L(,+|)r(i/ +^-)-'"^•'•' •'"^^ Xcos''^'*-'' 6cos-"-- sin6sin(j)dddcj}dt. Now,ifaand/3arepositive,itappearsfrom aconsideration of (1-e«'^) (l-e3'^)!dz This result iseasily proved bycontour integration. 398 THEORY OFBESSEL FUNCTIONS [CHAP. XIII round acontoyir consisting oftherealaxisandalarge semicircle aboveit,that /•°° (1-cos(at)} (1-COS(301 ,_f""sin(at)smj^t) Jo^^ Jot^ Hence thetriple integral under consideration isequaltothetriple integral evaluated in proving (1),andconsequently (2)isestablished inthesamewayas(1). Thereader willproveinlikemanner that,ifR(ju)andR(v)bothexceed I,then (3) rH.wj,w^,^ _j^ J and thismay beextended over therangeofvalues offiand vforwhich R{v)>l and R{fi+v)>1. Theintegrals/J"^^^^^^^^^,/^cit maybeevaluated inasimilar manner, buttheresults areofnogreatinterest*. 13"4.Thediscontinuousintegral ofWeber andSchafheitlin. TheintegralrJAai)JAJ>t) ^^^. Jo t inwhich aand haresupposedtobepositivetosecure convergenceatthe upper limit, wasinvestigated byWeber, Journal furMath. lxxv.(1873), pp.75—80,inseveralspecial cases, namely, (i)x=^=o,v=\, (ii)\=-\, /ti=0,v=±h. Theintegral wasevaluated, forallvalues ofX,^and vforwhich itisconvergent, by Soninet, Math. Ann. xvi.(1880), pp.51—52;buthedidnotexamine theintegralinvery great detail, nordidhelayanystress onthediscontinuities which occur when aand h becomeequal. Some years later theintegral wasinvestigated verythoroughly bySchaf- heitlinI,buthispreliminary analysisrests toasomewhat undue extent onthetheoryof linear differential equations. Thespecialcase inwhich X=wasdiscussed in1895byGublerJ^ whoused avery eleganttransformation ofcontourintegrals ;unfortunately, however,itseems impossible toadaptGubler's analysistothemore generalcase inwhich X=^0.Theanalysisinthe specialcase willbegiven subsequently (§13'44). *Some related integrals have been evaluated bySiemou, Progrnmm,Ltiisenschule. Berlin, 1890[Jahrbuchilber dieFortschritte derMath. 1890, p.341]. tSeealso §13-43 inconnexion with theresearches ofGegenbauer, Wiener Sitzungsberichte, Lxxxviii.(2),(1884), pp.990—991. XMath. An)t. xxx.(1887), pp.161—178.Thequestionofpriorityisdiscussed bySonine, Math. Ann. xxx.(1887), pp.582—583, andbySchafheitlin, Math. Ann. xxxi. (1888), p.156. §Math. Ann. xlviii.(1897), pp.37—48.SeealsoGrafandGubler, Einleitung indieTheorie derBesseVschen Funktioyien,ii.(Bern, 1900), pp.136—148. 13-4]INFINITE INTEGRALS 399 The firstinvestigationwhichweshallgiveisbased ontheresults of§13*2. Theconditions forconvergence are* [R{,jL+v+l)>R(\)>0, (a=b) itbeing supposed,asalready stated, thataandbarepositive. We shall firstsupposethat theformer conditions aresatisfied, andwe shall alsotake b<a.Theanalysisisgreatly shortenedbychoosing new constants a,/3,7definedbytheequations I2a=fjL+v-X+l, i\=y-a- /3, 7=I'+1,' j^=7—1. Itwillbesupposedthatthese relations holddown totheendof§13'41. Itisknown that Joi c-^+oJt since theintegralonthe left isconvergent;now,when chasanyassigned positive value, theintegralontherightisconvergentforcomplexvalues of6; wereplacebhyzandtheresulting integralisananalyticfunction of 2^when R(z)>0 and\I{z)\<c. Now] ^^6-* ^piz-,dt Jo [n>=o mir(y +m) J „,=om!r(7 +m)Jo provided thatf ^=0m!r(7 +m)Jo isabsolutely convergent;and itiseasytoshew that this isthecasewhen \z\<c. Hence, when 12^ |<c, (ia)«-^r(2a+2m) J,/ty-'^-^ ,„rom\r(ry+m)• (ft-+c-)'^+'«r(a-^+1) X2-Fi (a+m.h—/3—m;a—/S+1;a" ft-+c- *Itfollows from theasymptotic expansions oftheBessel functions thattheconditions R{lii+v+l)>R (\)--0 aresufficient tosecure convergence when a=b,provided that^-^zisanoddinteger. tCf.Bromwich, Theory ofInfinite Series, §176. 400 THEORY OFBESSEL FUNCTIONS[CHAP. XIII andthehypergeometricfunction ontherightmaybereplaced by* ,r(a-/3 +i)r(-i) cR^ /^_L_Li 1 /Q 3^M Now themoduli oftheterms intheexpansionof donotexceed inabsolute value thealternate terms intheexpansionof (1—\/x)~'^-'^"\ whereAisthegreaterof |2a ]and12/3-11; and, similarly,the moduli oftheterms intheexpansionof oi^i(a+7?i+I,1—/3—m;f;a?) donotexceed inabsolute value thealternate terms intheexpansionof (1-V*-)"^"^"'"VVa;- Hence theterms inthe infinite series which hasbeen obtained donot exceed inabsolute value theterms oftheseries *(-)>»( i^)Y+2»i-i {laf-^ r(2a+2m) Lir(l-/5-m)r(a +m+l)| + where x=c'^l{a^+c-).But this last series isabsolutely convergent when I2: 1<V(a^+C-)—c,and itrepresentsananalytic function ofzinthisdomain. Hence, bythegeneral theoryofanalytic continuation, _V(-)""(W+^'"-'' (lay-^ r(2«+2m) providedthat zsatisfies thethree conditions R{z)>0, \I{z)\<c, \z\< ^{ar+c=)-c. Now takeCtobeapositive number sosmall that 6<V(a-+C')-G, andtake <c^C,sothat also h<s/(a-+c")-c. *Cf.Forsyth, Treatise onDifferential Equations, (1914), §127. yi 13-4] INFINITE INTEGRALS 401 Then inthelastintegralformula wemaytake z=h,andwhen thishasbeen done, ifwensefonctions majorantes justasbefore, wefindthattheresulting series has itsterms lessthan theterms ofanabsolutely convergentseries r(|)(i-.v/X)-^-^- ^(_)m (i5)Y+2m-] Qa)--^r(2aH-2m) r=owITT(7+m) (a'+C'Y^''-'^ r(1-/S- III)r(a+m+\) I +V(h-^-m)T{a +m) where .Y=(7V(rt= +02). Hence, bythetestofWeierstrass, theoriginalseriesconverges uniformly withrespecttocwhen <:c^C,andtherefore thelimit oftheseries when c-* isthesame asthevalue oftheseries when c=0. Wehave therefore provedthat 00 „Zom\r{y +m)a^''+^'^ V{a-j3+1)' >2^ andtherefore J,.p(at)Jy_i{bt) ^^^ (_)m5v+2m-i r(2a+2m)V(i) _V „,ro'//i!r(7+III)2--^+v+2m-i (^a+^+2m i^(1_/5_,M)r(a+m+A) Ithastherefore beenshewn that ^^^Jo ^r-a-^"^^ 2v-«-^a"+^r(7)r(l-/3)-- ^V'^'^'aV' that istosay f^JAat)JAbt) bTilfi+h^-h^ +l) ^'^- Jo^^ 2^a-^+^ l^(z.+])r(i\ +*/.-iz. +i) providedthat <6<a,andthattheintegralisconvergent.This istheresult obtained bySonine andSchafheitlin. Ifweinterchange aand b,andalso/u,and v,throughoutthework,wefind that,when <a<bandtheintegralisconvergent,then ...[-J.-,{at)Jy-r(bt)a<^-Pria) ^'^^J. iv-«-^" "^~2v-«-^6-^'^->+U'(7-a)r(a-/3+l) Xoi^jfa,a-7+1 ;a-/^+1; ^,j- \y.B.F.26 402 THEORY OFBESSEL FUNCTIONS [CHAP. XHI j» Now itsohappensthattheexpressionsontherightin(1)and(3)arenot theanalyticcontinuationsofthesamefunction. There isconsequently a discontinuityintheformula when a=h;and itwillbenecessarytoexamine thisphenomenoninsome detail. 13*41. TJte critical caseoftheWeher-Schafheitlin integral. Inthecaseoftheintegral nowunderconsideration, when a=b,wehave, asbefore, .'ot c^+QJa t^ assumingthatR{im-\-v ^I)>R{\)>0, tosecureconvergence. Now consider ._,,Ja-(^ (at)Jy-^ (at)^ty-^^'^^' where ^isacomplexvariable withR(z)positive. WhenR(z)>2awemayexpand theintegrandinascending powersofa andintegrate term-by-term,thisprocedure being justified bythe factthat theresultingseries isconvergent. Wethusget,byusing §5'41, /,"^ ^1JJg-p (at) /.y-i(at) CO l*3C «=0 J Xe-^t(_)m (^^),-p+y+2;»-l^2a-Hom-l T(g-/g+ry+2m) m\r(a-^^-Hm+1)T(7+m)T(a-yS+7-hm) ^^(-)"^(^ay-^^y^-^-' V(2a 4-2in)T(a-yg+7-h2m) ,„t z''^^^^m !r(a-y8-I-m-h1)r(y -h7/i)T(a-/3-F7+m)" Now theintegralonthe left isananalyticfunction ofzwhenR(z)>0, and soitsvalue, when zhasthesmallpositivevaluec,istheanalyticcon- tinuation oftheseries ontheright. But,byBarnes'theory*, theseries ontherightand itsanalytic continua- tionsmayberepresented bytheintegral (^a)«-^+y^-i r(2a -f2s)r(a-/3-f-7+2s) —r2lTi] _V(-s)ds;^.^,a+,sr(a-/y-F 6-+1)r(7-hs)r(a-/3-h7+s) andthisintegral representsafunction ofzwhich isanal3'tic whenarg^^|<tt. Itissi:pposedthatthecontour consists oftheimaginaryaxiswithloopsto ensure thatthepolesofV(—s)lieontherightofthecontour, while thepoles ofr(2a+2s)andofT(a-yS+7+2s)lieonthe leftofthecontour. When \z\<2a wemayevaluate theintegral bymodifyingthecontour so astoenclose thepolesonthe leftofthecontour andevaluatingtheresidues *Proc.London Math. Soc.(2)v.(1907), pp.59—118. Seealso §§6'5,6-51svpra. 13-41]INFINITE INTEGRALS 403 atthem. Thesum ofthese residues forms twoconvergentseriesproceeding inascending powersofz;hence, whenR{z)> and |^ |<2a, r-_^^Jg.p (at)Jy^, {at) I=0(_yn(l(^)Y-a-^-m-l^m Y(ry-g--m)T(a+hu)" 2„^oviir{l-/S -hjn)r(7-a-im)T(7-/3-^m) 1^ (-)m (Ift)-"'-' ^Y-a-3+mp(^^^_^_ ^,^-)p(1^ 4.|^_1^+Ij^i^ "^2,„':o »i!r(^a-|/tf-|7-im+ 1)r(|/:^+|7-|a-i7Ai) r(ia^=4^ +*7-im)' NowR{y—a—,8)>0,and so,whenwemake zassume thepositivevalue candthenmake c^0,wededuce that ^ ^ Jo" ^^-"--^ 2r(i-/3)r(7-a)r(7-yS) providedthat R(a)>0, B(y-a-/3)>0. From theGaussian formula foroF^ia, /3;7;1),there istherefore nodiscontinuity inthevalue oftheintegral, thoughthere isadiscontinuityintheformula which expressesthatvalue ashincreasesthroughthevalue a. The resultmaybewritten inthealternative form (2)f^J^(-iJAA^dt (ha)^-i r(X)ra^, +y-p.+i,)~2ra,\+^,v-^fi +^)T(^x +^fi+hv+h)r^x+^/x-^v +hv providedthat R{fx+r+I)>R(X)>0. Iffji—visanoddintegertheintegral converges when0^R{'X)>—1; thiscasenextdemands attention. Weshallmake achangeinnotation bywriting a+pand a—p—1in placeof/jland vinthepreceding analysis;ifR(X)>0,wethen findthat " _^fJg+p(at)Jg-p-i (at)^^ /"'*^ 1r«i(la)-^+-'-' r(2a+2s)V{2a+2s-X)~ 2^'.' _xi2--+'^«-M" {a+p+s+l)V{a-p +s)T{2a+s)*^^^ _1^ {-y (|a)^-"»-i2'*^r(A,-m)r(ct-|\+|m)~ 2,„lomlT{p- hn+|X+1)T(-;j-hn+U)T{a+hX-^m) "^2„,timir{'p- \m+1)r{-p-Im)T(a-^m)' •26—2 404 THEORY OFBESSEL FUNCTIONS [CHAP. XIII andhence t^ 2r(p+^\+i)r{a +^\)r(^x-2J)' unless X=0.This should becompared with themoregeneral formulae obtained from§13'4,namely that,when h<a, (4)j^dt Y\2^1U-l\,-p-l\\ a-p: --, 2>^a''-P-^r{a-p) T{p+l\+l) and,when b>a, , fJg+p {at)Jg-p-i {bt), JO t 2^b'^+p-^+^r(a+p +l)T{lX-p)F,(a-l\p+l-^X; a+p^l;-^ Since X,^0,thefunctions ontherightin(4)and(5)donottend tolimits when a^b. Ontheother hand,when A,iszero, thecontourintegral becomes 1 /•=-*(^ay''+''-' [r(2a+2s)}-T(-s) 2^iJ-ooi ^'"+-*V{a+p +s+l)r(a-p +s)r{2a+s)*' andtheresidue ats=—ais(—)^/(2a). Itfollows that ' a^-PT(.-p).pl^^^['''-i'-'°-^^'a^)' (6)j^Ja+p (at)J„_^_i (bt)dt=\^_ypu2a), accordingasb<a,b=a,b>a.Since ,F,(a,-2r, a-p; l)=(-)PplT(a-p)/V(a), itisevident that tJievalueoftheintegrallulien b=aisthemeanofitslimits tuhenb—a—and b=a+ 0. The result oftaking X=lin(2)is dt_2sin^{v-^)IT (7)/J^{at)J,{ca)'~j= Jo't IT which isalsoeasily obtained byinsertinglimits inv$5"11(13);thisformula hasbeen discussed ingreat detail byKa^jteyn,Proc. SectionofSci., K.Akad. van Wet. te Amsterdam,iv.(1902), pp.102—103; Archives Neerlandaises, (2)vi.(1901), pp.103— 116. 13-42] INFINITE INTEGRALS 405 13*42. Specialcases ofthediscontinuousintegral. Numerousspecialcases ofinterest areobtainedb}^giving specialvalues totheconstants \, ju.,vinthepreceding analysis. Tosaverepetition,when three values aregivenforanintegral,the first isitsvalue forh<a,thesecond forh—a,andthethird forb>a;when twovaluesonlyaregiven,the first is thevalue forh^a,thesecond forh^a; andthevalues arecorrect forall values oftheconstants which make theintegrals convergent. Thefollowingarethemostimportant special cases* : "-- J^(at)J^{bt)^^^[i{h/aYlfM, Xr/^\•J, r/x~'sin{yu,arcsin(6/a)!, Ju.(at)sinbt,,.,V //J'^^ ^^^= "I«"si^i/^"^ ' [fji{b+V(6-^-a^),p•(1) (2)[Rif.)>0] (3)JAat)cosbt, |>-^cos{;.arcsin(6/a)}, --^-^7^at= -a"cosh/JLTT [[R(/x)>0] (4)IJ^(at)sinbt .dt= ,xorf^{b+ ^(b-'- a')]''' ' sinIfiarcsin(b/a)] v'(a-^-b') [R(f.)>-2] aJ"-cos}2f^7r y(Jf-a').{b +^(b-'-a')\'^' 'cos{/J,arcsin(b/a)] (5) J^{at)cosbt .dt=-Icew0, o/^sinh/xTT ^/(b'-a').[b+^/(b'-a')\'^ Specialcases ofprecedingresults are[R(p-)>-l] b-7,.• (6) (7)0; IJo(at)sinbt.dt= -jx, IJo(aOcosbt.dt= -Uo , 0. These twoformulae, which were given l>yWebert,JournalfilrMath. lxxv. (1873), p.77,areknown asWeheYs discontinuousfactors;theyareassociated with theproblemof determiningthepotentialofanelectritied circular disc;. *Numerous otherspecial eases aregiven byNielsen, Ann. diMat.(3)xiv.(1908), pp.82—90. Theintegralsin(4)and(5)divergeforcertain values ofixwhen a=h. tTheformer wasknown toStokes many years earlier, andwas, infact, setbyhim asa Smith's prize examination questioninFeb. 1853. [Math, andPhijs. Papers,v.(1905), p.319.] XCf.Gallop, Quarterly Journal, xxi.(188(J), pp.230—231. 406 THEORY OFBESSEL FUNCTIONS[CHAP. XHI Anotherspecialformula is (8)fJ^(at)J,., (bt)dt=\1/(26),.[R ifjC)>0] (O; and ifweputfi=l,weobtain Weber's result{ibid., p.80), (9) rMat)J,{bt)dt=\ll(2b)Jo U- 1/7 ,1/6. Theresult ofputting ;li=J^in(8)isknown asBirichleVs discontimiousfactor;seethe article byVoss, EncyclopadiederMath. Wins. ii.(1),(1916), p.109. Some otherspecial formulae havebeen found useful inthetheoryofFourier series by AV.H.Young, Leipziger Berichte,lxiii.(1911), pp.369—387. Another method ofevaluating (5)hasbeengiven byHopfandSommerfeld, Archiv der Math, v/iidPkys. (3)xviii.(1911), pp.1—16. Aconsequenceofformula(1)must benoted. When v>0,wehave, by§5*51(5), <»f^dt2e„^-. +„(.i-)=2^/JvHt)- 71=0 Jn r .'• t =1, andso (10) 1^.(^)1=$!, IJ.+i(aO1^1/^2, provided onlythat vhepositive;this isaninteresting generalisationofHansen'sinequality (§2"5)which wasdiscovered byLommel, Miinchener Abh. xv.(1886), pp.548—549. Thereader mayfind itinterestingtodeduce Bateman'sintegral *, [log (6/a),(11)f^J,{at)[l-J,{bt)]'^=^^' from theWeber-Schafheitlin theorem. 13*43. Gegenbauer's investigation oftheWeber-Schafheitlin integral. Inthespecialcase inwhich theBessel functions areofthesame order, Gegenbauerffound thatbyhismethod Weber'sintegralcould beevaluated inasimplemanner. IfR{'iv+1)>R (X)>R{v+h)wehave dt J,{at)J.{bt) {-lab)"I=^ /--J,[tV(a^+6—2abcos0)}.„., ,, ' 'sin-"cpdxpdtr{v+^)T{h).K Jot^-''{a'+b'-2abcos<p)i (abyT{v-IX+I)r sin-"(f>d(f) 2^r(J/+1)r(1)r(iX+i)j(a'+^-^-2abcos(/))''-*^+^ *Messenger, xli.(1912), p.101;foraproofoftheformula byanother method, seeHardy, Messenger,xlii.(1913), pp.92—93. tWiener Sitzimgsberichte, lxxxviii.(2),(1884), p.991. 13-43] INFINITE INTEGRALS 407 by§13'22.When h<a,theexpressionontherightis Nowfrom therecurrence formulae ^1(1-z^)-h^C,,^(^z)]={n-v2,M- 1)(1-^•^)-i«-iC^_,(^), ^{(1-z'-f-+>^ G,,^{z)]=-{n+1)(1-.-)^-+'^-i r>,+,(^), weseethat (n+l)|'(\-z^y-i C^n^,{z) dz =-C(1-z"-)"-^^i''+^ ~\{l-s")i"+^CV(2)\dz ={n+2,x-2v-1)f'^(1-^-•^)''-*a„'^(5) dz J-1 ="2':;!t -/!/'-^=>"""'i{(i -.')-"'c,,(.)}rf. sothat G'^n-i (cos (f))sin-"(f)d(f).C^l+l (cos (/))sin-"(f)d({) ^(n+2fi-2v-l)(n +2fi-l) {2v+7i+l){n+l) Hence itfollows that X,fJv-l\+H-IX;v+l;^^Y andthisagreeswith theresult of§13-4. Themethodgivenhere issubstantiallythesame asGegenbauer's;but heusedilightly morecomplicated analysisinorder toavoid thenecessityof appealingtothetheoryofanalyticcontinuation toestablish theresult over themore extendedrangeR{2v+1)>-R(X,)>—1. Byexpandingthefiniteintegralinpowersofcos cf),weobtain theformula (1) I^./.(at)J.(bt)^=-,^^,^^^^_^,,, ^^^^^~ ^(2v+\-X2z/+3-\ -,4a-6- \ ^•^^^1—4'4'^+l;(a^T6^0' which isvalidwhether a>bora<b.This result wasgiven byGegenbauer, andwith thisform oftheresult thediscontinuityismasked. Thereader willfind itinterestingtoexamine the critical caseobtained byputting6=ainthefiniteintegral. 408 THEORY OFBESSEL FUNCTIONS [CHAP. XIII 13"44. Ouhlers investigation oftheWeher-Schafheitlin integral. Theintegral .-00 J^(at)/,(ht)dt Jo willnowbeinvestigated bythemethod due toGubler*. Itisconvenient first toconsider themoregeneral integral (-JAo^t)J,{ht)^^ -' t^ eventhoughthisintegralcannot beevaluated inasimple manner byGubler's methods. ItisfirstsupposedthatR{v)>0, R(k)>^,R(fM—X)>—I;and, asusual, aandbarepositive, anda>b. From thegeneralisationofBessel'sintegral, given by§6*2(2),itisevident thattheintegralisequalto Mv/I2-"-^expHhtlz- -]}-dzdt. 27riJo ^^ Wetake thecontour asshewn inFig.29tomeet the circle j^ |=1andthe FiK. 29. lineR(z)=onlyatz=±i;andthen, forallthevalues ofzand tunder consideration, R{hbt(z-l/z)]^0; andtherepeated integral converges absolutely,since Joidt0^ ,-v-ldz\ isconvergent. Theorder oftheintegrations maytherefore bechanged, and wehave Jj,(at)J,{bt) t^ IfwewritedtLrr'r^M,,^i.27rij-COj t''H'-i)y—V—ldtdz. b(z-l/z)=-a(^-l/0, Math. Ann. xlviii. (1897), pp.37—48. 13-44] INFINITE INTEGRALS 409 andsupposethatthatvahie of^istaken forwhich j^j^1,wehave, by§13'2, dt=—~. z Xoi^iU-X+1,/i+X:/i+1 ;j-^-^ojc^^, b\^Kummer's transformation*. Next consider thepath describedby ^,when zdescribes itscontour. Since thevalue of^with thegreater modulus ischosen, thepathisthecurve on therightofthecircle inFig.29;andthecurve isirreducible because different branches ofz,quafunction of^,aretaken onthedifferentjmrtsofit.The curve meets theunit circleonlyate-'",where wistheacuteangleforwhich b—asin co. Nowboth theoriginal integral andthefinalcontourintegralareanalytic functions ofXwhenR{X)>—1,solongasa^^b. Hence wemaytakef \=0, provided thatR{/x)>—1;andthenwehave Next write^=zrandthen •2=br+a_T(bT+a) .~ T(ar+6)' tiT+b' andthetcontour isthatshewn inFig.30;itstarts from-b/a,encircles the origin clockwise, andreturns to—b/a; where thecontour crosses thepositive halfofthereal axis,wehaveargr-0. c,. dz 1adrbince ,0(l+n 2T(6T +a)' wefind(onreversingthedirection ofthecontour) that I"'/^(at)J,(bt)dt=^r~. f'"^'ri'-'-'^-i' (br+a)-i"'+'^+i> (ar+6)*c+'--i» dr J ZttI J-1,1a b" /'<o+' / b-x-iiv+ft+ii 27r?'o''+\' _i Va *Journal furMat Ii.xv.(183G), p.78,formula(57). SeealsoBarnes, Quarterly Journal, xxsix. (1908), pp.115—119. tIfX:^0,thehypergeometric function doesnotingeneral reduce toanelementary- function, andtheanalysis becomes intractable. 410 THEORY OFBESSEL FUNCTIONS [chap, xni Ifweexpandinascending powersofh-ja-andsubstitute thevalues ofthe Euler-Pochhammerintegrals, then Gubler's result //x+i^+1V-ii+i ^A-^^- ismanifest. Fig. 30. 13"45.Amodification oftheWeber-Schafheitlin integral. Theintegral .'0K,(at)J^(ht) ^^^ z whichconvergesiiR{a)>\I (b)\andR{v+1—\)>\R (fi)\,isexpressiblein terms ofhypergeometric functions, liketheWeber-Schafheitlinintegral,but unlike thatintegralithasnodiscontinuity when a=b. Toevaluate it,expand J^(bt)inpowersof6,assuming temporarilythat 16 j< ja jinorder that theresult ofterm-by-term integration maybea convergentseries. Byusing §13"21 (8)itisfound that ^^ Jot" n=on\l {v+n+l)Jo^ ' 6"r(Ay-^x4-^At+^)r(^1/-1\-i/i -I-i) '-<-X,4-;u,-|-l y—A-—/i+1 2v+\\-b- a- and, inparticular, (2) providedthat J?(i/+1)> ,i2(/a)jandR(a)>I(b)\.f>(«o^(.o..^-^-.e= ^^°>'<^^^pr^^). Formula (1)wasgiven byHeaviside* whenfj,=v=and A,isand—1. *Electromagnetic Theory,iii.(London, 1912), pp.249,268, 275. 13-45, 13-46] INFINITE INTEGRALS 411 13"46. GeneralisationsoftheWeber-Schafheitlin integral. Toobtain thevalues ofintegrals containingthree Bessel functions under theintegral sign,take theintegral f'^J^{at)JAbt)^^ Jot'^ replacebby^/(b-+c--2bccos(}>),where band carepositive, multiply by sin-"(f)i(b"+c--2bccos(f))^''andintegrate.Itisthusfound that [^J^{at)J,{bt)J,(ct )^^ (hbcY r»[^J^{at)J,{zTt).,,, , I, t^^^^= r(.+'i)r(i)Jo Jo—^t^~-«^^-''^#^^^' where^=s/{b-+c--2bccos(f)); andtheintegralontherightisabsolutely convergentif R(v)>-^,R(fM +v+2)>R{\ +l)>0. Changetheorder oftheintegrations ontheright;then theresult ofthe integrationwithrespecttotisanelementaryfunction of-arifX-i- 1'+1=+yu-, bytheformulae J^(at)J^(^t) Itfollows that (1) J^{at)J,{bt)J„(ct)t'-''(H J ^ (UcY inwhich thevalue ofAis-^r(.+ i)r(i) ly^'-^'-''"-^'''''^^'~'-' ^"^"^^^^' 62+c^-a- 0, arccos^ ,ir 26c accordingasa-islessthan, between, orgreaterthan thetwonumbers {b-c)\{b +cf, providedthatbothR(fj,)andR(v)exceed —i. Inparticular (2)j^/.,, (at) ./.(60/.(ct)I?= ^^^,^\^'^,^ ^^^^j^^sin-</>c^</.. MultijDly by(1"+'anddifferentiate under theintegral signwithrespectto a ;andwethen obtain theinterestingresult that, ifR(v)>— -J-, dt 2"-!A^"-! (3) 1^•^'<««-^'<*«-''(^')j.-.-(„io)-r(. +i)r(i)- when a,b,carethesides ofatriangleofareaA;but ifa,6,carenotsides ofatriangle,theintegraliszero. Thisformula isduetoSonine, Math. Ann. xvi. (1880), p.46;otheraspectsofithave been investigated byDougall,Proc.Edinburgh Math. Soc.xxxvii.(1919), pp.33—47. 412 THEORY OFBESSEL FUNCTIONS [CHAP.XIII Ithasbeen observed byMacdonald* thattheintegralonthe leftin(1)is always expressibleinterms ofLegendrefunctions. Theexpression maybe derived from theintegralontherightinthefollowingmanner: When a,b,carethesides ofatriangle, bythesubstitution sinI<^=sin^Asin6 wehave rA (a--h--c-+2hecos(f>y-''-^sin-" </>d(j> .A=(2bcy-'-^I{cos (f>-COSAy-"-^sin-"(j)d(fi Jo rhTT=2-''-'^''-^bcy-''-'sin-''-'hA.(l-sin-| Asin-^)''-Hin-"6'cos-'^--'-i^^6' JO =i'^-'{bcy-^-'sin''^-' ^A^^'''^f^^^~''\ ,F,(^+vA-i']f^ +^;sinnA), i(i^+i) and therefore, ifR(yu,)andR(v)exceed —^,and a,b,carethesides ofa triangle, wehave (4)j^J,(at) ./.(60-L(ct)t^-^dt= -^^:^^,,,, P:,->^^^)• If,however, a^>(6+cf,andwewrite a-—b'-—c-=26ccosh S^-, wehave I(a--6--c=+ 26ccos</))'^-"-isin-"^(Z(/)Jo =(26c)'^-''-iI"(cosh S4+cos</))'^-'-isin-"</)c7<^ =(260cosh^V-'-'^;f^-fv-1(I'M- 1) X.,F,{^^^+1,'—f^;»^+1;sech-^^), sothat,when a->(6+c)-,wehave /-\rT/^\r/j^N r/^x^i 7^(6ey~^cos^'7^. sinh'*-*c^^.i-j^. ,^.^ (o) J^{at)J,ibt)JAct)f-''dt=^-^, Q,(cosh^). Inlikemanner, wededuce from^13*45(2)that (6)j^K(at)J.(bt)J.(ct)t^»dt=- --^-j^^-i^-j-^-Q^_^(A ), where 2bcX=a- -\-b-+ c-;and inthisformula a,b,cmaybecomplex, provided onlythatthefournumbers R{a±lb±ic) arepositive;thisresult isalsoduetoMacdonald. *Proc.London Math. Soc.(2)vii.(1909), pp.l-i2— 149. 13-46] INFINITE INTEGRALS 413 [Note. Theapparent discrepancy between these fornnilae andtheformulae ofMac- donald's paperisaconsequenceofthe difi'erent definitions adoptedforthefunction Q^"^ ; see§5-71.] Other formulaeinvolvingthree Bessel functions maybeobtainedbytaking formuhx§11"G(1),replacingzbyx,multiplying by 2/p{xcos6)1x'^ and integrating-. Itisthusfound that /^. . dx (7) ./^(«cos^cos(I>)/^(«sin(^sin4))Jp(«cos^)—^jJo"^ . _cos'^ cos'^Osin"6sin"4>cos''6= ^W-'-Tip +l)[V{v +l)Y^ X00 n= r(^a+|/'4-ip-lX +n+1) X2F,(^111+v+p—\p—\— /x—v ———+»+1,^~ n;p+l;cos- whenX2-^1(—n,fx+ i>+n+l;v+1:sin-(/>) X2F^{-n, /j,+v+n+l;v+I;sin-^) R(/i+V+p+2)>R(\)>-^ andcos isnotequalto+cos(^±0). Somespecial cases ofthisresult havebeengiven byGegenbauerinaletter toKapte}-n, Proc. SectionofSci., K.Acad, canWet. teAmsterdam.,iv.(1902), jip.584—588. Some extensions offormula(3)havebeengiven recently byNicholson*. Iftti, rto,...a,„,arepositivenumbersarrangedindescendingorder ofmagnitude itiseasy toshew that, if then (8)dt . 71=1nJJaJ) ^-=0; 1=1t thesjjnplest method ofestablishingthisresult isbyinduction, bysubstituting Gegenbauer'sformula of§11"41[ontheassumptionthatR{i')>—h]for ./^((/„i_ir) J,.{a,„t), andthenchangingtheorder oftheintegrations. When «!, a.., ...«„iaresuch thattheycanbethelengthsofthesides of apolygon,theintegralisintractable unlessm=8(thecasealready considered), or III=4. *Quarterhj Joiinud, xlviii.(1920), pp.321—329.Some associated integrals willbediscussed in i^13-48; 414 THEORY OFBESSEL FUNCTIONS [CHAP. XHI Whentti, a.2,as,a^canfoi'm thesides ofaquadrilateral, wewrite 16A-=IT(«!+Oo+tts+ttj— 2a,i), 71=1 sothatAisthearea ofthecyclic quadrilateralwith sides a^,a.2,a3, a4. Theintegralcanbeevaluated inasimpleformonl}-* when v=0:but todeduce itsvalue, itissimplestfirst toobtain anexpressionfortheintegral when R(v)>|,anddeduce thevalue forv=byanalyticcontinuation: the value oftheintegralassumes different formsaccording asf th+a4$a2+ tts, i.e.accordingas A-$ajcioa^a^. We write ct-=a.;-+o/—2a2^scos^,andreplace J^,(a.,t) J^(aJ) by Gegenbauer's formula, sothat^ Jo^^^^^^^¥^=V\^^\)V^, Jo^^sm-c^^^^ _(a,asr(a^a,r;^f ,^y._^.i.-j r^._ ,^_„yi.-*sin^<A# where thelower limit isgiven by•57=ai—a^andtheupperlimitbyot=aj+a^ oraa+«3•whichever isthesmaller. Wewrite Ts-- (o,-«,)- (fli+a4)-- (ao-^3)-' sothattheupperlimit for a-is1orA\\i{a^a.2asa^\thisexpressionwillbe called\\k. Wenowcarryouttheprocessofanalyticcontinuation(unless aj+04=00-^ a;,, when theintegrals divergeattheupperlimit if t-=0),andweget InJf)(ant)tdt«=i =-.I[[(«i+"4)'-^']{^'-(Oi-a,y] [zT-^-(a,- 03)-} {(rt2+fls)'^-t^i]-^t:r(^^ TT".' 1rlorl/i- £;^.I Hence 1 j^Njaia-iOzCii)rx 4Itt^A VA (9) InJo(«nO^C?i= n=\K .TT^/^^a-.a^asai) \y/(aia20sa^)/' whereKdenotes thecomplete elliptic integralofthe firstkind,andthatone whose modulus islessthanunityistobetaken. *Forother values ofvitisexpressible asahypergeometric function ofthree variables. tWe stillsupposethat«j^«._,^ 03^04. 13-47] INFINITE INTEGRALS Nicholson hasalsoevaluated415 {JAat)Y^, whenR{v)>0 anda>0.Thesimplest procedureistoregardtheintegralas aspecialcase ofthe last, sothat itisequalto 1 1"in «,-, ,M isin'-''(j)(16 andhence* (10) ^13'47. I^/^ediscontinuousintegrals ofSonine andGegenhauer. Several discontinuousintegrals,ofamoregeneralcharacter thantheWeber- Schafheitlintype,havebeeninvestigated bySoninef andGegenbauer;!:;some modifications oftheseintegralsareofimportanceinphysical problems. The firstexample^which weshall take isduetoSonine, namely (1)\'j.<fit)^^^'^'t^»dt^Jo'^^ (^2+^-)-" 0, («.<6) Tosecureconvergence,aand baretaken tobepositiveandR(v) >R(fx,) >—1; ifa=b,thenwetakeR(v)>i?(/x+1)>0.Thenumber 2isanunrestricted complex namber, andtheintegralreduces toacaseoftheWeber-Schafheitlin integralwhen ziszero. Theintegralsinvolvedbeing absolutely convergent Ij,weseefrom§6'2(8) that,if 0,then j^Xbt^^^t^'^t^-^dt 27rm'^+i' _„;J^{bt)f'^'u-"-'exp (o-— b'-)uala{u—t^+2' ududt uexp•la liudu. *Anarithmetical error inNicholson's work hasbeen corrected. Theresult forvalues oiR(v) between andJisobtained byanalytic continuation. tMath. Ami. xvi.(1880), p.38etseq. XWiener Sitzungsberichtc, lxsxviii. (1884), pp.990—1003. §This formula isalsoinvestigated byCailler, il/e'm. delaSoc. de2^hys. deGeneve,xxxiv. (1902—1905), pp.348—349. IITheconvergenceisabsolute onlywhenR{v)> R(ijl+1}>0;forvalues ofvnotcoveredb}' thiscondition, theformula istobeestablished byanalytic continuation. 416 THEORY OFBESSEL FUNCTIONS [CHAP.XIII When a<hthecontour involved inthelastintegral maybe deformed intoan indefinitely greatsemicircle ontherightoftheimaginary axis,andthe integral alongthis iszero; but,when a^6,wehave toapply §6"2(8),and thenweobtain theformula stated*. Arelated integral (2)/;J.(^o^i^fy<-cu=%|^')f-'-v^. w(«=+6=)! maybeevaluated inasimilar manner. Wesupposethataandharepositive f,andthatR(/x)>—1;^\in4ivaluating theintegralitisconvenient tosupposethat!arg2'!<i7r, though wemay subsequentlyextend therangeofvalues ofzto\a.rgz\< ^ttbyanalytic continuation. From§6*22(8)itfollows thattheintegralonthe leftof(2)isequalto t-+z-~ ^Jo .0—ha[u+ S^^lo'''^"""^Pdudt duu {a-+¥)uaz' 2a '2u by§6'22(8);andthis istheresult stated. Nowmakeargz^^±\ir.Ifweputz=iy,wherey>0,wefindthat Jo \^yJ' bi^{\/(a-+62\)V-/X-1=i7r.-i-''--i''- l^^^^^^l[/.-.-. {yV(a^+6^)i-tF._,_, {y^/(a^+¥)}l providedthatR(v)<1 ;and itissupposedthatthepathofintegrationavoids thesingularityt=ybyanindentation above thesingular point,andthat interpretationisgiventov'(i-—y-)which makes theexpression positive when t>y. Ifwehadputz=-iy,weshould havehadtheindentation below thereal axisandthesignofiwould havebeenchanged throughout (3). Inparticular (A\rT(ht\exp{-«\/(^— .V^) ]. ;._exv\+il|^l{a^+b^)] ^^^ ]^"^^^^—w^^'— '— 7u,^:rb^r~ where theupperorlowersignistakenaccordingastheindentationpasses above orbelow theaxisofy. *Forphysical applicationsofthisintegral, seeLamb, Proc.London Math. Sac. (2)vii.(1909), pp.122— lil. tWith certain limitations, aand bmaybecomplex. 13-47] INFINITE INTEGRALS 417 The lastformula (withthelowersign*)hasbeen used inphysical investigations by Sommerfeld, A7m. derPhjsikunciChemie, (4)xxviii.(1909), pp.682—683;seealso Bateman, Electrical andOpticalWave-Motion (Cambridge, 1915), p.72. Ifin(1)wedivide by6'^andmake h-^i),weobtain Sonine's formula providedthatfOO andR{\v—\)>R{iju)>—1;thismighthave been establishedindependently bythesame method. Similarly,from(2)wehave ifa>andR{ix)>-I. In(5)replacevby2v,aby2sin6andintegratefrom ^=to6=^tt.It follows that n\rJ-''Ml±^^ rwdt= ^AtL±J:} |-^^^..-M-:(2^sm^)^^ . '^^ Jo '{t''+z''-yTT^^-'-'^-i.'o sin^+^d* this isvalidwhenR{v—V)>Rijx)>—1. Theintegralontherightiseasily expansibleinpowersof^:;buttheonly caseofinterest iswhen 2/-= 2//-+3,andwethenhave (S^ f^^^!M^l+fl)} ...-.^.-^^^-i)H(^z^^^^ Jo^if-\-z^* '^^-2z^^7^^'^-''^' sothat (9) I""^^Y ('^^^-^"y-^da=l-'^!rPp H.(2z) ; andthese arevalid ifR(v)>hThe lastformula was established ina different manner (when y=l)byStruvef; andfrom itwededuce theimportant theoremthatij:,luhenv>^ andx>0, Il^{x)ispositive.Struve'sintegralis ofconsiderable value intheTheoryofDiffraction. Some variations ofSonine's discontinuousintegralareobtainable by multiplying byh'^'^^andthenintegratingwithrespecttobfrom tob. Itisthusfound that theupperlimit inthelastintegral beingbora,whichever isthesmaller. *Mythanks arcduetoProfessor Love forpointingouttomethedesirability ofemj^hasiziiig theambiguity ofsign. tAnn. derPhijsikniul Clicmic, (3)xvii.(1882), pp.1010—1011. +Cf.§10-45." w. v..F. 27 418 THEORY OFBESSEL FUNCTIONS[CHAP. XHI Ifi<a,theintegralontheright seems intractable, but,whenb>a,we putu=asin6anddeduce that (10) rj,,,(tt)'^^^^M^t^dt=^^^^^M,^ ^Jo {t'-hz"')^" b^+^ z"' providedthatR{v+1)>R(/u,)>—1;this isoneofSonine'sintegrals. Ifwereplaceabyi*in(1)andthen takea^bandintegratewithrespect toufromato00afterdividing by «<-""',wefind that, tvhen zisrestricted tohe positive, u^"-^ 2"-"-' Jo (v^+b-)" by§13'3(4),andthence weseethat (11) )^^m(^«; (^2_,.^2)j.+if «^^ 2^-'r{v)'^^ ^' providedthat a<handR(v+2)>R(fj,)>-1;therestriction that ^is positive maynowberemoved. Formula (10),whichmaybewritten intheform (12)]/^^^^) it'^z^)^^tat- ^^ ^^, where i^(v+2)>J?(/a)> and6>a, hasbeengeneralisedintwoways by Gegenbauer*, bytheusual methods ofsubstitutingNeumann'sintegraland Gegenbauer's integi'al (cf.§13-1)forthesecond Bessel function. The firstmethodgives (lo) JJ^(6«) (^2^^2^i(A+W^^^ = ^io io^'^(^^^-^^M-T^iM^-^^^^cos(X-.)<^#rf^ 2>^-a^(/A)/^(a^)JA(a^) providedthat 6>2aand 22(i/+X+f)>i2(/i)>0. *Wiener Sitzungsberichte,lxxxviii.(2),(1884), pp.1002—1003. 13-48] INFINITE INTEGRALS 419 If -S3-=\J{a"+c-—2ttccos(/)),thesecond methodgives rCi'+DrCDJo Jo^ -ar"(f-+2=')i''^^ ~ h>^ z" z"' ifb>a+candR(2v+^)>R{/j.)>0. Byinduction itfollows that, ifh>Sa, 'Jjaz)' z"(1-5) Jjbt)^— , -^ f^-'dt^ /^^^n where theproduct appliestonvalues ofa,and R(nv+In+^)>R{fM)>0. Iftheinduction ofthesecond method isused after applyingthe firstmethod once,we find stillfurthergeneralisations. Thespecial case of(15)when 2-9-O is (16) f^^,(50n[./.(«0]^—^^^=~-l~^^^[^)\-' thishasbeenpointed outbyKluyver,Proc. SectionofSci.,K.Akad. van Wet. teAmster- dam,XI.(1909), pp.749—755. 13'48.Theproblem ofrandomflights. Aproblemwhich waspropounded byPearson*(inthecase oftwo-dimen- sionaldisplacements)isasfollows : "Aman starts from apointandwalks adistance ainastraightline; hethen turnsthrough anyanglewhatever andwalks adistance (/inasecond straightline.Herepeatsthisprocessntimes. "Irequiretheprobabilitythat after these nstretches heisatadistance between randr+Zrfrom hisstarting point,0." Thegeneralisedform oftheproblem,inwhich thestretches maybetaken tobeunequal, saya^,a^,..., ctn,hasbeen solvedbyKluyverfwith thehelpof thediscontinuousintegralswhich were discussed in§13*42; andsubsequently Raylei^i gavethe fulldetails oftheanalysisoftheproblem (which hadbeen examined somewhatbriefly byKluyver), andthenobtained thesolution ofthe corresponding problemforflightsinthree dimensions. Ifs,„,istheresultant ofa^, a.,,...,a,„(m=1,2,...,/? —!),and ifd,„isthe *Nature,lxxii.(1905), pp.294,342(seealsop.318); Drapers' CompanijRe.searcli Memoirs, Biometric Series, in.(1906). tProc. Section ofSci.,K.Akad. vanWet. teAmsterdam, viii.(1906), pp.341—350. XPhil. Mag. (6)xxxvii. (1919), pp.321—347.[Scientific Papers,vi.(1920), pp.604—626.] 420 THEORY OFBESSEL FUNCTIONS [CHAP.XIII angle between s^andcim+i, then, inthetwo-dimensionalproblem,allvalues of theangle 6,nbetween —ttand ttareequally probable. Now letPni't^'; ch,«2>•••)«m)denote theprobabilitythat after )istretches thedistance from thestarting pointshallbelessthan r,sothattheprobability thatthedistance liesbetween rand r+Sris dPn(r; a^,ao, ...,an)^ dr Itisthen evident that Pn(r;a,,a.2, ...,an)= .-^-^;^i||••• /Id^n-i dOn-o...dO,dO^ , where ^i,^o,...,^„_2assume allvalues between —ttandtt,while 6n-i isto assumeonlysuch values asmake* Sn^r, foreach setofvalues of^i,^o,...,dn-2- Now(§13-42) rrj,(rt)j,{s,,t)dt=\l' ^';<;;^Jo [0, {Sn>r) and so,ifthisdiscontinuousfactorisinserted inthe(n—\)-twple integral,the rangeofvalues of6n-imaybetaken tobe(—tt,tt). Wechangetheorder oftheintegrationswithrespectto^,j_iandt,and, rememberingthat S"ii=Sji_2+ft'H ^S)i_i ft,jcost/ji_i, weget Ji(rt)Jo{Snt)dtddn-i='27rr \J,(I't)Jo(s„_i t)Jq(a„t)dt .-TT.'O .0 by§11"41(16).Wenextmake thesubstitution S'n—i^^S",i_2Tft'»i_i ^5)j_2 ftji,— 1cosC/ji_2, andperformtheintegrationwithrespecttodn-2-Byrepetitionsofthispro- cesswededuceultimatelythat r--c n Pn(r ;fli ,«o,...,an)=r \J,(rt)UJ^{a,n t)dt, .' M=1 andthis isKluyver'sresult. W^eshallnowconsider thecorresponding problemforspaceofpdimensions. Inthisproblemitisnolongerthecasethat allvalues of6„iareequally likely. Ifgeneralised polar coordinates (inwhichd,,,isregardedasaco-latitude)are used, theelement ofgeneralisedsolidanglecontains 6^onlybythefactor sii\P~^ 6^1d6m>and 6,nvaries from tott.Thesymmetrywithrespecttothe polaraxisenables ustodisregardthefactordependingonthelongitudes. *Itistoberemembered that5„iisafunction ofthevariables di,6^,...,^^-i• 13-49]INFINITE INTEGRALS 421 IfP„(9'; ai,ao, ...,a^Jj)denotes theprobabilitythatthefinal distance is lessthan r,wededuce, asbefore, that P„(r;a„a.,...,an\jy) [i{-kp-i)1W) .'c.'o .'0.'W=] v/here theintegrationwithrespecttoO^-iextends over thevalues of^„_i which make s„<r. Thediscontinuous tactoi- which wenowintroduce is andthen, since, by§11'41(16), weinfer that F,,(ra, ,a,,...,a, |p)=r[Ti^Wr^|J(irtP'^ J,,{rt)U^^i^^^]'^'- When thedisplacements a^,cu, ...,«»areallequaltoa,andnislarge,wemay approximatetothevalue oftheintegral byLaplace's* process. Theimportant partoftheintegrandisthepartforwhich tissmall, and, forsuch values oft, sothat(§13-3) P, (7-;a,a,..., a \j))~jT^/J(^rt)i'-' Jip(rt)exp(-'-|^')dt Thisprocessofapproximationhasbeen carried much furtherbyRayleighin thecasesjj=2,jj=3,while Pearson haspublishedvarious arithmetical tables connected with theproblem. 13'49. Thediscontinuous integrals ofGallop andHardy. Theintegral ^ {z+tr (x+ty isconvergentifaand 6arepositiveandR{/x+i')>-I;when a=bthelast condition must bereplaced hyB.{fi+v)>0. Thespecialcase oftheintegralinwhich/x=0,i'=^hasbeen investigated byGallop, Quarterly Journal, XXI. (1886), pp.232—234; andthecase inwhich a=bhasbeen investigated byHardy,Froc. London Math. Soc.(2)vii.(1909), pp.469.Theintegralis obviouslytobeassociated with thediscontinuous integralsofWeber andSchafheitlin. *Latheorie analytiquedrsProbabilites (Paris, 1812), chapterin.The process may be recognised asasomewhat disguised form ofthemethod ofsteepestdescents. 422 THEORY OFBESSEL FUNCTIONS [CHAP.XIII Toevaluate theintegralinthegeneral case, themethod discovered by Hardyiseffective; supposethata^b, andatfirst letustakeR{v)>—^, R{/jl)>^,sothat Poisson'sintegral maybesubstituted forthesecond Bessel function and alltheintegralswhich willbeused areabsolutelycon- vergent.Write tinplaceoit+^,and letz—^=Z,sothattheintegraltobe evaluated becomes -r(.+1)r(i)11.C~izv^'^^^^^''''"^^''''''^^^^^ 2.(16)" [^ ^ '''^''^cos{btcos(ji)cos(bZcos(f>)sm''^(f>d(l,dt{Z+tyt" (16)" f"[-J,{a{Z +t)] r(i;+i)r(i)Jo .0t>^ = i2aYr(,W^TV^) /o^''-''^"^^^'-''''^'''^'^)^^'^'^' byaspecialcaseof§13"4(2). Thisintegralisexpressibleinasimple manneronlywhen/i=:|,acase considered byGallop,orwhen a=6,thecaseconsideredbyHardy. WeeasilyobtainGallop'stworesults (1) f^^^"^^^^ Mbt)dt='7TUbz), {b%a) J—00 Z-TI f"sina(2^ + r/7.^ 7.f^cosuz .du,, andHardy'sformula ...rJ.{a(^+1)]J.[a(^+t)] ^r(;^+.)r(i) ^^^J-. {z+tr i^+ty r(/.+i)r(^+i) ^aj (z-^y+^'-h Thereader willfind itinterestingtoobtain(1)byintegrating gai{z+t) /'Joi^i)dt z-k-t round thecontour formed bytherealaxisandanindefinitely great semicircle above it;it hastobesupposedthatthere isanindentation at—2when zisreal. Theintegi'al J_3o Z+t hasalsobeen consideredbyGallop. Toevaluate it,weobserve that *1-' z+t z+t' 13-5] INFINITE INTEGRALS 423 andsotheintegral maybewritten intheform ro /"» {-sina{z+1)]Jo{bt)dt+ sina(^+1)Jo(^0f^^ j-00 Jo +J"sin«(£_M) JJ^^^_2J• sin«(^+i) j^^^^^^ .'-00 ^+^ Jo Z+t rCO raraj=2cosaz sinaiJq(6^)fZ^+^ cos if{z+t)jQ (bt)dtdu Jo . .'-00 ra /"CO—2z\ cosu(z+t)Jo(bt)dtdu .'o.0 Too raC'^=2cosa^ sina^Jo(6^)dt+2z jIsin «2;sinutJq(6^)cZ^cZm. Jo JJ Hence, whena>b, r*\t\sma(z +t)J., 2cosaz /"«sinit^ ("i)—^ ^^ Jo{bt)dt= ——^—Y-+22--——j^dit J-^ z+t^ ^V{a-- b-) JiV(w'-b) 2cosaz ^rarecosh«/6 _,^,,^ = V(^^6-^)-^-"Josm(^6cosh^)rf^, but,when a<b, /-\ ["^\t\sma(z +t) ,;,.,, ^(o)j^ j~^^'-Jo(bt)dt=0. 13-5. Definite integralsevaluatedbycontourintegration. Alargenumber ofdefiniteintegralscanbeevaluatedbyconsidering integralsoftheforms ^. j(z)H^^'^iaz) dz,^.U(z) '^^(bz)JZ,») (az) dz, taken round suitable contours; itissupposedthat4^{z)isanalgebraic function, andthataispositive. Theappropriatecontours areoftwotypes.Wetake the firsttypewhen {z)hasnosingularities except polesintheupper half-plane;thecontour is taken tobealargesemicircle above therealaxiswith itscentre attheorigin, togetherwith thatpartofthereal axis(indentedattheorigin)whichjoins theeiids ofthesemicircle. Wetake thesecondtypewhen^{z)hasbranchpointsintheupperhalf- plane;thecontour isderived from the firsttypebyinserting loops starting fromandendingattheindentation, onelooppassinground eachbranchpoint, sothattheintegrandhasnosingularityinside thecontour. Amorepowerfulmethod (cf §13"1) which iseffective inevaluating integralswith Bessel functions under theintegral signistosubstitute forthe Bessel function oneoftheintegralsdiscussed in§6'5,andchangetheorder of 424 THEORY OFBESSEL FUNCTIONS [CHAP.XIII theintegrations;since theintegrandin§6'o (7)is(x"'^), quafunction of^, where 8isanarbitrarilysmallpositive number, thedoubleintegral usually converges absolutely when theoriginal integraldoes so,andtheinterchange producesnotheoretical difficulties. 13*51. HankeVsintegrals involvingoneBesselfunction. Before Hankelinvestigatedthemore abstruseintegralswhich willbe discussed inChajDter xiv,heevaluated alargeclass ofdefiniteintegrals* by considering taken round the firsttypeofcontour described in§13"5. Inthisintegral,a ispositive,misapositive integer (zero included), risacomplex number with positive imaginary part,and \R(v)\<R{p)< 2m+^. The firstinequalitysecures theconvergenceoftheintegral when theradius oftheindentation tends tozero; and(asaconsequenceofJordan'slemma) thesecondinequalityensures that theintegralround thelargesemicircle tends tozero astheradius tends toinfinity. Theonlysingularityoftheintegrandinside thecontour isthepointr.It follows that 1pscP-'{ir,<^' (ax)-ep"' if,<'' (axe^')}, 1 /'"•+'^"-^.g,"' ((i^)c^^ J^iJQ (^2_r2)'»+i ^~2'KiJ (^f.-r'.yn+r~ Itfollows from§3-62 (5)that xc~^dx (1)["[(1+gCp-")-') J,(ax)+i{l- e(p-'')-0 F,(ax)]- J (^'(x^-?-2)w+i This result canbeexpressedinaneater formbywritingr=ik,sothat R(k)>0.Itisthusfound thatf (2) I^[cosl(p-v)-7T../,(ax)+sinl(p-v)77. F,(ax)]t^^^^.^^ *Hankel's workwaspublished posthumously, Math. Ann. viri.(1875), pp.458—401.Apartial investigationoftheintegral with v—n,p=2h+2,m=2n wasgiven byNeumann, Theoric der BesseVschen Functionen(Leipzig, 1867), p.58. fTheevaluation ofintegralsofthischaracter which contain onlyone ofthetwoBessel functions iseffected in§13-G. 13-51] INFINITE INTEGRALS 425 Thereader should notice thefollowing special cases ofthisformula : (3)j^{cosVTT .J,(ax)-suiVTT.K {a:c)} ^^.>_^j^,yn^i= ^^ .„,,^.^^,»' f"^x^+^Jy(ax)dx a'»k"""'K^_,„{ah) ^'^> J (.«2+F)'"+i~ 2'» .m ! Theformer isvalidwhen-2)n-f</i(i/)<1,andthelatterwhen-1<^(i^)<2m+-^. Foranextension of(4)tothecasewhenmisnetaninteger,see§13-6(2). Thespecialformula TNr-xJ^{ax)dx _ hasbeenpointedoutbjMehler, 3/ath. Ann. xviii.(1881), p.194,and Basset, Hydro- dynamics.,II.(Cambridge, 1889), p.19;while Nicholson, Quarterly Journal, slii. (1911), p.220,hasobtained another specialformula /'"Yo{ax)dx _Ko{ah) ^^)jo:^+i^~^' byacomplicatedtransformation ofrepeated integrals. Someintegrals resemblingthosejust given maybeestablished here, thoughitismost convenient toprove them withoutusing Cauchy's theorem. Thus. Nicholson hasobserved that Jn{ax)dx _2f"^ f^-""cos(axcos6),^^ 7fI 1T^U.l/1//^ X-+IC- TTJJo ^'+1^' 1a^ .' = '^-j^[I,{ak)-lMo{p.k)\, by§10'4(11), providedthataandR(k)arebothpositive;sothat (7) (^^^^^^ ^{Io(ak)-I.,(ak)]. Moregenerally,ifR(v)>— |,wehave /.(ak)-L.{ak)=j.,^^2itra^ f'^-"^--^ sin-ddd, and since, byaspecialform of(2), VJ_cc .t"+ /i'"" A; providedthatR{y)<'2andaispositive,itfollows that /.(a^)-L.(aA-)= -j.^^^^^^,^j^ j_.—^1^^^^'^^ _ki~''r'Jy{ax)-t in,,(ax) TTj_cc^'"+k- I'—v r^ dx j[{I+e^-^)J.(ax)+i(1-e^-')ll Act A')]^^^, TTJQ 426 THEORY OFBESSEL FUNCTIONS [CHAP. XHI andsowehave theformula ftCCf4/yt ^__ (8) [cos\v'K.J^ {ax)+sin\v'k.H^(ax)]^„ ^.,=— ^^[I^(ak)-L^{ak)\, where a>0, B,(k)>and-^<R(v)<2. Thechangeintheorder ofthe integrations presentsnogreattheoretical difficulties. Asomewhat similarintegralis r"^x^Ky (ax)dx Jo x'+k'' AvhichconvergesiiR(v)>—\andR(a)>0. Ifwechoose ksothatR(k)>0,wehave, by§616(1), x''K^{ax)dx _r(v+^)pp(2a)''cosxu.dudx '^^+k' ~~r(i7 JoJo (x"-+k^{u'+a'y+'- fJo _TTr(jM- i)r°°(2aye-''^du -~2krJl) Jo(w^+a-^r+i TTZjgV-1 [II.^,(ak)-Y_,{ak)],4cos I/TT whenweuse§lO'-il(3).Hence, whenR(v)>—^, (9)I" ^-^.(a^Oc/a;^^r^^-^^^_-j,_ ^^^^^ j ^'+^^- 4cos I'TT^ ' andtherefore, when i^(v)<|, These formulae (whenv=0)areduetoNicholson, andthe lasthasalso beengiven byHeaviside. Theintegral f"=^>!^ ^Jox^+k^x" hasbeeninvestigated byGegenbauer*. Toevaluateit,wesupposethat R(v)>—^andthataandR(k)arebothpositive; wethenhave Jox'+k^x"T(v+^)r(l)Jo'(> x-+k' i)n''./ andsor(v+^)r(^)kj, (11) f" '^f^-=-^ {/,(ak)-L,(ay^OI- *WienerSitzungshericlite, lxxii.(2),(1876), p.349. Gegenbauer's result isincorrect because heomitted toinsert theterm-L,,(ak);andconsequently theresults which hededuced from his formula arealsoincorrect. Asimilar errorwasmade byBasset, Proc. Camb. Phil. Soc. vi.(1889), p.11.The correct result was given byGubler, ZilrichVierteljahrsschrift,xlvii. (1902), pp.422—424. 13-52] INFINITE INTEGRALS 427 The condition R{v)>—hmaynow bereplaced bythe lessstringent condition R(v)>—#,byanalyticcontinuation. Anintegral whichmaybeevahiated intheform ofaseries bythismethod is sinhax IJJ^,{bx)x''*Ulx,sinh 7r.r which isageneraHsationofNeumann's integral described in^13-2;itissupposedthat IR(a)\+\I{b)\<7rand //(,-)>-1. .^ , . 1Tsinh rt~ rv-/,»/7 s,17 °2771Jsmh TTz round thecontour used inthis section, wefindthat thedefiniteintegralisTvitimes the sum oftheresidues of (12) ^^ J,(b.v)x''*^dx=-2(-)»-•vi-'+i sinH«.A'^ (Hi). /smh ttx tt,,=isinh ttz atthepoints /,2i,3i,Itfollows that sinhax-r ,-, ^ ..,, ,2 The series converges rapidlyifbisatalllarge. Anintegral expressibleasasimilar series wasinvestigated byEiemann, Ann. derPkysik andChemie, (2)xcv. (1855), pp.132—135. 13"52. Thegeneralisation ofHankeVsintegral. Letusnext consider theintegral 1fz^-'HJ'^ (az)dz 27riJ {z'+k-y+'' This differs from Hankel'sintegralincontainingthe(complex) number/x inplaceoftheintegerm.The conditions forconvergence (with thesecond typeofcontourspecifiedin§13'5) are* a>0, \R(v)\<R(p)<-2R{^) +?^. Thecontour ischosen with alooptoexclude thepoint ik,asshewn in Fig.31,andthen there arenopolesinside thecontour; andtheintegralround thelargesemicircle tends tozero astheradius tends toinfinity. Hence /T ,-«>/pp-i (If 2^1Jo^^''^ ^"*'^~'*'"''"'' ^'''" ^'""''"'^^{af^+k^r^^ _1r^"'+^z''-^H^^'^(az)ch ~2^iJo (z-^+k^y+^• Now =—le(ip-'^)'^*^^^^-^— . 27ri], {z^+k^r^^•''r(ip-/.)r(/t +i) *Asin§13-51, wetakeR{k)>0. 428 THEORY OFBESSEL FUNCTIONS [chap.XIII Hence, whenweexpand ^^'^' (az)inascending powersofz,wefindthat 7j-gi(p-»'-2fi)ir? sinvir .r(/J,+1)(^a)"kP+''-^'^-'- SV(h_p+iv +m).(laky =0w!r(y +??i+1)r(^p+^1/- /A+7?i) andtherefore r^'xP~'^dx (1)I[cos(^p-|i/--f.i)IT.J,{ax)+sin{Ip-Iv- fx)it.Y^(ax)],j,2\^+i 2siny7r. r(//.+ 1) lr(v+i)r(ip +^v-fMyn2' T{i-v)r(^p-^v-fiy'^'{2p—y , a^k- Fig. 31. Itisnatural toenquirewhether theintegralofthistypewhich contains asingleBessel function cannot beevaluated;itseems that theonlyeffective method ofevaluatingitisthemethod which willbeexplainedin§13 '(J. 13'53. HankeVsintegrals involving aproduct ofBesselfunctions. Integrals resemblingthose of§13'51, exceptthattheycontain aproduct ofBessel functions instead ofasingleBessel function, havebeeninvestigated byHankel*byapplying Cauchy'stheorem totheintegral ir;i> {az)dzif inwhich a'^h>0,misapositive integer,7'isacomplex number with a positive imaginary part,^^denotes anycylinderfunction oforderfi,and \R{v)\+\R{fi)\<R{p)<2ni +4>. ^*Math. Ann. viir.(1875), pp.461—467. 13-53] INFINITE INTEGRALS 429 [When (^^/ni«aBessel function ofthe firstkind, \R{fj.)\ maybereplticed by—It{fj.)in thisinequahty.] When a=h,thepresenceofanon-oscillatory term* intheasymptoticex- pansionoftheintegrandshews thatwemustreplace 2m-f4by2m+3inthe inequalityinorder tomake theintegral, when taken round alargesemicircle above thereal axis,tend tozero astheradius tends toinfinity. Thecontour tobetaken isthat of§13"52;and ifweproceedinthe manner ofthat section, wefindthat (1)-^. I" ['2^{hx) jy,'" (cix)-e^-i9^^(&«e-0 ^^'" (a^-e'^Ol rf~'^^^ 27riJo''^^ ' ' ^^,--,-. . ^^i^^2_^.2yn+i Numerousspecialcases ofthisresult aregiven byHankel. Itmust 1)6pointedoutthat,whenp=2?H+3and«=6,tlieintegral ronnd thelarge semicircle tends toanon-zero limit astheradius tends toinfinity; and,ii -g?^{az)=c,H^m {az)^c,ff^i'-) (az), wethen obtain thenewformula (2)2^.j^[%\ {ax') H,i^) {ax)+<W^{axe-^) ^J'^ (axe-i)](^^23^2)^+1 Theparticularcaseof(1)inwhichp=2,n—and'&'^isaBessel function ofthe firstkind deservesspecialmention;itis r"^ • ccclor (3) Jy.{hx)[cosl(fx-v)'7r.J^ (ax\+sm^(/j,—v)tt .Y^(ax)]~ ^ providedthata^b->0 andR{/j,)>\ H(v)j-2. IfwetakefM=vandR(p)> -1weseethat ["r/ Xr/7 ^^^f^^'{i7riJ,(br)HJ'>(ar), accordingasa^6. Theexistence ofthedisconti niityintheexpressionforthisintegralwas pointedoutbyHankel. Ifwemodifyformula (3)weseethat, ifa^6>andR{k)>0,then [^X (^) ^o— 7i^M(bx)[cos]j{P'—v)7r. J^(ax)+sin|(^—y)tt .F^{ax)]dx =r^{bk)K,{uk). *SinceH^'-') (az) H^>'^i {a~2-~— ^c-^'^ ''"''when j£ |ilarge. i 430 THEORY OFBESSEL FUNCTIONS [CHAP.XIII Moregenerally, taking equation (1)withm=andWfj,=J^,wehave (^) ^TTT'^—\GO&\{p-^ fJL-v)Tr.J^{ax) +s\u\ip-{- ij,-v)Tr.Y^{ax)]dx =-I^{hh)K,{ak)kf>-\ Inthisresultreplace phyp-\-v,aby\J{a'^-\-c-—2accos6),where a—c> b, multiply bysin-"^/(a^+c^—2accos6)^",andintegrate withrespecttofrom tott;wefindfromGegenbauer's formula, §11"41(16), (7) I~ m(V ^AC'^) j-^^gi(p+fj,)TrJ^ (ax)+sin^(p+/i)ttY,.(ax)]dx Jo X-+fC =-I^ {bk)I,(ck)K,(ak)kp-\ Thisprocess mayberepeatedasoften asweplease ;andwefind that, if a>b->r 2c,then (8) f" ^"^"'^'mC^^)jjj^(^^^^^-^ |-^Qgh^\p+pi-¥{N-l)v]7r.J, {ax) J X+k- „=i +sm^{p +p,+{X—1)2^iTT .Y^(ax)]dx =-4(bk)fl/,(c„k).K,{ak).k''--. «=i Again, byconsidering 2^-f,^>[n^'^(^^)]^''''(«^)^^ round thecontourpreviously used,where both band/xdiffer inthedifferent factors oftheproduct, weobtain theslightly moregeneralresult (^> J,-r^,U^^>^(^^''-)] -l^os i^ip+^/j.- p)7r.J,{ax) 4-sin^{p+'Sfi-}.•)IT .Y^(ax)]dx=-[U I^(bk)]K^{ak)k"'- providedthata>^[R{b) andR{p-h-(a*))>R(v)r lip+1/ii—i^isaneveninteger, th^'integralonth^eftinvolves functions ofthe firstkindonly;aresultinvolving"theintegralsofproductsoffunctions ofthe firstkind ofthistypewasgiven byGegenbauer, whooverlooked the necessityforthisrestriction (cf§13"51). Anextension ofHankel's results isobtained byconsidering 1 f.o-i-^.l^x/(^^+r -)]H,'^{az) round thecontour, where a^ft>0,misapositive integer, and \R(v)\<R(p)<2m-i-^/i +R{M'). Itfollows that 1 /d 2m+i.„it \^.(Ij 13-54]INFINITE INTEGRALS 431 and, inparticular, aresult obtained inamuch more elaborate manner bySonine, Math. Ann. XVI.(1880), pp.56—60. 13*54. Generalisations ofNicholsonsintegral. Aninteresting consequenceofMehler'sintegralof§13"ol(5)isdue to Nicholson*, namely that,when aandkarepositive, Themethod bywhich thisresult isobtained isasfollows : Jo p~->rh' =-r [^-r^ Jo[a\/(r+^'- -P^''cos</>)}^dp.TTJJP'+k~ Thisrepeated integ)-al mayberegardedasanabsolutely convergent double integral,since theintegrandis(p~-)whenpislarge.Nowmake achange oforiginofthepolarcoordinates bywriting pcos(fi=k+rcos0,psin=rsin6, andwehave zz/ 7^r/ /N1rrJAar) rdddr[^J,{ar)rdr andthis istheresult tobeestablished. Togeneralisetheresult consider -z^-'H,^'^(az)dz (^+4kY^' taken round thecontour shewn inFig.32. Fig. 32. Itissupposedthataispositive, and, toensureconvergence, \R{v)\<R(p)<iR{fi) +\K *Quarterly Journal, xlii.(lUll), p.224. 432 , THEORY OFBESSEL FUNCTIONS[CHAP.XIII Itisalsosupposedthat |argk\< ^ir,andtheloopsinthecontour surround thepoints Byanalysis resemblingthat of§13"52, thereader willfindthat (2) [cos{\p-h^v-2^l)7^.J,{ax) +Bm{lp-^v-^,Ji)7^.Y,{ax)^]JftI,^,_,,, ,, ^, ,„„,.„„,,^ ._,._^,,. ..,,_,_, ^^_^^^^^^^^ TT(^^2)"-^->.'- ^i 5 ^cQsr_r:^ ^ 2sinz^7r.r(yu,+ l)[„,=o"i!r(y+7?i +l)r(^/j+^z;-/i+i??i) 4 00 '^ "cosi TT;«=ow!r(-y+m +l)r(^|0-^i/-;i +i??i) 4 Iftheseries ontherightarecomparedwith thosegivenin§5"41, itisseen thattheformer isexpressibleasaproductofBessel functions ifp—2=?/=/i,+| orif/3—4=1/= yu.+|,while thelatter issoexpressible Up—'2=—v= ijl-\-\ orifp—4=—t-=/A+|. Thecorresponding integralwhich contains asingleBessel function willbe considered in§13'6. 13*55.Sonine'sintegrals. Anumber ofdefiniteintegrals,ofwhichspecialforms weregiven by Sonine, Math. Ann. xvi.(1880), pp.63—66,canbeevaluatedbythemethod ofcontourintegration. Themostgeneralcontourintegraltobetaken is rpiu+k) •liriJ(z+ky round acontourconsistingofthepartsofthecircles \z\=8,\z\=E, terminated bythelinesarg(—^)=±tt,andthelineswhichjointheextremities ofthese cu'cular arcs*. Itissupposedthat ;nisanintegerandkisnotanegativerealnumber. Theintegralround z=8tends tozeroasS^-0, providedthatR(p)>\R(v)\, andtheintegralround\z\=Rtends tozeroasii^-x,providedthat R(p)< m+f. ByCauchy'stheorem wehave 1 27rij *Cf.Modern Analysis, §6-2; or§7*4supra. 13-55] INFINITE INTEGRALS 433 andthuswehave " 7^ 1("^+A;)"'+^^^' ^'^^^^^^ip+v)'n-+2icosvttcosp7r} +iYy(x)sin(jO—v)tt]o^a;. Inparticular, takingm=0,weget (2) ki'-^H/^ (k)=- .—[J^(ic){sin(p+v)7r+2icos z^ttcosott] +t'F^ (,r)sin(p— z/)tt]c/^'. Ifweconsider theintegral wefindthat (3) kfl-'H,^-^ (k)=-^ -^—[/,{w){sin (/J+;')TT-2icospttcosott! —il"^ (a;)sin(p— z/)tt]dx. Ifwetakep=1,t-=0,weget {A\ Tn,\2r»sin(£+A;). .. , TT.'X-TIC (5) F„(A;)=-^ f"^ cos(^^-)^^_ The lasttworesults areduetoSonine*. Moregenerally, taking p=v+\and-l<R{v)<h, weget (6)-—^^J^^a^)dx=:^^ k^[JAl-)±iY,.{k)],TTj .le+a; 2icos v-tt aresult alsoduetoSonine. -D -J.- cos(a;+A-) 1r^./ ,x ,Bywr.t,„g -_-^^=--^_ j^„„ J(,,+i.),ft, andu>«igtheformulae(6)and(7)of§13-42, Sonine deduced from(5)that (7) F„{h)=-^r'^dx+- 1^^sin(kcos6)d0, ITJoX+k TT .' andhence from§§3-56(2),9-11(2), 2f^ 2r^"' (8)Yn(k)= IOn(a;+/t)^0(^)f^^+—sin(A'cos^—|?i7r)cos7i^rf^. ''^.' Tj *SeealsoLerch, Monatshefte fUrMath, undPhys.i.(1890), pp.105—112. w.B.F. 28 434 THEORY OFBESSEL FUNCTIONS [CHAP.XIII 13*6.A7iewmethod* ofevaluating definite integrals. We sliallnowevaluate various definiteintegi-als bysubstitutingforthe Bessel function, under theintegral sign,thedefiniteintegralof§6'5,and reversingtheorder oftheintegrations. Asafirstexampleconsider theintegralofHankel'stype af~'^J^(ax), inwhich itisatfirstsupposedthat R{v)>(), R(2fi+2)>R(p +v)>0: andaisareal(positive) number, inorder thattheintegral mayconverge. Theintegralisequal tof 1["p''r(-g) x^^-^^axy-^^d-~r'Tj-g) 1(ah- 1) When this isevaluated(byswinginground thecontour soastoenclose the polesontherightofthecontour) wefindthat xp''-J^(ax)dx 2''^'r(fx +l)r(v+l) "~-'^'[ 2'2^•''+^'4 + 22^+3-pr(^ +2+iv-^p)-^^'V^^-^-^^^^2'^^"2'4/- Thehypergeometric functions ontherightarereducible toBessel functions incertain circumstances;theformer ifp=v+2orp=v+2/ji+2,thelatter ifp=2+ I.. Bythepriucipleofanalyticcontinuation(1)isvalidwhen -R{p)<R(p)<2R(p,) +h ,/•°° A-^+^j;{ax)dx a'^k"-!^Inparticular, taking p=v+2,wefindthat aformula obtained byanother method bySonine, Math. Ann. xvi.(1880), p.50; itis validwhen -l<R{v)<2R{ix) +%. *Thismethod isduetoLerch, Rozpravy,v.(1896),no.23[Jahrhiichiiher dieFortschritte der Math.(1896), p.233]; heshewed that ^..w sr(s+i) butnoother usehasbeenmade ofthemethod. tThechange oftheorder oftheintegrations maybejustified withoutdifficulty. 13-6] INFINITE INTEGRALS 435 Aformula ofsome interest isobtained bymaking />=1, yu-=—|,the hypergeometricfunctions thenreducingtosquaresorproductsofBessel functions;andanother suchformula isfound bymaking p=l—v, /u,=v—h. Itisthusdeduced that(cf.§5"41) — ,, providedthatR(v)>—1;andthat —/ (4)j^ (^^+k^r^f=— r(2.+ l)~^^('M-)^Aiak), providedthatB(v)>—I- Next consider af~^J^(ax)dx {x'+U-'r-^^ mwhich a>and jarg^'j<^vr.Itisfirst tobesupposedthat R{v)>Q, R(^fj.+4^)>R(p +v)>0. Theintegralisequalto j_r^p.-r(-.) x^-^(^axy^- 27r^jo J-^iFiv +s+l)(x'+4>k'r^' 27^^jo j-ooi r(z/+5+l)(x'+U^)^' ^^r- r{-s)r{ip +iv+^s)r(^+i-ip-iv-ls ) l7^(^'^/2)''-^'^- sin(1/3+1^/-2/x)TTr(/x+1)V (a/l7V2r+^-r(ip +ii.+l/.i) +2_„^ow! r(z^+//;+1)r(j/3+^i^- /i+|»i) Xcos(^p +lv— /j,+I/zOtt (-)"' (ak|^/2y^+'-p+*"' V(p,+m+1)" ^owi!r{2p-y +hp+2m+3)r(2/x- 1/3- |j/+2m+3)_* Thisexpansionisarepresentationoftheintegral when theconditions tobe laidonp,,vandpare ^R{^Ji) +yi->R{p)>-R{r). Now takethecases inwhich the first series reduces toaproductofBessel functions, namely p—'2=v=p.+\orp—4'=z/=/x+2- By§5'41wethen obtain theformulae ._.{"^x''+^J^{ax)dx {^ay s/ir ji\ir (j\ <«) j„(^+4,l.y^*=2T(2iF%7Ti)•^'-' <"^'>^^'- ("^)- Theformer ofthese isvalidwhen R(i')>—^,thelatterwhenR(v)>^,and, inboth, a>and |argA; |<itt. •2H--2 436 THEORY OFBESSEL FUNCTIONS [CHAP.XITI Finally,asanexample suggested by§13'55, weshall consider ai'~^ J^,(ax)doc (x+ky+' inwhich a>and |argk\<tt.Itisfirst tobesupposedthat R(v)>0, R(fi+l)>E(p +v)>0. Theintegralisequalto r(-s) xp-'(iaxy+^ 1pp 27riJo j-dsdx -l^rr(-.)r(p+.+2.)r(^4-1-p-.-2.) ,^^„..^^_._. ^^ ^^,p-M-l^(-)"^daXO"^"" r(p+i;+2m) m^o wz!r(j^+m+1)r(p+1/-p,+2m) _" (^a^O'^+^-P+'» r(/u,+?n+1)sin^(p+ i^- At- :?n)ttsin(p+I'— p.)TT .r(/i+1) m=wi !r(1 /x+iI'-ip+im+f)r(i /A-iJ.-1p+^m+f) The first series reduces toJt,(ak)when/x=0,andthesecond series isthen expressible byLommel's functions(cf.§10"7). Inparticular wehave ,_.['^'x"J^{aa)dxirk" r_, ,,.„/txt providedthat—^<R{v)<^. Thereader willfindthatalargenumber oftheintegralsdiscussed inthis chapter maybeevaluated bythismethod. 13*61. Integrals involving products ofBesselfunctions. Ifanintegralinvolves theproductoftwoBessel functions ofthesame argument (butnotnecessarilyofthesameorder),itislikely.that theintegral iscapableofbeingevaluated either byreplacingtheproduct byNeumann's integral (§5'43) andusingthemethodjust described, orelsebyreplacingthe product J^{x)Jyix)by \_p»r(-g)r(/x +t/+2g+i)(^^-)^+-+^- 27rij_oe,r(/x +5+i)r(i. +s+i)r(/x +i/+.s'+i)^' inwhich thepolesofr(— s)areontherightofthecontour while those of r(/x+i/+2s+ 1)areontheleft;thisexpressioniseasilyderived from§5'41 byusingthemethod ofobtain-ing §6'5. Thereadermayfind itinterestingtoevaluate 3(^~^Jft.{ax)Jy{ax)dx /: bythese methods. The result isacombination oftwofunctions ofthetype ^F^ ,and the^ finalelement ineach function isa^k^. 13-61] INFINITE INTEGRALS 437 Aaotherintegral formula, obtainable byreplacing J^{bjx)byanintegral,is r«p-i (1) /XJ^{ax) Jt,{blx) dx Jo "2-^''-''+^r(v+i)r(i^+iv-ip +i)-"-H"^^'~2—+^'^—2—+^'-16- 16 This isvalidwhen aand6arepositive and Thegeneral formula wasgiven byHanumanta Rao, Messenger,XLVii. (1918), pp.134— 137; specialcaseshadbeengiven previously byCailler, Mem. delaSoc.dePhys.deGeneve, XXXIV. (1902—1905), p.352;Bateman, Trans. Camh. Phil. Soc.xxr.(1912), pp.185,186; andHardy, whodiscussed thecaseoffunctions oforders ±|,(see §6'23). Aninteresting exampleofanintegral* which contains aproductis lim Iexp{-p'x') J^{x)/i_, {x)— 5"^ +4^!^l,)^^^^^'si8-i^(^^ +^)-iri2-vy^ whichmaybeAvaitten intheform r°° /„„xfT-/Nr /x ^icsin I'TT 1dx +Jv(l- ^)It(2)-f("+1)-f(2-")-2logipl. Itiseasily provedthat r=»^ sfr/^r /x -g^i^;sinyTrldx exp {pX)r'/„, ,.,1.T^ /9_„, Xr,/„ ,i,\"'^"^ 1/-i+cc*r(-s)r(25 +2) rfs ^-rrij^-^i s{s+l)V{v +s+\)V{-l-v +s}(2^;)--' 1^ (27i+l)!(2p)-2" =-"S 4„riw.(n+1)!r(y+n+1)r(2-1/+n)' andthisseries isanintegralfunction ofIj}). Toobtain anasymptotic representationoftheintegral,validwhen jjaiis small and ]argpj<jtt,weobserve that the lastintegrandhasdoublepoles atand—1,andsimple polesat—f,—2,- |,—3,— *This integral wasbroughttomynotice byMrC.G.Dar.vin, whoencountered itina problem ofDiffusion ofSalts inacircular cylinder ofliquid. 438 THEORY OFBESSEL FUNCTIONS Hence wefind J_f^+--'-'T(-s)T(2s +2) ds_ S-jriJi^^i s(s+l)r{v+s+l)r(2-v^s) (:2py' sinVTT[chap.XIII ^^^^^^^^--^^{ir(2)-f(p+l)-ir(2-j,)-2\0g2p][l +2v(l-v)p^] (-Y(273Yr(hi) (1-2v)sinVTT,^ 27ri'(l-z;)^'^ „r32n(n-2)r (v+I-^n)F{2-v-in).(n-2)! andso (2)limexp(-2f-x-) J,(x)./i_^ (a;)— 5 '^" +smVTT 4<7n>{l—v){l+2\ogi8-yJr(v +l)-f{2-v)] sini^TT,1 1„(1—27/)sinVTT„ --^;^{ir{2)-^{r(v+l)-ylr{2-p)-2\og2p]f--^ 27rWl-I.)^ +S(_)n(2^)«r(l>l) Inthespecialcase v=0,wefindthat (3)lim 6^0^ f^^-^+ Iexp(-f-af) Jo(*•)Ji(*•)-;;^X- lf+s-^[r(m+f)l3(2/9)=»«+3 Zo7r={2m+1)2(27W+3).(2m+1)!" 13'7.Integral reprenentations ofproducts ofBesselfunctions. FromGegenbauer'sformula of§11-41(16)aninterestingresult isobtain- ablebytakingthecylinderfunction tobeofthe firstkindandsubstituting theresult of§6'2(8)forthefunction under theintegral sign. Thisprocedure gives 2^r(.+i)r(i)':^^'^^ and ifwechangetheorder oftheintegrations, wefindthat (1) J,{Z)J,{z)^^^.\ exp-ji^i^r~\-^''\J)J- 27rlJn- C—Xl V2t This result isproved whenR{v)>—\and ^<Z,buttheformer restriction mayobviouslybereplaced hyR{v)>—1,andthelattermayberemoved on account ofthesymmetryinzandZ.Itisalsopermissibletoproceedtothe limitbymaking \z'\-^\Z\. 13-7, 13-71] INFINITE INTEGRALS 439 Byusingtheresults of§6"21(4)and(5),wefind inthesamewaythat providedthatR(p)>—land |^ |< |Z |. Theformula(1)wasobtained byMacdonald, Froc.London Math. Soc. xxxii.(1900), {)p.152—155,from thetheoryoflinear differentialequations, andhededucedGegenbauer's integral byreversing thestepsoftheanalysis which wehavegiven. Theformulae(2)and (3)were given byMacdonald, though they arealso tobefound inamodified form in Sonine's memoir, Math. Ann. xvi.(1880), p.61. Afurther modification oftheintegrals ontherightin(2)and(3)wasgiven bySonine, theobjectofthechange beingtoremove theexponential functions. Forphysical applicationsoftheseintegrals,seeMacdonald, Proc. London Math. Soc. (2)XIV.(1915), pp.410—427. 13'71. I'heexpression ofK^{Z)K^,{z) asanintegral. Weshall next obtain aformula, due toMacdonald*, whichrepresentsthe product K^{Z) Ky{z)asanintegral involvingasinglefunction ofthetype K^,namely Thisformula isvalid forallvalues ofvwhen IargZ\<7r,jargz\<7rand |arg(Z+z)\< ^tt; but itisconvenient toproveitwhen Z,zhavepositivevalues X,x,and toextend itbythetheoryofanalytic continuation; theformula, which is obviouslytobeassociated with§11"41(16),isofsomeimportanceindealing withthezeros offunctions ofthetypeK^(z).Itispossibletoprovetheformula without tierather elaborate transformations used inproving §11"41(16); thefollowing proof,which differs from Macdonald's, isonthelines of§2'6. By§6-22(7) wehave y-^{,(X)K, (x)=\j^re-('+«)-A-cosh ^-..co^hu^nci^^ 'iJ-00j—GO Q-2i'T-Xcosh(T+U)-xcosh{T-U)dJJdT. 2j-c. _. If(A'e^+ .re~^)e^' betaken asanewvariablev,intheintegral IQ~Xco9MT+U)-xcosh(T-U) ^JJ *Proc. London Math. Sor.xxx.(1899), pp.1G9-*171. 440 THEORY OFBESSEL FUNCTIONS [CHAP.XIII 1fZ2+a;^+2Za;cosh2^1 itbecomes exp—^j^Hrdv V' andsowehave ^dT,VK,(Z)/C{x)=^I"J"e-2.r-(x./oco.h22' exp- ||_^^'l and,onperformingtheintegrationwithrespecttoT,weatonce obtain Macdonald's theorem when thevariables Xandxarepositive. 13"72. Nicholsonsintegral representations ofproducts. Weshallnowdiscuss aseries ofintegral representationsofBessel functions which aretobeassociated withNeumann'sintegralof§5"43. Theformulae ofthistypehave beendeveloped byNicholson*, andthe twowhich aremosteasily provedare (1) K^{z)K, (^)=2fK^+^ {2zcosht)cosh(/J.-v)tdt Jo =2ir^_„ (22cosht)cosh{/j,+v)tdt, Jo when Iargz\< ^tt,while/xandvareunrestricted. Toobtain these formulae weuse§6'22(5)which shews that K^(z)K,(^)=T Ie-~^""^^t+coshu) coshfitcoshvudtdu.^J—00J-CO Therepeated integralisabsolutely convergent,and itmayberegardedasa doubleintegral.Inthedoubleintegral make thetransformation t+u=2T,t-u=2U, and itisapparentthat K^(z)K,{z)=^jre-2^'^°^h^«°«i>^cosh/i(r+ U)coshv(T- U)dTdU. But 2coshfi(T+U) coshv{T-U) =coshifi+v)Tcosh(/Li—v)U+cosh(yu,- i/)jTcosh{fu,+v)U +sinh{/x+v)Tsinh(//,-v)U+sinh(/u,-v)Tsinh(/x+v)U. Theintegrals correspondingtothelasttwoofthese fourtermsobviously vanish; and, ifweinterchangetheparametricvariables 2"andUintheintegral correspondingtothesecond ofthefourterms, weobtain theformula K^{z)7C{z)= \\ \e-^'«°«i^Tcoshucosh{fi+/')Tcosh(,m-v)U dTdU. Ifweintegrate withrespecttoUweobtain thefirstform of(1),and ifwe integratewithrespecttoTweobtain thesecond form of(1). *»QuarterUj Journal, xlii.(1911), pp.220— 223, 13-72, 13-73] INFINITE INTEGRALS 441 Theformula (2) I^{z)/,{z)=-'f/^+,{2zcosd)cos{fj^-v)e dd, TTJ which isvalidwhen R(fx,+ v)exceeds —1, isatonce deducible fromNeu- mann's formula. Ifwetakefjb=andchangethesignofv,wefindthat (3) Ioiz)K,{z)=~i'^K,(2zcos0)cosvdcW. Moregenerally,ifwetakefi=—mandthenreplace /nand vbymand— i', wefind that, ii \R{v—m)'<1,then /„,{Z)K,{Z)=^^^-^K^^ra (2^COS6)COS(m+v)0cW.(4) ^, Ifwecombine(8)with§6'16 (1)wefindthat ^,,„ ,,2rCiz+i)r^-r(4zY cos(hcos^)cos i^^ ,,. providedthat—|<72(z^)<1;andinparticular Ja(u)du (5) I,{z)K,{z)= .'o \/(u-^+42^)' aresult ofwhich amoregeneralform hasbeengivenin§13'6,formula(3). 13'73. Nicholsonsintegral* forJ^-{z)+IV{z). Theintegral, correspondingtothosejust discussed, whichrepresents J^-{£)+IV (yz)isdifficult toestablishrigorously.Itisfirstnecessaryto assume that theargumentispositive (=x),and itisalsonecessarytoappeal toHardy's theoryofgeneralised integrals,orsome suchprinciple,inthecourse oftheproof Take theformula(§6-21) I,'OC+77?" ZT^d) ix)=— . e-^sinhw-utv^yj^ -TTl'-co From themanner inwhich theintegrandtends tozero as\iv\-*xon thecontour, itisclear thatwhen anexponentialfactorexp{—\w'-}isinserted, theresulting integral converges uniformlywithregardtoA.,and soitisa continuous function ofX.Hencef i/^(i) (x)=lim— .exp|-\w-]e-^sinhw-w« ^l^^,^ *Phil.Mag. (6)xix.(1910), p.234; QuarWrlij Journal, XLii.(1911), p.2'21. tHcardy, Qnarterhj Journal, xxxv.(HlOi), pp.22—66; Trans. Camb. Phil. Soc. xxi. (1912), pp.1—48.Inthisintegral (asdistinguished from tho.se whichfollow)thesignlim iscommuta- tivewith thehitegral sign. 442 THEORY OFBESSEL FUNCTIONS[CHAP.XIII ByCauchy's theorem, thecontour maybedeformed into the line I(w)=^ir,solongas\hasanassigned positivevalue;writingt+^Triforw, weget if^(i) (x)=lim^4^fexp{-\(t +^Trif]e'-i-cosh^-.^ (^ K-^+OTTi J-OD inHardy'snotation. Inlikemanner Avni /-co TTt J-x^ withanimplied exponentialfactorexp{—X,(u— ^iri)-]. Since therequisite convergenceconditions arefulfilled when X>0,wemay regardtheproductofthetwointegrals .'—CO J—CO (inwhichei{t)ande.^iu) stand fortheexponential factors)asadouble integral J-coJ—X Wethus findthat J—XJ—X with theimplied exponentialfactor exp[-X(t+Itti)--\(u-lirif]. Make thesubstitution t+u=2T,t—u=2Uandthen .'—X.'—X withanimplied exponentialfactor exp{-2XT-'-2\(U+liriy]. Inview oftheabsoluteconvergenceoftheintegral,itmaybereplaced by therepeated integralinwhich theintegrationwithrespecttoUisperformed first, sothat \_JJ—X +rre-2'-xsmhrsinhC^+2.-r^^(^2T .'J-X withanimplied exponentialfactor ineach caseequalto exp{-2\T--2X{U +i-rriy}. 13-73] INFINITE INTEGRALS 443 We firstconsider theintegral rexp{-2X{U+l7riy} e2ixsinhrsinhf/f^[^^ J—cj:) inwhichTispositive. WhenTispositive,the [7-pathofintegration maybedeformed intothe contour/([/')=^tt;ifwethen writeU=v+^iti,where visreal,theintegral becomes exp[-2X(v+irif]e-2^«i"hrcosh,;^^y x> =2exp(2\7r-) exp(-2\v-)cos4>7r\v .e'-^^'"^^^^^^'^ "dv Jo =2exp(2\7r2)/g-^'^sinhrcosht.^^ -2exp(2\7r-)({1-exp(-2Xv-) cos47r\v}e""'*^'^'"h2'cosh ^^^^ .'0 Toapproximatetothelatterintegralwhen Xissmall, weusethe inequalities 0^1- exp(-2Xv^)cos 4>'TrXv =1-exp(-2Xv')+2exp(-2Xv')sin-27rXy ^2Xv-+8tt-XH\ sothat, forsome value ofbetween and 1, rexp{-2\(tr+l7r02}e2txsinhrsmhr;^^ =2exp(2X7r-)|"^ g-2a:sinhrcoshu ^^-(2\+Stt-X')\y2g-2xsinbrcoshr^y J .' =2exp(2\7r-) Ifwetreat theintegralK,(2a'sinhT)-(2\+Stt-X-) ^.f^,7i^(2*'sinhT)0' Cx> exp{-2X(f7+i7rt)-|e'^'^si»ii ^^"'1^ c'dU J—cc inasimilar manner, wefind itequalto 2Ko{2xsinhT)-2X0,J^K^{2xsinhT)]-, where ^6^i;$1,providedthatTispositive. rco Oncollectingtheresults andrememberingthat means thesamething asliirT'l,wefindthat lim limexp(2X77-)/i{2xsinhT) (2X+877=X-)^exp(2\77-) ',^-^A^^(2*'sinhT) -f-lim lim A^.+05-».+07^0{2xsinhT)-2X^11^,if^ (2ci;sinhT)}e-'~''^dT \-''TdT. 444 THEORY OFBESSEL FUNCTIONS .[CHAP.XIII Now, quafunction ofT, (1^,K^{2xsinhT)\=[(logsinhTy}, whenTissmall, andsowemayproceedatonce tothelimitbymakingB-^0, since theintegralisconvergent;and, since theintegrals ,^ \~K^{2xsinhT)\e±^-^clT [Oh-- Jm=o areconvergent,theresult ofmakingA,-* is 1-tt'{J,- {cc)+F,^{x)]-r/iTo(2^sinhT)(e"2"^'+e^"'^)dT. J Itistherefore proved that,when x>0, J,'(x)+17(x)=A1^Ko(2^'sinhT)cosh2vTdT. Ifwereplacexby^,both sides ofthisequation becomeanalyticfunctions of2,providedthatR(z)>0.Hence, bythetheoryofanalytic continuation, wehave theresult (1) JJ"(2)+17(2)=— ^rKo(2zsinht)cosh 2vtdt, providedthatR{z)>0. Anotherintegralformula which canbeestablished bythesamemethod* is (2) J,{z)^i^-F,{z)^-^h^=-^rK,{2zsinh e-'"'dt. Toprovethisformula, wefirstsupposethat^isapositivevariable (which wereplace byx),andthen /.(-)^^-F.(..)^4^OV OV 2ir''^^^dv"^^dv) ~.G jI(«-1-tri)e'^fcosh«-cosh«) g-K'+«) .(o/^^u)27r-i -^l^r r(2^^+"^Oe^'"'"''"''^''°''^e-^"^-(f^^^ t^)—00 ,'—00 202.-"T^^f ((2f^+TTl)e-2'*sinhTsinhUg2.T ^^f/'^J Forthefulldetails oftheanalysis, seeWatson, Proc. RoyalSoc. xciv.a,(1918), pp.197- 13*73]INI'INITE INTEGRALS Now,Tbeing positive, wehave (2C/+iri)exp[-2\{U+l irif]e2'>8i"i>Tniuhu^ijj "? '{v+iri)exp[-2\(v+-rri)-]e--i^sinh rcosb^-^y445 and, since ^e-2.^sinhrcoshv jg.^^f^j^function ofv,itmaybeprovedthatthe lastintegralis 27ri Ie-2^sinhTcoshV(ly_^Q(^x), where theconstantimpliedinthesymbol (A.)isafunction ofTsuch that itsintegralwithrespecttoTfrom toxiisconvergent. Inlikemanner, r{2U+iri)exp{-2X(f7+1 irif]e-2«^i"'^rsinhu^m = 12Vexp(-2\v-)e---^«">'! ^<=»^*i "dv=0. ./-CO Hence itfollows that 4/• TT.'oifo(2«sinh^)e-2''^c^r. Theextension tothecase inwhich theargumentoftheBessel functions is complexwith apositiverealpartismade asin(1). Itshould bementioned thatformula(2)isofimportanceinthediscussion ofdescriptive propertiesofzeros ofBessel functions. Thereader mayfind itinterestingtoprovethat CV andhence that(22-^2)J„(^)^^-n(.)^ Cv(^)dv (3)j;(z)^LlJ£> _.r,' (z)^-^=_JL /"" (^2cosh2T-v^)Ku(2.rsinhT)e-'^"'^dT. Cv Ov nz-J(, Other formulae whichmaybeestablished bythemethods ofthissection are (4) j^{z)j,{z)+r^{z)rAz). =\/A'^_^(2ssinh0.{e^''+"^'cos(^-r)7r +t'-('^+ '')'}c/<,^y {^) J^{z)i\{z)-J,{z)Y^{z):4sin{ii—^i^K J{2ziimht)e^''-^"^^dt] these arevalidwhenR(z)>0 and \R(ijl-v)\<1 ;theydonotappeartohave been previously published. 446 THEORY OFBESSEL FUNCTIONS [CHAP.XIII 13"74. Deductions fromNicholsonsintegrals. Since -^o(f)isadecreasing*function of^,itisclearfrom§13"73(1)that Jj'{x)^-Y,^{x) isadecreasingfunction ofxforanyrealfixed value ofv,when xispositive. Since thisfunction isapproximately equalto1l{Trx), when xislarge,we shall investigate X[JJ'{x)+17{x)] andprovethat itisadecreasingfunction ofxwhenv>\, andthat itisan increasingfunction oixwhenv<\. Itisclear that ^\J^{x)+YHm =-„f"[K,{2xsinhT)+2a;sinh TK^' {^^sinhT)}cosh 2vTdT 8 TT^ d+-JK,{2xsmhT)cosh2z^r-^{tanhrcosh2i/r}dT,K,(2xsinhT)tanhTcosh 2vT onintegratingthesecond term intheintegral byparts.Hence ^Jx{J.^{x)+Y.^(x)]] =--,rKoi2xsinhT)tanhTcosh 2vT{tanhT-2v tanh2vT]dT. Now\tanhXT isanincreasingfunction ofXwhen X>0,and sothe last integrandisnegativeorpositive accordingas2y>1or<2i^<1;and this establishes theresult. Nextweprove that,when x'^v'^0, {x^--p^)^[J,^{w)+Yj^{x)] isanincreasingfunction ofx. Ifweomit thepositivefactor 8(x^—v^)~^Itt^ from thederivate oftheex- pressionunder consideration weget I[xK,{2xsinht)+2 {x"-- v"")sinh t.Ko{2xsinh0}cosh 2utdt,Jo andtoestablish thetheorem stated itissufficient toprovethat thisintegral ispositive. *This isobvious from theformula 13-74] INFINITE INTEGRALS 447 Wetwiceintegrate bypartsthe lastportionofthesecond term inthe integralthus 1v' sinh t/C ("^-xsinht)cosh Ivtdt Vsinh tsinh '2vtK^' {2xsinht) dJo-VI—[sinhtlu (2a.'sinh0}«inh -Iitdt f^^ =-y -y-{sinh^AV (2a;sinh^)|sinh2vtdt JoM z=—V \2xsinh tcosh tK,^{2xsinhi)sinhIvtdt —Xsinh ^cosh tK^^{2xsinhi()cosh 2vt + .^• I;^[sinh^coshtK^{2xsinh^)]cosh 2vtdt "^ = :[^cosh 2f7^0(2a;sinh+2*=sinh <cosh- tK^'(2xsinh0]cosh 2vtdt: Jo thesimplificationafter thesecondstepisproduced byusingthedifferential equation ^/lo" (z)+K;(z)-zK, {z)=0. Theintegralunder discussionconsequentlyreduces to [-2xsinh- 1K^{2xsinht)—2./.-sinlrtKo {2xsinhi)]cosh 2i^tdt a-sinlr^ rr,r.•1,x ,o ,—Ao(2a' snih t)cosh Zvt )shtJ cos —2sinh" tcosh 2vt+v:]— ^,—' cosh 2vt dt(coshi ==X IA'o(2a;sinht)[tanh^tcosh 2f^+2vsinh^ isech ^sinh2vt] dt,a; A'o(2a;sinht)Jodt andthis ispositivebecause theintegrandispositive;hence thedifferential coefficient of (x^--v^)i{J^^(x)+Y,'(x)] ispositive,andtheresult isestablished. Since thelimits ofboth thefunctions a;{.7,^(x)+F,-^{x)\, {x'-v^)i{J,'(x)+}V(.^O} are2/7r,itfollows from thelasttworesults thatwhen a;^i'^|, n^.-^>^^H-)+lV(.^)>^.{X-— 1^2)1 ^(1) Anelementary proofofthe lastinequality (with various relatedinequalities) was deduced bySchafheitlin, BerlinerSitzungsberichte,v.(1906), \).8(5,from theformula (cf.§5-14) =%-^+62)-.r[|^^+''#; (-^O }' +2 (l- I)'g';^ (,^)+r^y2 (,,) where'g'^{x)=aJ^ (,v)+b)\{x). 448 THEORY OFBESSEL FUNCTIONS [chap.XIII Thenextconsequencewhichweshalldeduce from theintegralsof§13*73 isthat,when vispositive, d\\(v)„ dJ^jv) ^»'(^)— :jJ*-(»')— 1—>0- dv dv Toobtain this result, weobserve that theexpressiononthe leftmaybe written intheform +J.{^)dv dx-Y^{x)dv dJ„(x) dx X=v TTV TTJQ TTVk\(2vsmh T)e-"'^dT t1-/e-'KJ'Ivsinh^^dt But, foreachpositivevalue oft,2vsmh{ltlv)isadecreasingfunction ofv, and so,sinceK^ix)isapositive decreasingfunction ofitsargument,weseethat 1-1e-'Ko(2vsinh^\dt isadecreasingfunction ofv,andtherefore v\J^{v) ,- l^{v)dv dv rj dY^jv).dJ.{v)[' =lim =0,^lim " 4 lOoTry- byusingtheasymptotic expansionsof§8*42;andthisestablishes theresult stated. 13*75. Theasymptotic expansion ofJ^{£)+F/{z). Itiseasytodeduce theasymptotic expansionoft/^-(^)4 Y^'^z) from Nicholson's formula obtained in§13"73, namely /^^{z)-vYy^ iz)=— ,rK^(2zsinh cosh 2vtdt : forw^ehave,by§7*4(4), cosh2^^.,->,nApm)^^^.^^,„,^ ^coshf ,„=o (2m)!^ \R^\<m- COS I'TT^^i^^^^^^^^2^sinh=n- (2^)!^^'°^ ^' where i--i^i- o/ x ^cosIt{vrr) and,when visrealandpissolargethatp+^> v,Upliesbetween and ^^^^^2^i'sinh^^^ 13-75, 13-8] INFINITE INTEGRALS 449 Weatoncededuce theasymptotic expansion JJ^{z)+17(z)~ ;^,im^^2- [" K,(2zu) U--du,^m=0 {,-^ni)i Jo that istosay,by§13-21(8), (1) J.^(z)+r,'(.)~^211.3...(2m-1)1*1^^; this isproved whenR(z)>0,but itmaybeextended over thewiderrange Iargz\Kir] and,ifvisrealandzispositive,andj)exceeds v—\,theremainder afterpterms isofthesamesign as,andnumericallylessthan, the{p+l)th term. 13'8.Ramanujans integrals. Someextraordinary integrals havebeen obtained byRaman ujan*froman applicationofFourier'sintegral theorem ftoCauchy'swell-known formula which isvalid ifi^(^+i^)>1.Theapplication shews that I 0, {\t\>'ir), where tisanyrealnumber. Byexpandinginascending powersofxandy,andthenapplyingthis formula, itisseen that (1)fJ-^'-^e^'^i^ = U.-t^+yli a)"'""'''"""'"•^-+'[^!2cos\t{ofe-i" +fei")\l ii—7r<t<TT;forother realvalues oft,theintegraliszero. Inparticular (2) rJ^+^{x)J,_^{x)d^=J,^,{2x). J—'» Invi6w oftheresearches ofMarch, Aim. derPhi/sik undChemie, (4)xxxvil. (1912), pp.29—50andRybczynski, Ann. derPhysik ^md C/iemie, (4)XLi. (1913), pp.191—208,it seemsquite likely that, inspite oftheerroneous character oftheanalysisofthese writers|, theseintegrals evahiated byRamanujan mayprovetobeofthehighest importanceinthe theoryofthetransmission ofElectric Waves. * Quarterly Journal, xlviii.(1920), pp.294—310. tCf.ModernAnalysis, §§9-7, 11-1. tCf.Love, Phil. Trans, oftheRoyalSoc.ccxv. a,(1915), pp.123—124. W.B.F. 29 CHAPTER XIV MULTIPLE INTEGRALS 14"1.Problems connected withmultiple integrals. Thedifference between thesubjectsofthischapterandthelast ismore thanoneofmeredegree produced bytheinsertion ofanadditionalintegral sign. InChapterXliiwewereconcerned with thediscussion ofintegralsofperfectly definite functions ofthevariable andofanumber ofauxiliary parameters;in theintegralswhich arenow tobediscussed thefunctions under theintegral signaretoagreaterorlessextent arbitrary. Thus, inthe firstproblemwhich wallbediscussed, theintegralinvolves afunction which hasmerelytosatisfythe conditions ofbeingasolution ofapartialdifferentialequation,andofhaving continuous differential coefficients atallpointsofrealthree-dimensionalspace. Insubsequent problems,which aregeneralisationsofFourier'sintegral formula, thearbitraryelement hastosatisfy evenmoregeneralrestrictions such ashavinganabsolutely convergent integral,andhavinglimited total fluctuation. 14"2.Weber's infinite integrals. Theintegralswhich willnowbeconsidered involve Bessel functionsonly incidentally;but itseems desirable toinvestigate them somewhatfully because manyoftheformulae ofChapterXiiimay easilybederived from them, andwere, infact,discovered byWeber asspecialcases oftheresults of this section. Weber's researches* arebased uponaresult discovered byFourierftothe effect thatasolution oftheequationofConduction ofHeat du d-u d^u d-u is u=\rrr^{a:+2X^/t,y+2Y^%z +2Z^/t) Xexp{-(X-^+Y'+Z')]clXdYdZ, where <l>isanarbitraryfunction ofitsthree variables. Weber firstproved that, if^{x,y,z)isrestricted tobeasolution ofthe equation 5'^ a-<i> d-"^,,. *Journal filrMath. lxix. (1868), pp.222—237. tLaTheorie Analytique delaChaleur(Paris, 1822), §372.Thesimpler equation withonly onetermontherighthadpreviously been solved byLaplace, Journal deI'Ecole iwlytechnique, vm.(1809^ pp.235—244. 14-1, 14*2] MULTIPLE INTEGRALS 451 then (2) u=exp(-k-t)<i>(x,y,z), providedthat <I>hascontinuous firstandsecond diiferential coefficients, and theintegral convergesinsuchawa}'*thattransformations topolarcoordinates arepermissible. Themethodbywhich thisresult isestablished issuccessful inexpressing amoregeneral triple integralasasingle integral [cfequation (4)below]. Ifwechangetopolarcoordinates bywriting 2X\Jt=rsin6cos<^,1Y^Jt—rsin^sin^,2Zsjt—rcos6, weget u=T-—-7 ^{x+VBmd cos<i>,y+rsin6sincf),z+rcos6) (47r^)- Jo JJ-TT XexpI— J-, ]r'^sin6dxpdddr. Now consider thefunction ofr,zr(?•),definedbytheequation (r)= /'t>{x+rsindcos(b,y+rsin6sincf),z-\-rcos6)sin6d(f)dd. JJ-TTC7^ ^' .' Itisacontinuous function ofr,with continuous firstandsecond differential coefficients when rhasanypositivevalue;andtheresult ofapplyingthe operator 1d/„(/ tow(r)isdi'\drj Weproceedtoshew thatthelastintegraliszero. Ifwemake useofthe differentialequation (1),which^satisfies, wefindthat Toavoid thedifficulty!caused bytheapparent singularityofthe lastinte- grand onthepolar axis,weconsider theintegraltaken over thesurface ofa sphereWith theexceptionofasmallcapofangularradius 8ateachpole; since theintegrandonthe left isbounded atthepoles,theintegralsover the capscanbemadearbitrarilysmall bytakingSsufficientlysmall. Ifweperformtheintegrationofthesecond termontherightwithrespect *Asufficient coudition isthat <i>should bebounded when thevariables assume allrealvalues, infinite values ofthevariables being included. Cf .thecorresponding two-dimensional investigation, ModernAnalysis, §12-41. tThisdifficulty wasoverlooked byWeber. 092 452 THEORY OFBESSEL FUNCTIONS [CHAP. XIV to(f),weseethat itsintegralvanishes becaused^/d(j>issupposedtobeaone- valued function ofposition.The firstterm ontheright gives 1 sni^^77ouT-S clef), andthiscanbemadearbitrarilysmall bytakingSsufficientlysmall since o<l>/(sin^a^) iscontinuous andtherefore bounded. 1df„rft3(r)^^l(-'-¥^V^-« canbemadearbitrarilysmall bytaking8sufficiently small, andtherefore it iszero. Consequently (3) *i^+i=,-»W=0. ,..Asinkr+Bcoskr sothat CT(r)= ,r whereAandBareconstants;since bt(r)and itsderivate arecontinuous for allvalues ofr,AandBmust have thesame constant values forallvalues ofr. Ifwemake r^-O, weseethat B=0,A=4<'Tr^{x,y,z)/k. Hence* u=exp(—— jsinA;r .rdr=exp(—k'-t)'t>{x,y,z),2k^/{'Trt') j and thisestablishes Weber's result. Asimilarchangetopolarcoordinates shews that, if4>{x,y,z)isasolution of(1)ofthetypealready considered, and if/(?')isanarbitrarycontinuous function of/',then 00 Too Too 00 .'—a>J—CO(4)) ( Ici>(Z,F,Z)fy[{X-xf+(F-yy+(Z- zy]-\dXdYdZ =^^ \f(r)smkr .rdr. .0 Thereader willhave nodifficultyinenunciatingsufficient conditions concerningabsoluteness ofconvergencetomake thevariouschangesinthe *This integralismost easily evaluated bydifferentiating thewell-known formula Iexp{- 17)coskr .dr=J{irt) exp (-K^t) with respect tok. 14'3] MULTIPLE INTEGRALS 453 integrations permissible. Onesuch setofconditions isthat^should be bounded asthevariables tend toinfinity, andthat f{r)=0{r-v), (r^O); f{r)=0{r-% (r-*x), wherejtx 3,q>\. Asomewhatsimpler formula established atabout thesame timebyWeber* isthat, if?«(/•,6)isafunction ofthepolar coordinates(r,6)which hascontinuous firstandsecond differential coefficients atallpoints such that0-^r-^a, whose value attheoriginisu^, andwhich isasolution oftheequation l^u d^u,„ then /u(r,6)dd=Stti^uJq(^t), y—77 when ^r^a.Theproof ofthis islefttothereader. 14'3. General discussionofNeumannsintegral. Theformula (1) [^udurfF{R,^).J,[u sl{R'+r'-^Rr cob{^-(^y^^Rid^dR)Jo J J-77 wasgiven byNeumann inhistreatise fpublishedin1862. Inthisformula, F(R,4>)isanarbitraryfunction ofthetwovariables {R, 4>),andthe in- tegrationovertheplaneofthepolarcoordinates (R,^)isadoubleintegration. Inthespecialcase inwhich thearbitraryfunction isindependentof<I>,we replacethedoubleintegral byai-epeated integral,andthenperformthein- tegrationwithrespectto^;theformula reduces to (2) rudurF(R) Jo(uR) Jo(ur)RdR=F{r), Jo .'o aresult whichpresentsacloser resemblance toFourier'sintegral Jthan(1). Theextension of(2)tofunctions ofanyorder, namely (3) rudurF(R) J,(uR)J,(ur)RdR=F(r),Jo Jo waseffectedbyHankel§.Inthisresult itisapparently necessarythatv^—\, thoughamodified form ofthetheorem(§§14'5—14'.52)isvalid forallreal value^^^fV;whenv=±\, (3)isactuallyacase ofFourier's formula. Theformulae(2)and(3)are,naturally, much moreeasytoprovethan(1) ; andtheproofof(3)isofpreciselythesame character asthat of(2),the •Math. Ann. i.(1869), pp.8—11. tAllgemeine Lostvigde.iProblentes iibcrdenstationdren Tcmperatnrzustand einea homogencn Korpers, welcher vonzwei nichtconcottrischen Kugelfldchen begrenzt wird (Halle, 1862), pp.147— 151. Cf.Gegenbauer, Wiener Sitzungsberichte, xcv.(2),(1887), pp.409—410. tCf.Modern Analysis, §9"7. §Matft. ^HH.vm.(1875), pp.476— 483. . . 454 THEORY OFBESSEL FUNCTIONS [CHAP. XIV arbitrariness oftheorder oftheBessel functions notintroducing anyadditional complications. Following Hankel, manywriters* describe theintegrals (2)and(3)as "Fourierintegrals"or"Fourier-Besselintegi-als." Onaccount ofitsgreater simplicity,weshallgiveaproofof(3)before proving (1);andatthisstageitisconvenient togiveabrief account ofthe researches ofthevarious writers whohaveinvestigatedtheformulae. Ashasalreadybeen stated, Hankel wasthe firstwriterftogivethe generalformula(3).Hetransformed theintegralinto limCRF{R)dR J,{uR)J,{ur)udu ^.coJo .0 k =limRF(R)[RJ,+,{\R)J,{\r)-rJ,+,{Xr)J,{\R)-^"^ A-*oo .' K-—r,2 andthenappliedthesecond mean- value theorem totheintegrand justas intheevaluation ofDirichlet'sintegrals. Substantiallythesameproof was given bySheppardj who laid stress ontheimportantfactthat thevalue oftheintegral depends onlyonthatpartofthei^-rangeofintegration which isintheimmediateneighbourhoodofr,sothat thevalue oftheintegralis independentofthevalues whichF(R) assumes whenRisnotnearly equaltor. Adifferent mode ofproof,based onthetheoryofdiscontinuousintegrals, hasbeengiven bySonine§, whointegratedtheformula(§13'42) aftermultiplication byF(R)RclR, from tox,soastoget r-'+if"I" J,+i(ur)J,{uR)F(R) RdRclu =fR''+^F(R) dR; .'0Jo Jo andthen,bydifferentiatingboth sideswithrespecttor,formula(3)isatonce obtained; butthewhole ofthisprocedureisdifficult tojustify. Aproofofamoredirectly physicalcharacter hasbeengiven byBasset ||, but,accordingtoGrayandMathews, itisopentovariousobjections. Aproofdepending onthetheoryofintegral equationshasbeen constructed byWeyllT. The extension ofHankel 'sformula, which iseffectedbyreplacingthe *Seee.g.Orr's papercited later inthissection. tAstatement ofamode ofdeducing (3)from(1)when visaninteger wasmade byWeber, Math. Ann. vi.(1873), p.149,butthiswasprobably laterthan Hankel's researches, since itis dated 1872, while Hankel's memoir isdated 1869. +Quarterly Journal, xxiii. (1889), pp.223— '244. §JIath. Ann. x\i.(1880), p.47. IIProc. Camb. Phil. Soc. v.(1886), pp.42-3—133. SeeGrayandMathews, ATreatise onBessel Functions (London, 1895), pp.80—82. HMath. Ann. lxvi.(1909), p.324. 14-3] MULTIPLE INTEGRALS 455 Bessel functions byarbitrary cylinder functions, wasobtained byWeber*, and itwillbediscussed in§§14'5—14"52. AnattempthasbeenmadebyOrrftoreplacetheBessel functions byany cylinderfunctions, thew-pathofintegration beingacontour which avoids the origin*,butsome oftheintegralsusedbyhimappeartobedivergent,soitis difficult tosaytowhat extent hisresults arecorrect. Thesame criticism appliestothediscussion ofWeber'sprobleminNielsen's treatise. Itwillbe shewn(§14'5) thatif,asNielsen assumes, thetwocylinderfunctions under theintegral signarenotnecessarilyofthesametype,therepeated integralis not,ofnecessity, convergent. Itshould bestated that, ifrbeapointofdiscontiimityofF{R),the expressionsontherightin(2)and(3)must bereplaced by:!: l[F{r-0) +F{r+0)\, justasinFourier's theorem. Forthemore recent researches byNeumann,thereader should consult histreatise Ueber dienach Kreis-, Kugel-unciCylinder -functionen fortschreitenden Entwickelmigen (Leipzig, 1881). Neumann's formula(1)wasobtainedbyMehler§asalimitingcase ofa formula involving Legendrefunctions;infact,itwasapparentlywith this objectinview thatheobtained theformula of§5"7l, lim P,,[cos{zin)]=/„{2), but itdoes notseemeasytoconstruct arigorous proofonthese lines(cf. §14-64).Amore direct method ofproofisgiveninadifficult memoir by DuBoisReymond||onthegeneral theoryofintegrals resemblingFourier's integral.Theproofwhich weshallgivesubsequently (§§14'6 etseq.)isbased onthese researches. SubsequentlyErmakoffHpointedoutthat theformula isalsoderivable from aresult obtained byDuBoisReymondwhich isthedirect extension to twovariables ofFourier's theorem foronevariable, namely =J_ ["p j"ryir(X,Y)cos[a{X-x)-\-i3{r- i/)].{dXdF)dad/S. Ermakoff deduced theformula bychangingtopolarcoordinates bymeans of thesubstitution a=ucos 0), (3=usinco, andeffectingtheintegrationwithrespecttoco. *Math. Ann. vi.(1873), pp.146—161. tProc. BoyalIrish Acad, xxvii. a,(1909j, pp.205—2-18. JThevalue oftheiutegral atapointofdiscontinuity hasbeenexamined withsome careby Cailler, Archives desSci.{Soc. Helvetiqne), (4)xiv.(1902), pp.B47— 350. §Math. Ann. v.(1872), pp.135—137. IIMath. Ann. iv.(1871), pp.3G2— 390. HMath. Ann. v.(1872), pp.639— G40. 456 THEORY OFBESSEL FUNCTIONS [CHAP. XIV If(?',<^)and(R, <i>)bethepolarcoordinatescorrespondingtotheCartesian coordinates(x,y)and{X,Y)respectively,theformal result isfairlyobvious whenweI'eplace^(X,Y)byF{R,^) ;buttheinvestigation bythismethod isnotwithout difficulties, since itseems tobebynomeanseasytoprovethat therepeated integraltaken overaninfiniterectangleinthe (a,/3)planemay bereplaced byarepeated integraltaken overtheareaofanindefinitely great circle. Ifthearbitraryfunction F{R,^)isnotcontinuous, thefactorF{r, (f>) which occurs ontherightin(1)must bereplaced bythelimit ofthemean value ofF(R, 4>)onacircle ofradius 8with centre at(r, cf))when8^0. This was, ineffect, proved byNeumann inhistreatise of1881, andtheproof willbegivenin§§146—14"63. Thereadermight anticipatethisresult from what heknows ofthetheoryofFourier series. Aformula which ismore recondite than(3),namely (4)I\IJ^^ J.iu-r)-^^^^^F{R)dudR=F{r\ hasbeenexamined byBateman,Proc.London Math. Soc.(2)iv.(1906), p.484;cf.§12'2. 14'4. HankeVsrepeated integral. ThegeneralisationofNeumann'sintegralformula which waseffectedby Hankel (c£§14"3)inthecaseoffunctions ofasingle variable, maybeformally stated asfollows : LetF(R)beanarbitrary function ofthereal variable Rsubjecttothe condition that rF{R)s/R.dR Jo eocists and isabsolutely convergent; and lettheorder voftheBesselfunctions benot* lessthan—|.Then TOO rx (1) udu \F(R)J,(uR)J,{u7-)RdR=l{F(r+0)+F(r-0)}, J .' providedthat thepositive number rliesinside aninterval inwhichF{R) has limited totalfluctuation. Theproofwhich weshallnowgiveissubstantiallyHankel'sproof, and it isofthesamegeneralcharacter astheproofofFourier's theorem;itwillbe setoutinthesamemanner astheproofofFourier's theoremgiveninModern Analysis, Chapterix.Itisfirstconvenient toproveanumber oflemmas. *Itseems notunlikely that itissufficient forvtobegreater than-1;buttheproof forthe more extended range ofvalues of i>would bemore difficult. 14-4, 14-41] MULTIPLE INTEGRALS 457 14*41. Theanalogue oftheRiemann-Lehesgue lemma. Aresult, which resembles thelemma ofRiemann-Lebesgue*inthetheory ofFourier series, andwhich isrequiredintheproofofHankel'sintegral theorem isasfollows : Let-f IF(R)\/R.dR exist,and(ifitisanimproper integral)let ithe absolutely convergent ;and letv^—\.Then, «s A.-^cc , F{R) J,(XR)RdR=o(1/^/X). Itisconvenient todivide theproofintothreeparts;inthe firstpartitis assumed thatF(R) \/R isbounded, andthat bisfinite;inthesecondpart therestriction that bisfinite isremoved;andinthethirdparttherestriction thatF(R)\/Risbounded isalsoremoved. (I)Lettheupper bound of\F{R)\/R\beK.Divide therangeofin- tegration (o,b)into nequalintervalsbythepoints cc^,Wo,...Xn-i{^o= ^, Xn=b),andchoose 7isolargethat 111=1 where eisanarbitrarilysmallpositive number and17,^andL^aretheupper andlower bounds ofFiRJ^/'R inthemih interval. WriteF{R) sJR=F(R,„_^) \/R„i-i +Wm(R),sothat,whenRliesinthemth interval, |«,„(R)\-^ U,,^—L,„.Now,whenv^-h, both ofthefunctions ofa;, rx x^J^{x), tiJ,(t)dt, .'o arebounded when x^0,eventhoughtheintegralisnotconvergentas^-*oo . LetAandBbetheupper bounds ofthemoduli ofthese functions. Itisthen clear that F{R)J,{\R)RdR\ 2F(R,,_,) s/R,,_, /,(XR)^R.dRm=l .Xm-l +S I'" oy,,,{R)J,{XR)sJR.dRm=l.'Xm-l 2B'*Ae A-,„=! \'A. 2BnK Ae *Cf.ModernAnalysis, §9-41. tTheupperlimit oftheintegral may beinfinite;and«^0. Theapparently irrelevant factor Rpreserves theanalogy with§1-1-3(3). 458 THEORY OFBESSEL FUNCTIONS [chap. XIV Bytaking Xsufficiently large (nremainingfixed after ehasbeen chosen) the lastexpressioncanbemade lessthan2Ae/'\/\,andsotheoriginal integral is(1/a/X). (II)Iftheupperlimit isinfinite, choose csothat Jc^\F(R)\^/R.clR<6, andusetheinequality F(R)J,{\R)RdR then, proceedingasincase(I),weget F{R)J,(\R)RdR€ ;F(R)J,(\R)RdR' +A_F{R)\^R.dR; IJa2BhJl2Ae Thechoice ofnnowdependsonethroughthechoice ofcaswellasbythe mode ofsubdivision oftherangeofintegration (a,c) ;butthechoice ofnis stillindependentof\,andsowecaninfer that theintegral (with upperlimit infinite)isstillo{l/\/\). (Ill)IfF{R) \/R isunbounded*, wemayenclose thepointsatwhich itis unbounded inanumber pofintervals 8such that t\\F{R)\s/R.dR<e. sJs Byapplyingtheargumentsof(I)and(II)tothepartsof(a,b)outside these intervals, weget 1'f(R)J.(XR) RdR I<25-^(4±iM +3^,^JaIA,* VA, whereKisnowtheupper bound of [F(R) \\/Routside theintervals S.The choices ofbothKandwnowdependone,butare stillindependentofX,so thatwecan still infer thattheintegraliso(l/VX). 14'42. TheinversionofHankel'srepeated integral. Weshall nextprove that,whenv^—\,and IF{R)\'R.dRexists and Jo isabsolutely convergent,then ndu\ F{R)J,{uR)J,(ur)RdRJo f=limF(R)] J,{iiR)J,{ur) k-^aa J [Jo providedthat thelimit ontherightexists. *Cf.ModernAiudysis, §9-41.•)uduyRdR, 14-42, 14-43] MULTIPLE INTEGRALS 459 Foranyassignedvalue ofX,,andanyarbitrary positive number e,ex hypothesithere exists anumber /3such that r\F{R)\^JR.dR< /3 whereAistheconstant defined inS14"41.2.4-\' 'V{R,u)du\dR- \\ \ (/){R,u)dR\Ifwewrite F(R)J^(uR)J,{ur)uR=(f>(R, u), itisclear that* (""j\^c})(R,u)duldR- 1^\f (f)(R,a)dR\du IdR-r\ffpiR, u)dR\du ^r \f^(R,u)Idu]dR+r\rl<^(R,«>ldR]du -^ri'^^\F(R)\^R. dudR+[^ I4-\F{R)\'^R. dRdu Jp.0V' JJpV^' Since thisresult istrue forarbitrarilysmall values ofe,weinfer that I{R,u)dudR= 1r(jyiR,u)dRdu, Jo•' . theintegralonthe leftexistingbecause theintegralontherightisassumed toexist. Iftheintegralonthelefthasalimit asX,^x,itisevident. from thedefinition ofaninfiniteintegralthat I"" udurF{R) .7,(uR) .7,(ur)RdR . .0 =limruduT F(R)JJuR)J,(ur-)RdR A.^.XJ .0 =limrF{R)\r7,(uR) .7,(ur)udu\RdR, andthis istheinversion formula which hadtobeproved. 14^43. Therelevantpartoftherange ofintegrationinHanhel'srepeated integral. Nextweshallprove that, inHankel'sintegral,theonlypartofthei?-range ofintegrationwhich contributes anythingtothevalue oftheintegralisthe partofthepathintheimmediatevicinity ofr,provided merelythatF{R) s/R hasanabsolutely convergent integral. *Thejustificationoftheinversion oftheorder ofintegrationfor n,finite rectangle whose sides are\and/3presents nogreat theoretical diriiculties. 460 THEORY OFBESSEL FUNCTIONS [chap. XIV Toeffect this, itissufficient toprove that, ifrisnotapointofthe interval*(a,h),then fudu (F{R) J,(uR)J,(ur)RdR=0. ./ Ja Weinvert theorder oftheintegrations,asin§14'42, andwefind that,if thelimits ontheright exist, uduIF(R) J,{uR) /.(ur)RdR Ja =lini I^F{R) \l\j,.(uR) ,/,(la-)udiilRdR K^:r..n (,.'o .1 bJO =lim F{R) [RJ.+, (7^R) J.(Xr)-rJ,+, (Xr)J,(XR)] Km A./,(Xr) f'~^^-^ J,+,(XR)dRx^x Jn-ti-—r-X.RdR R^-r' Since both theintegrals-limXrJ^^,(X7') C^J;f^^^JAXR)dR. ^F(R)Ri rF(U) JaR'-dR,f^F(R)Rh dR r^Ia-R'-r' areexhypothesi absolutely convergent,itfollows fromthegeneralised Riemann- Lebesgue lemma(§14'41) thatthelasttwolimits arezero; andso .'uduF(R)/,(uR) ,/,(ur)RdR^O providedthatrisnotsuch thata^r^b. 14"44. Theboundednessuf IJ^(uR) J^(a7^)uR^dudR. JaJ Itwillnowbeshewn that, as\^~ cc,therepeated integral hTA Jt,(uR)Jt,(ur)uRdudR aJ remains bounded, providedthataand bhaveany(bounded) positivevalues. It ispermissible foraand btobefunctions ofXofwhich one(orboth)maytend toras \.^00 . Letusfirstconsider theintegralobtainedbytakingthedominant terms oftheasymptotic expansions, namel}' 9 rbrA , /cos(uR—hi'TT—iir)cos(ur—^vir—4-7r)dudR TT'^rJaJo 1f''[s\nX(R—r) cos[X(R+r)—vtt]—cosTTT TT\/rja 1 TT\/'rR-r AC'-)-) sinX X{a-r) XdxR+r A(Hr) cos(X-vtt) Kia+r)^'dR dx+cosvirlogb+r a+r Itispermissiblefor litobeinfinite. 14-44] MULTIPLE INTEGRALS 461 The firstintegralisbounded becausesin a? X-dxisconvergent;andthe \^Cos \'(jC VIT) secondintegralisbounded because I— ^ dx\%convergent;andso J X theintegralnowunder consideration isbounded, and itslimit, asX-*x,is thelimit of _1 providedthat thislimit exists. Butwemaywrite h,-Adx+cosVKlog A(f(-)-) ^ a+r J^(uR) Jv(ur)uR-dudR aJ TT'\/rJ ab('00 [WuJAnR)-h{in-)^!{Rr) —cos{uR—\viT—jtt)cos{ur—\v7r—jTr)]dudR b"» ll-rruJ, (uR) J,(wr)^J(Rr) —cos(uR—\v7r—jTt)cos{ur—^vrr—jtt)]dudR H T- I ICOS(i^i^— |i'7r—Itt)COS(«r—i/TT— jtt)c?uc?ii. TT\/rJ„J Now, oftheintegralsontheright,thefirst istheintegi-alwithrespectto Rofanintegral (with respecttoit)whichconverges uniformlyinanypositive domain ofvalues ofRand r,andsoitisacontinuous (and therefore bounded) function ofrwhen rispositiveandbounded. Thethirdintegralhasbeenshewn tobebounded, and itconvergestoa limitwhenever •Hb-r) sinX da \{a-r) X does so. Thesecondintegral maybewritten intheform 4>v--1-h''00 y^TT \/r.aJ\uRsin{uR— |i/7r—Jtt)cos{ur—\v7r—^ir) -\co?,{uR—\vTr—jtt)sin{ur—^vir—\it){0{llu-) dudR 4z;2-l ['' 47r\/r Ja^cf,^{X)+lcfy,{\) +cP,{X) dR, where (j)^(X), (po(X)and^3(X)arefunctions ofXand 7^which tenduniformly tozero asX-^00 ..• 462 THEORY OFBESSEL FUNCTIONS [CHAP. XIV Hence, forallboundedpositivevalues ofa,h,r,theintegral rbr\1 J^(uR)Jv(ur)uR^ diidR JnJ isbounded asA.^-oo;and itconvergestoalimitwhenever r\{b-r) sin a; ,ax Jk(a-r)^ does so. 14"45. Proof ofHanleV sintegraltheorem. Now that allthepreliminary^ lemmas havebeenproved,theactualproof ofHankel's theorem isquite simple. SinceF(R)haslimited fluctuation inaninterval ofwhich risaninternal point,soalsohasF{R) \/R ;andtherefore wemaywrite whereXi(-^)^^^%2(-^)^^^monotonic(positive) increasingfunctions. After choosingapositivenumber earbitrarily, wechoose apositive number Ssosmall thatF(R)haslimited total fluctuation intheinterval(r—B,r+8) andalso %i(r+8)-%i(r+0)< e-|xi0'-0)-%i(r-8)<e] X2{r+8)-X2{r +0)<e]' ;^,(r-0)-^.C/'- 8)<ef' Ifweapplythesecond mean-value theorem, wefindthatthere exists a number |intermediate invalue between andSsuch that /i(R)J^(iiR)J^(ur)u>JR.dudR =Xi(^'+^>)\ \J^("^)J^("0u^R.dudR +[Xi i^'+S)-%i(^*+0)}rIJ.(uR)J,(ur)u\/R.dudR. Jr+fJ t"sin cc Since• I dx-^^-tt,JoX asX,-^00,8remaining fixed, itfollows from§14'44 thatthe firstterm onthe righttends toalimit asX-*gowhile 8remains fixed. Andthesecond term ontherightdoesnotexceed Ceinabsolute value, whereCistheupper bound ofthemodulus oftherepeated integral (cf.§14-44), Hence, if Km I J^(uR)J^(ur)uR^dudR =C\/^r, itfollows that limr 1x^(R)JAi(R)J.(ur)uR'^dudR A.-».00 .'rJo exists and isequaltoC^Xi (^+0)l\/r. 14-45] MULTIPLE INTEGRALS 463 Wetreatx^i-^)^'^^similar manner, and alsoapplysimilarreasoningto theinterval. (;—h,r) ;andweinfer that, if lim \ IJ^{uR) ./,(ur)uR^duclR=C.,/\/7; then lim I IF{R)J,{aR)J^{ur)uRdudR exists and isequalto C,F{r +0)+C,F{r-0). Wenowhave toevaluate Ciand C.,.Bythetheoryofgeneralised integrals*,wehave ri rccrr+S , ,JAyR)Ju{iir)nR^dRdu \/r JoJr fr+S=limexp(—|)-it^)IJy(uR)J^,(ur)uR'-^ dRdu I"exp(—jj-U-)J^(uR) J^(ur)uR^dudRrr+S /'x=lim p^O.r.' by§13-31(1). Now, throughouttherangeofintegration, /.^^'\_Pn ,n/.V2M .-...,.^^'^!l+0/)]exp(|" V2/V v/(7ri?r)' ^^ ^^ -t- \2if)' j,2psJ{Rr) [ 4<p' J asj9^0. Hence 6'i=lim^——exp -^-—,—;H^ }dR andsimilarly1r*^"'=Km—r- exp(—X-)dx=\, 1/""Co=lim—f— exp(—^•^)dx=^. We^iave therefore shewn that limr^ fV(i?) ./,(ui^) ./,(ur)uRdudR exists and isequalto ^\F(r+0) +F(r-0)}. *Hardy, Quarterly Journal, xxxv.(1901), pp.2-2—66.Foradifiereut method ofcalculating C,andCo,see§14-52. 464 THEORY OFBESSEL FUNCTIONS [CHAP. XIV But, ifthislimit exists, then, by§14"42, rudurF{R) J,(uR)J,{ur)RdRJo JO also exists and isequaltoit;and sowehaveproved Hankel's theorem, as stated in§14"4. Theuseofgeneralised integralsintheproof ofthetheorem seems tobedue to Sommerfeld, inhisKouigsberg Dissertation, 1891. Forsomeapplicationsofsuchmethods combined with thegeneralresults ofthischaptertotheproUentedesmoments ofStieltjes, seearecent paper byHardy, Messenger,XLVii. (1918), pp.81—88. 14'46. NoteonHankeVs proof ofhistheorem. Theproof given byHankel ofhisformula seems todiscuss twopoints somewhat inadequately.The first isinthediscussion of lim I/F{R)J^{uR)J^{ur)uI{dudR, which hereplaces by \^x, Jli--r- Inorder toapproximatetothisintegral, hesubstitutes thefirstterms oftheasymi^totic expansionsoftheBessel functions withoutconsidering whether theintegrals arising from thesecond andfollowing terms arenegligible (which seems afatalobjectiontotheproof), andwithout consideringtheconsequences oi\Rvanishingatthelower limit ofthepath of integration. Thesecond point,which isofasimilar character,isinthediscussion of lim I IJ^(uR)Jy(ur)uRdudR;A^* Jr+f J afterpi'oving bythemethod justexplainedthat this iszero if^tends toapositive limit and is^if^=0,hetakes itforgranted that itmust bebounded if^-»-0 asX-^x;and thisdoesnotseemprima facieobvious. 14*5. ExtensionsofHankeVs theorem toanycylinder functions. Weshallnowdiscussintegralsofthetype uduF{R)^,( uR) "W,(Mr)RdR, inwhich theorder voftheunrestrictedcylinderfunction'^^,(2')isanyreal* number. Thelower limits oftheintegralswillbespecified subsequently, since itisconvenient togivethem values which dependonthevalue ofv. Fordefiniteness we-shallsupposethat ^,.(z)=a{coscc.J^,(z) +s\n(i.Y^ (z)], where crandaareconstants. *Thesubsequent discussion issimplified andnogeneralityislostbyassuming thatv^O. 14-46-14-51] MULTIPLE INTEGRALS 465 TheanalogueoftheRiemann-Lebesgue lemma(§14-41), namelythat Ja provided that [F(R)^R.dR Ja exists and isabsolutely conver^gent, mayobviouslybeproved bypreciselythe methods of§14-41, provided thata<:6<oo,and ^a^ if^y^1, \a>0ifv>h Thetheorem of§14-44 hastobemodifiedslightlyinform. Themodified theorem isthattherepeated integral •br^ (uR)'^,{ur)u^/R,dudR aJT isbounded asX-^cc while tremains fixed; asin§14-44,aand hmaybe functions of\which have finite limits asX,-^oc .Thenumber rispositive, thoughitispermissibleforittobezerowhenO^t-^^, Also therepeated integralandtheintegral I dx J\(a-r) ^ bothconvergeorboth oscillate asA,^qo . [Note.Ifthetwocylinder functions intherepeated integral were notofthesame type,i.e.ifweconsidered theintegral ' f^ -^^(uR)^^(wr)?ts/R.dudR, aJr itwould befound thattheconvergence ofthisintegral necessitates theconvergenceofthe integral fK(h-r) 1-cos.r, / dx; andso,ifX(a—r)-^-0 asX-^x,therepeated integralisdivergent*.] 14-51. TheextensionofHankel's theorem when "^v%^. Retainingthenotation of§§14-4—14-5,weshallnowprovethefollowing theorem. Let^j F(R) \/R.dR existand beanabsolutely convergent integral, and letO^v^l.'J^hen (1) l^adurF{R) 9^,(uR) ^^,(ur)RdR *Thispoint wasoverlooked byNielsen, Handbuch derTlworie derCylinder Junhtionen (Leipzig, 1904), p.365, inhisexposition ofHankel's theorem. w.B.F. 30 466 THEORY OFBESSEL FUNCTIONS [chap. XIV providedthat thepositive number rliesinside aninterval inwhichF(R) has limited totaljiuctuation. Asin§14"42, wemayshew that Iudu Jo JouduF{R)9^, (uR) "W,{ur)RdR Jo =lim\^F{R)\^'i^, (uR) "W,iiir)uRdudR, X-»• JO . i providedthatthelimitontherightexists. Butnowweobserve that '^,{uR)''^,{ur)udu 1 R'-r'uR'^^,+1{uR) "W,(m-)-ur%\+, (ur) "W,(uR) X 2a"sinasin(a+vtt) R^"-r^" TTsinvir R''r''(R^-r')' Hence weinfer that, ifrisnotapointoftheinterval{a,h),then F(R) f^ '^^(uR) "&,{iir)uRdudR TTsin y-TT J„R''r''(R^-r^) asX-^00 ;andsothelastrepeated integralhasalimitwhen X.-^oo . Now choose anarbitrary positive number e,andthen choose 8sosmall thatF(R)haslimited total fluctuation intheinterval(r—B,r+8)andsothat F(R)-F{r+0)\<eifr<R^ r+8, F{R)-F{r-0) I<eifr-8^R<r. Now take rF{R)^'i^,{uR)'^,{ur)ududR,Jo Jo anddivide thei?-pathofintegrationintofourparts, namely (0,r- 8),(r-8,r),(/,r+8),(r+S,x). Applythesecond mean-value theorem asin§14'45, andwefindthat F{R)rIK(iiR) "^^(wr)uRdudR 2cr2 gjj^ Qjgjjj^Qj_^j^^^fJ'*'"'* TTSm 1^73-+IR""-r'&v +i^(r+0)Vr.r+Sr\i'o ]r+slR''-'r''(R^-7-) '&',{uR)%{ur)uR^dudRF(R)dR + +v,F(r-0)Vr.fr<^,(uR)^,(ur)uRhladR Jr-sJ 14-51] MULTIPLE INTEGRALS 467 where |rj\hasanupperbound which isindependentof\andwhich is arbitrarilysmallwhen eisarbitrarilysmall. Theintegralsontheright convergetolimits when \-*x,and so,by makinge-^ after A,^x,weinfer that udu F(R) 'W,(uR) '<S,(ur)RdR Jo isconvergentandequalto 2a--sinasin(a+vtt) f'-^R-"-r-" TTSniVTT JI,^v ,yr^T^lT> TxF(R)dR /,)R''-'r''{R'-t'-)^' +i^(/-+0)s/r.liinrf' "(h(kR)%(ur)uRHRdu +F{r-0) ^/r.lim["1' '/^,(tiR)W,(xr)uR^dRdu, providedthat thelimits ontherightexist. Toprovethatthelimits existand toevaluate themsimultaneously,take F{R)=R"when r<R<r+8 andF(R)=forallother values ofR. Wethus findthat •00fr+S 7-'+Mimr['^lP,{uR)9S,{ur)uRhlRduS^O JJr =limf"/'^%{uR)9/,(ur)uR-'+'dRdii,&^i)Jo Jr•-crr+S S^[)JO Jr providedthat thisrepeatedlimit exists;andsimilarly Jr,-+ilim I" f'r^,.{uR)%\{ur)uRHRda =limf"|' '6^,{uR)9^,(m^)uR''+'dRdn. Forbrevitywewrite hinplaceof /'+8.Wethenhave %,{uR)%{ur)uR^+'dRdu =["{¥+'y/,+1 {uh)-?-'+i%%i {ur)] 'W,{ur)dii J ^ =limr{6''+>'2?.+:(u6)-r''^>^+,(«r)}'^,(wr)-!*, since thesecond ofthese threeintegralsisconvergent,andthethird isabso- lutely convergent when <p<1—z'. Now the lastexpressioncanbereplaced byacombination ofthefour integralsofthetypes Ju{xr) du 30-2 468 THEORY OFBESSEL FUNCTIONS [chap. XIV andthese are allabsolutely convergent. Theymaybeevaluated ascases of Weber's discontinuousintegralof§13"4,andhence wefindthat '' {6"+^ '^.+z (uh)-V^^K+, (ur)]^,{ur)^* o--?-"sin(a+pir)sin(a+vtt).V{}>-\-\— p) 22psinpTTsinfTT .r(i/+l)r(|0+1) X62P.^,{v^\- p.,+i.^V,.r(.+i)r(2p)- a-r"sinasin(a+pir+vtt).F(1—p) 2-fsin(pTT+vk)sin i/tt .r(1— I/)r(i/+p+1) ]f.v+2p iFiil-p, -V-p\l-v;^J- ] Thelimit ofthisexpression,whenp-^0,isreducible to (r-r"sinasin(a+vtt) TTsin t'TT^V h-J^rvr log(l-2) +21og^-:^,i^(l,-.:l-.;^;) +^TTcota—Jttcot(a+i/tt)—-^{\)-\-y^{—v) \, aftersomealgebra ;andthelimit ofthelastexpression, when 6-*-/•+0,is simply \a-r^. Inlikemanner itmaybeshewn that andsowehaveprovedthat uduF(R)^,(uR)^,(ur)RdR =la^-{F(r +0)+F{r-0)]2a-sinasin(a+vtt) TTsinVTTR'"-riv R''-'r''{R^-r'-)F{R)dR, providedthat ^v<:^,F{B.)\% subjecttotheconditions stated in§14-4',and *^„{z)=a-[cosaJ^(z)+sinccY^ (z)]; andthis isthegeneral theorem stated atthebeginningofthesection. 14*52. ]Vehe7-'sintegraltheorem. Itisevident from§14-.51 that, if 1F(R)'JRdR exists and isabsolutely Ja convergent,where a>0,then (1) limrF{R) [RW.^, (\R)^^(Xr)-r^,^, (Xr) <^,(Xi^)]^^, =|(7={i^(r +0)+i^(r-0)}, providedthat rliesinside aninterval m.whichF{R)haslimited total fluctuation andF{R)isdefined tobezerowhen $i^<a,iftheorder ofthe cylinderfunctions liesbetween —\and\. 14-52] MULTIPLE INTEGRALS 469 Weshallnowestablish thetruth ofthisformula forcylinderfunctions of unrestricted order. Let[RW,^, (\R)W,(Xr)-r%^, (Xr) "&,(XR)]^A^= <j>^(^^,..^y Itisaneasydeduction from therecurrence formulae that ^.{R,r; \)-<P,_,(R, r;\)= j^S'^f.-dXR)%M\7^)+''S',.,(X7-)%M\R)], and so,bytheanalogueoftheRiemann-Lebesgue lemma(§14-41), wehave (2) lim["[$,(R,r ;X)-<P,_,(R,r ;X)]RF(R) dR=0. Hence, byadding uprepetitionsofthis result, (3) hmr[<I>,{R,r; X)-<t>,±n (R,r ;X)]RF(R) dR=0, where 7iisanypositive integer. Choose nsothatoneoftheintegersv±nliesbetween +|,andthenfrom(1) limr^,^n{R,r; X)RF{R)dR =la'{F{r+0)+F(r- 0)], A-».oo Ja and so,forallrealvalues ofv,wededuce from(3)that (4)limr^,{R,r; X)RF{R)dR=la^{F{r +0)+F(r-0)\.K^^x Ja This result ispracticallyduetoWeber*, and itwasobtainedbythemethod indicated in§14'46. Toobtain theresult inWeber's form, let P,(,)=F,(r)/,(z)-./.(r)F.(z), ^'^ l"^.(^)=Y.(R)J.(s)-J.{R)F,(z). Then_ XI'W,(uR)"7^--^^(ur)-^^au du=(R'- ?•-)W,(U7-)W,(uR)udu, 1 andtheexpressiononthe left isalsoequalto u[iiC+i i'lR)%(wr)-r%+, {ur)W,{uR)] =uR[Y, (R)/.+!(«E)-J.(R)F,+i (iiR)] [F,(r)J,(ur)-J,(r)F,(ur)]-ur[Y^ (r)J^^,(ur)-J,(r)F,^,(ur)][F,(R) ./,(uR)-J^(R) F..(uR)] =uY,(R)Y,(r)[RJ.+, (uR)J,(ur)-rJ,+, (ur)J,(uR)\ +.^i {./,(R) F.,(r)-./,(r)F,(R)][RJ,+, (uR)F,(ur)-RY,+,(uR)J,(ur) —?'Fi,+i (ur)J^(uR)+r./^+i (ur)Y^{uR)] -\u[J, (R)F,(r)+J,(r)F,(R)][RB,+^ (uR)D,{ur)-rD,+, {ur)D,(uR) -RD,+, (uR)D,{ur)+rD,^, (ur)D,(uR)] -nJ,(R)J,(r)[RY,+, (uR)YAkv)-rY,+, (ur)F,(uR)], where p.(.)=J.(.)+F.(.), \DAz)=J.(^)-YA^)- *Math. Ann. vi.(187H), pp.14(3—161. 470 THEORY OFBESSEL FUNCTIONS[CHAP. XIV Nowsupposethat '^ f(R)BdR exists and isabsolutely convergent;andconsider \iml'^f{R)\ '^,(ur)%\(uR)uRcludR. Carryouttheintegrationwithrespecttou,andreplacetheintegrated part bythesum ofthefourterms written above, dividedbyR^—r^. ^., J,{R)Y,(r)-J^(r)l\(R) si-—r- ishounded near r,andhaslimited totalfiuctuationinanyhounded interval containing r,itfollowsthat theintegrals correspondingtothesecondgroup ofterms tend tozeroas\-^y:,hythegeneralised Riemann-Lehesgue lemma. Correspondingtothethirdgroupoftermswegetapairofintegrals which happentocancel. When we-use(1),wearether^efore leftwith theresult that limrf{R)[^'^^(ur)^,(uR)uR .dudR =il^.HO+IV(r)}.{/(r+0)+/(r- 0)}, that istosay (6) l'^udu\"f{R)%\{ur)%\{uR)RdR =i{/.M'O+F.Hr)}.{/(r+0)+/(r- 0)1, inwhich thecylinderfunctions aredefinedby(5),andrliesinside aninterval inwhich /(-R)haslimited total fluctuation. Apartfrom details ofnotation, this istheresult obtainedbyWeber inthe case offunctions ofintegralorder. 14"6. Formal statementofNeumannsintegraltheorem. We shallnow statepreciselythetheorem which willbethesubjectof discussion inthesections immediately following.Itisconvenient toenunciate thetheorem withDuBoisReymond's* generalisation,obtainedbyreplacing theBessel function byanyfunction which satisfies certaingeneralconditions. Thegeneralisedtheorem isasfollows : (I)Let'^{X, Y)heahounded arhitrary function ofthepair ofreal vaj'iables {X,Y),which issuch that thedouhleintegral rr^iX,Y).{X'+rO"'•(dXdY) exists and isahsolutely convergent. *2Iath. Ann. iv.(1871), pp.383—390.Neumann's formula(cf.§14-3)isobtained bywriting g{t)=Jo{t), andtheconditions (I)— (III)aresubstantially those given inNeumann's treatise publishedin1881. 14-6, 14-81] MULTIPLE INTEGRALS 471 (II)When "^'(X, F)isexpressedintermsofpolar coordinates, letithe denotedhyF(R, <I>),and letF{R, <I>)have thepropertytJiat(forallvalues of^bettveen ±tt),F(B, <I>),quafunction ofR,haslimited totalfluctuationin theinterval(0,oo);and lettJiisfluctuation andalsoF(+0,<I>)beintegrable functions of^. (III) IfQ{R,^)denote thetotal fluctuation ofF(R, <I>)intheinterval {±0,R),letfl(R, 'i')tend tozerounifoj'm.hjwithresjjectto <l>asR^^O, throughoutthewholeoftheinterval (—tt,tt),with theexception* ofvaluesof<P inanumberofsectors thesumoftuhoseangles maybeassumed arbitrarily small. Since \F(R, ^)-F{+0,(i>)\^n(R, <t>),thiscondition necessitates that F(R;^>)^-F(+0,<I>)uniformly exceptintheexceptioncdsectors. (IV) Letg{R)beacontinuousfunction ofthepositivevariable R,such thatg{R)\/R isbounded bothivhenR-^0 andivhenR-^cc . f-K f" dt Let q{t)tdt=G{R), and letG{t)—heconvergent. .'o' -'o t ThenI"udu["["^^(Z,F).g\u^(X'^+F'^)}.(dXdY) isconvergent, and isequalto .'i- 'Where^F {+0,<!>)means^ ^rFi+o,^)d^.Ztt J-„ Before provingthemain theorem, weshallproveanumber ofLemmas, justasinthecaseofHankel'sintegral. 14*61. Theanalogue oftheRiemann-Lebesgue lemma. Correspondingtotheresult of§14-41, wehave thetheorem thatifTis anunbounded domain\ surroundingtheorigin, ofwhich theoriginisnotan interiorjwintoraboundary point, then, as\^cc, (dRd^) 'j^FiR,ct>)GiXR)^-^^=o(l). *Theobjectoftheexceptionistoensure thatthereasoningisapplicabletothecase(which isofconsiderable physical importance)inwhich <if(A',Y)iszerooutride aregion bounded byone ormore analytic curves and is,say,apositive constant inside theregion, theorigin being onthe boundaryoftheregion. tThediscovery thattherepeated integralisequaltoanexpression involving themean value ofF(R, <f>)when theoriginisapoint ofdiscontinuityofF(B, 4>)wasmade byNeumann, Ueber dienach Kreis-, Kugpl- undCylinder-functionenforUchreitenden Entivickelungen (Leipzig, 1881), pp.130—131. XForinstance Tmight bethewhole oftheplane outsile acircle ofradius 5withcentre atthe origin. 472 THEORY OFBESSEL FUNCTIONS [CHAP. XIV Itwillbeobserved that this isatheorem ofamuch weaker character than the theorem of^14'41, inview ofhypothesis (II)of§14-6. Thereason ofthis isthefactthat O(XR)maybe*(v^X)forcertain values ofR,and thisseems tomake argumentsofthe typeused in§14"41inapplicable. Toprovethelemma, supposefirstthatTisbounded. Then, foranyvalue of<t>,F{R, <t>)maybeexpressedasthedifference foftwo(increasing) mono- tonic functions'x^i(R, <l>),%2{R, *^).whose sum isthe total fluctuation of F(R, <l>)intheinterval(0,R). IfRnandR^aretheextreme values ofRforanyparticularvalue of<i>,it follows from thesecond mean- value theorem that, forsome value ofR2between RoandR^, j^X^(R,^)G(XR)^=X,(i^o,<^)j^G(>^R)^+Xx{R.^)j^G(XR)^^ =X^{R„c^)\ G(t)^+xAR.,<^) Git)"^.JKE,t J>^E,t [^ dt . Since G(t)—isconvergent,ifeisanarbitrar}^ positive number, wecan choose X,solargethat rG{t)^<e, forallvalues of|notlessthan thesmallest value ofRq.Also Ixi{R> <E>) I^(Xi{R,^)-IF(+ 0,<!>)}+IIF(^ 0,^) I ^Xi{^,^)-hF{+0,^) +l\F(+0,^)\, andsimilarly whence itfollows that F(R,^)G(XR)^^^^T R ^2er[x,{co,^) +x-^('^^^)+F (+0,^)I }d^^ J—TT J—71 and, sinceF{{0,^)isbounded, thiscanbemadearbitrarilysmallbytaking esufficiently small, and itisindependentoftheouterboundaryofT.Hence wemayproceedtothelimitwhen theouterboundarytends toinfinity. *This isthecasewhen g{R)= Jf,(R);thenG(R)=EJi (R).Itisbjuomeans impossible that some oftheconditions imposed onF(R, 4>)aresuperfluous. tCf.ModernAnalysis, §3-64. 14-62] MULTIPLE INTEGRALS' 473 Weinfer that, ifThasnoouterboundary,themodulus of canbemadearbitrarilysmallbytaking \sufficiently large;andthis isthe theorem tobeproved. 14'62. TJieinversionofNeumannsrepeated integral. Weshallnextprovethat theexistence andabsoluteconvergence oftheintegral aresufficientconditions that Cudur r^V{X,Y).g[usJ{X''-+Y')].{dXdY)Jo J-r.J-X =limf"r^(X,Y)rg{uW(X"~+Y-'}]udu{dXdY), providedthat thelimit ontherightexists. Foranygivenvalue ofXandanyarbitrary positivevalue ofe,there exists anumber(3such that whereAistheupper bound of jg{u)jf^u. Wethenhave r\^F{R, ^)g{uR) udu .R(dRd^) -frrF{R,^)g{uR)R{dRd^)udu J(JJ-TTJ r-jT roorxrf"fF(R,^)g{uR) udu .R(dRd^)J-nJ13Jo "^^ F(R,(p)g{uR)R(dRd(^)udu J-n ^aT rr:F(R,^)\u'^duR^dRd^)- J-ttJpJ +a[j^r\F{R,<P)\Ri(dRd<l^)uiduJo J-ITJfi^^<€. Since this istrue forarbitrarilysmall values ofe,weinfer that c r-rr C'-i- F{R,^)g(uR) R(dRd^) udu J-TTJ rf"rF{R,^)g{uR) udu .R(dRd^^), J-TTJJo=lim theintegral ontheleftexistingbecause thelimit ontherightisassumed to exist. 474 THEORY OFBESSEL FUNCTIONS [CHAP. XIV Hence itfollows that, ifthelimit ontheright exists, then udu F(R,^)g{uR)R(dRd^) .'0 J-irJ =limrrF{R,^)G(XR) A-».30 J-rrJ(dRd^) R 14'63. Theproof ofNeumamn'sintegraltheorem. Wearenow inapositiontoprovewithoutdifficultythetheorem due to Neumann stated in§14'6.We firsttakeanarbitrarilysmallpositive number eandthen choose thesectors inwhich theconvergenceofH(R, <I>)tozero is uniform, insuch aw^aythatthesum oftheiranglesexceeds 27r—e.Wethen choose 8sosmall that II{R, <!>)<einthese sectors Avhenever R^8;andwe taketheupper bounds of n(R,^)+\F{R,^)\andf"(?(w)— ! tobeBand C. Wethenapplythesecond mean-value theorem. Wehave r^ dR jj,{R,^)G(XR)'^ r^ dR C^ rlTl=X^(+0,^) /^G(XR)^+(x.{8,<!>)-X.(+0,<I>)1j^G(xRf-^, where ^^^8. Now Hence"^ dR G(\R)RAS^j, G{u)U<2G. * dR f"^ diiF(R,^)G(\R)~=F{+0,^) G{ii)- +v, -tt J u where\r]\islessthan2eCinside thesectors inwhichconvergenceisuniform, and islessthan2BG intheexceptionalsectors. Hence itfollows that itJ . w R Hence, forlargevalues of\<27r.2eC+€.2BG =2eC{27r +£}. F(R, <!>)G(XR)^^^^1^^-27ri¥li?'(+ 0,<^)f'(?(u)— nJ Ji Jq u that istosay<2eC(27r +B)+o(l), lim A-*•00j^j^F(R,^)G(\R)^^^^^-27rmF(+0,^)j'^G(u)du I ^2eC{27r+B). 14-63, 14-64] MULTIPLE INTEGRALS 475 Now theexpression ontheleft isindependentofe;andsosince eisarbitrarily small, weinfer thatthelimit iszero. That istosay, K-^y. J-nJ{) -tt exists and isequalto J '' Applyingtheresult of§14-62, weseethatNeumann's theorem hasnow beenproved. Inthespecialcase inwhich g(a)=J(,(i(), wehave ru Itg{t)dt=tiJi{u),Jo sothat G(u)=mJ"i(u), 1r=°G(u)du r^,r'/ X, 7and\;={-Jo(u)]du=1. J u J Hence wehave (1)rudiir!^"i'(A;Y).J,[uv'(-Y^+701•(cLXd Y) =27rItl^ (+.cosa>,+.sin<l>). IfAvechangetheorigin, wededuce that (2)rudur r^¥(X,Y).J,[u^^{iX-.Ty +(Y-yy]].{dXdY)Jo J-V.J-X=27riH^ {x+cosO,y+sin^), andfinally, changingtopolar coordinates, (3)rudu\^rF(R,^)Jo[u^(R' +r"'-2Rr cos((l>-(f))]RdRd^Jo J-TTJ-00=27rmF(r, </>), whereJi"li^(r, 0)nowmeans themean ofthevalues ofF{R, <!>)when{R,^) traverses thecircumference ofanindefinitelysmall circle with centre(r,<^). 14*64. Mehler'sinvestigation ofNeumannsintegral. Neumann'sintegralhasbeendeduced byMehler* from theformula ^(0^(j))=i^!}^r[^f{S,^) F,,(cosy)sin^d(S^dS byalimiting process;inthisformula cos7=cos6cos@+sin sin cos(O— cj)). Theformula isobtainedf byconstructingasolution ofLaplace's equation, valid inside asphereofradius k,which hasanassignedvaluef{6, (j))onthe surface ofthesphere. *Math. Ann. v.(1872), pp.135—137;cf.Lamb, Froc.London Math. Soc.(2)ii.(1905), p.384. tCf.ModernAnabj^is, §18-4. 476 THEORY OFBESSEL FUNCTIONS [CHAP. XIV Thelimiting processusedbyMehler isthatsuggested bytheresult of §5'71;theradius ofthesphereismadeindefinitely large,andnew variables R,raredefined bytheequations R=kS, r=kO, sothatR,raresubstantially cylindricalcoordinates ofthepointswithpolar coordinates (k,©, <l>),{k,6,cfi);thefunction ofposition /(®,^)isthen de- noted byF(R, <t>),andP,i(cos7) becomesapproximately equaltoJo(117:7 /k), where OT-=R-{-r--2Rrcos(^- <^). Wearethus ledtotheequation K-^'X M=0*7'" .'0J-TT K' Ifnowwewrite'h/k=u,andreplacethesummation byanintegration (taking 1/kasthedifferential element), weget F(r,6)=^["udurrF{R,^)Jo(u^)RdRd^, which isNeumann's result. But thisprocedurecanhardlybemade thebasis ofarigorous proof,be- cause there aresomany stepswhichrequire justification. Thus, although weknow that X(pY^±irr /(e,^)P„ (cos7)sin0cZ<&de isapotentialfunction (when r<k),which assumes thevaluef(6, </>)onthe surfece ofthesphere,thetheorem thatwemayputp=kintheseries necessitates adiscussion oftheconvergenceoftheseries onthesurface ofthe sphere;andthetransition from thesurface ofaspheretoaplane, bymaking /c^00,with thecorrespondingtransition from aseries toanintegral,iffone ofconsiderable theoreticaldifficulty. Itispossiblethatthemethod which hasjustbeen described isthemethod bywhich Neumann discovered hisintegralformula in1862.Concerninghis method hestated that"DieMethode, durch welche ichdieseFormel soeben abgeleitet habe, istnichtvoUstandig strenge." CHAPTER XV THEZEROS OFBESSEL FUNCTIONS 15'1.Problems connected tvitli thezerosofBesselfunctions. There arevarious classes ofproblems,connected with thezeros ofBessel functions, which willbeinvestigatedinthischapter. Weshallbegin byproving quite generaltheorems mainlyconcerned with thefactthatBessel functions haveaninfinityofzeros,and withtherelative situations ofthezeros ofdifferent functions. Next,weshallexamine therealityofthezeros ofBessel functions (and cylinder functions) whose order isreal,and discuss theintervals in which therealzeroslie,either byelementarymethods orbytheuseofPoisson- Schafheitlinintegrals. Next,weshall consider thezeros ofJ^{z)when visnot necessarily real,andproceedtorepresentthisfunction asaWeierstrassian product. Wethenproceedtothenumerical calculation ofzeros offunctions ofassigned order, andfinallyconsider therates ofgrowthofthezeros with theincrease oftheorder, andthesituation ofthezeros ofcylinderfunctions of unrestrictedly largeorder.Afulldiscussion oftheapplicationsoftheresults contained inthischaptertoproblemsofMathematicalPhysicsisbeyondthe scopeofthisbook, thoughreferences tosuchapplicationswillbemade inthe course ofthechapter. Exceptin§§15'4—15"o4, itissupposedthat theorderv,ofthefmictions under consideration, isreal. Thezeros offunctions whose order ishalfanoddinteger obviouslylend themselves todiscussion morereadilythan thezeros ofother functions. In dx andbyRayleigh*;andmorerecently Hermitei* hasexamined thezeros of Jn+\{x).Thezeros ofthisfunction have alsobeen thesubjectofpapers by Rudskiij: whoused themethods ofSturm; but ithasbeenpointedoutby Porter andbySchafheitlin§thatsome ofRudski's results arenotcorrect, and, inparticular,histheorem that thesmallestpositivezero ofJn^^ (^)lies between\{n+1)ttand\(n+2)ttisuntrue. Such atheorem isincompatible with theinequality givenin§15"3 (5)andtheformrdae of§§15'81,15"83. *Schwerd, DieBeugungserscheinungen (Mannheim, 1835);cf.Verdet, Lemons cVOptiquc rjujsique,I.(Paris, 18G9), p.260; Kayleigh, Froc.London Math. Sac. iv.(1873), pp.95—103. tArchiv derMath, undPInjs. (3)i.(1901), pp.20—21. XMem. delaSoc.R.desSci.deLie/ie, (2)xviir.(1895), no. 3.SeealsoPrace Matonatijczno- Fizyczne,iii.(1892), pp.69—81. [Jahrbnchilber dieFortschritte derMath. 1892, pp.107—108.] §Porter, American Joiirwil ofMath. xx.(1898), p.198; Schafheitlin, Journal fiirMath, cxxii. (1900), p.304.particularthezeros of—^-^^—^have beeninvestigated bySchwerd 478 THEORY OFBESSEL FUNCTIONS [CHAP. XV 15"2. TheBessel-Lommel theorem onthezerosofJ^,(z). ItWcasstated byDaniel Bernoulli* andFourierf that Jo(z)hasaninfinity ofreal zeros;andaformalproofofthis result byananalysisofParseval's integralisduetoBessel^.ItwassubsequentlyobservedbyLomniel§that Bessel's argumentsareimmediately applicabletoPoisson'sintegralfor J"^(z), providedthat—|< i'^|.Astraightforward applicationofRolle's theorem to x^''J^{a:)isthen adequatetoprove Lommel's theorem that J^(2)hasan infinity ofrealzeros^for anygivenrealvalue ofv. TheBessel-Lommelinvestigationconsists inprovingthatwhen—|<i;^|, and ;/;liesbetween mirand(m+|)7r,then J^(x)ispositiveforeven values ofm,(0,2,4,...),and isnegativeforoddvalues ofm,(1,3,5,...). Since J^(x)isacontinuous function ofxwhen x^O,itisobvious that J„(x) hasanoddnumber ofzeros ineach ofthe intervals(|7r, tt), (|7r, 27r), (Itt/Stt),.... Some morepreciseresults ofasimilar character willbegivenin§§15*32—15"36. ToproveLommel's theorem, letx=(ni+1$)ttwhere ^^^1;then,by obvious transformations ofPoisson'sintegral, wehave 2{l7ryr2m+9 cos^ttu ^''^""^~ r{v+^)r{i).(2m +dy]o {(2m+ey- 1^-]^-"''"' r2m+9 cQg l,jj.y^ andso sgnJ.(x)=sgnj^|^2m+6)"--..^j^-'^"- Now thelastintegral maybewritten intheform 7)1 r cos TTlf where (-)-.,.= \ ^^_^^^^_^^^-___du, Ifnowwewrite u=2r—1±U,andthenput {(2m+ey-(2/--1+uy}"-^-[{2m+ey-(2r-1-uy]''-^ =f,. (U), itisclear that v,=ff-iU)sin^TrU.dU, .'o and, since IIv^^,fr{U)isapositive increasing•[function ofr. *Comm. Acad. Sci.Imp.Petwp.vi.(1732—3) [1738], p.116. tLaTheovie Anah/tiquedelaChaleur(Paris, 1822), §803. iBerliner Abk., 1824, p.39. §Studicn liber dieBesseVschen Functioiwti(Leipzig, 1868), pp.65—67. IIThis isthepointatwhich thecopdition v$^isrequired ;thecondition v>-^ensures the convergenceoftheintegral. %Thereader willprovethiswithout anydifficulty byregarding;•asacontinuous variable and then differentiating/^ (6^)with respecttor. 15-2-15-22] ZEROS OFBESSEL FUNCTIONS 479 Itfollows that andso sgnJ^(i/iTT+I^tt)=sgn[(-)^ [vj+(i',„- v,,,^,)+(v,„_„-v,„_3)+...]] =sgn(-l)'^, since v„/isobviouslynotnegative. That istosay,when—|< z^^|, +,-(m=0,2,4,...) andfrom thisresult Lommel's theorem follows inthemanneralreadystated. Thezeros ofe/j(x),aswellasthose ofJ^(.r),havebeeninvestigated bjBaehr, Archives JVeerlandaises, vii.(1872), pp.351—358,withthehelpofamethod which resembles the Bessel-Lommel method. Baehr's result fort/j{.v)isthat thefunction ispositive whenx liesintheintervals(0, tt), (f7r, Stt), (fir, 5tt), ...,andthat itisnegative whenxliesin theintervals(i^Tr, -Irr), {l-rr, -irv), (-V-TT, 67r),....ThefunctionJi{x) hasalsobeen investi- gatedinthiswaybyC.N.Moore, Annals ofMath.(2)ix.(1908), pp.156—162. Theresults juststated areofalessexact nature than theresults obtained withtheaid ofslightly more refinedanalysis bySchafheitlin(§!^15-33—15*35). .ItwasnotedbyWhewell, Trans. Camh. Phil. Soc. ix.(1856), p.156,that /q {'*')liasa zerobetween 2and2^72,andthatthefunctionH(, (2)hassome real zeros. 15'21. Thenon-repetition ofzerosofcylinder functions. Itiseasytoprovethat %^y{z)hasnorepeated zeros, with thepossibleex- ceptionoftheorigin*. For,if ^t^^,(z)a.nd '^Jiz) vanishedsimultaneously,itwould follow, byrepeateddifferentiations ofthedifferentialequation V^^^(z)—0, that allthedifferential coefficients ofW„(z)would vanish atthecommon zeroof ^^(z)andWJ(z),andthen,byTaylor's theorem,^;, (z)would beidenticallyzero. 15'22. Theinterlacing ofzerosofBesselfunctions. Itwillnowbeshewn that if>.1,jv,-!,•••arethepositivezeros ofJ^(if), arrangedinascendingorder ofmagnitude, then, ify>—1, This result issometimesexpressed bysayingthatthepositivezeros ofJy(x) areinterlaced with those ofJ^^i (x). Toprovetheresult weusetherecurrence formulae d d^[x-^J, {x)]=-a•-^/,+l {x\^\x^^^ /,+! {x)]=A-+> ./,(.c) ; the first ofthese shews thatbetween each consecutivepairofzeros of OP""./^(,7)there isatleast onezero ofa;""1/^+1 (a;),andthesecond shews that between each consecutivepairofzeros ofa'""^^t/^+j ix)there isatleastonezero oix"^'^Jyix); andtheresult isnowobvious. *This isaspecial case ofatheorem proved bySturm, Journal deMath. 1.(1S3G), p.109. 480 THEORY OFBESSEL FUNCTIONS [CHAP. XV Ifi^^— 1,thezeros areobviouslystill interlaced butthesmallest zeroofJv+i{x)is nearer theorigin than thesmallest zero ofJt,{x). The result concerning interlacingofpositivezeros isobviouslytrue foranyreal cylinder function* ^^(x)andthecontiguous function"^^,^(^')- Thisfundamental andsimple propertyofBessel functionsappearsnever tohave beenproveduntil about aquarterofacentury agof, when four mathematicianspublished proofsalmostsimultaneously;theproofwhich has justbeengivenisdue toGegenbauer;): and,Porter|;theotherproofs,which areofaslightlymore elaborate character, weregiven byHobson|| and vanVleckir. IthasbeenpointedoutbyPorter that, since 7-/\ r /\2(^+1)7- /\J^{x)+J^+o {x)=J^+1 {x), atanypositivezero ofJ^{x) thefunctions.7^+1 (a^)andJ^+o(x) have the samesign;butatsuccessive zeros ofJ^(x)thefunction i/^+i (x)alternates in sign,andsothere areanoddnumber ofzeros oft/^^., (x)between each con- secutivepairofpositivezeros ofJy(x) ;interchangingthefunctions/^+2 (^)and J^(x)throughoutthisargument, weobtain Porter's theorem thatthepositive zeros ofJ^+o{x)areinterlaced with those ofJ^,(x). 15•23.Dixon stheorem ontheinterlacing ofzeros. Aresult ofaslightlymoregeneralcharacter than thetheorem of§15"22 isdue toA.C.Dixon**, namely that,whenv>—\,andA,B,G,Dare constants suchthatAD=f^ BC,thenthepositivezeros ofAJ^.{x) +BxJJ{x) are interlaced with those ofCJ^{x)+DxJJ (x),andthatnofunction ofthistype canhave arepeatedzeroother thanx=0. Thelatterpartofthetheorem isanimmediateconsequenceoftheformula, deducible from§5'11(11), J^(x), xJJ (x) d{J,ix) ]d\xJJ {x)] dx'dx i fortheintegralispositive when xispositiveandtheexpressionontheright would vanish atarepeatedzeroofAJt,{x) +BxJJ (x). *Arealcylinder function isanexpressionoftheform aJ,(.T)+^Y^(x} inwhich a,j3and varereal,andxispositive. tCf.GrayandMathews, ATreatise onBessel Functions (London, 1895), p.-50. +Monatshefte filrMath, uiidPhys.viii.(1897), pp.383—384. §Bulletin American Math. Soc. iv.(1898), pp.274—275. IIProc. London Math. Soc. xxviii.(1897), pp.372—373. ITAmerican Journal ofMath. xix.(1897), pp.75—85. **Messenger,xxxii. (1903), p.7;seealsoBryan,Proc. Camb. PhiLSoc.vi.(1889), pp.248—264.jJ,-(t)tdt=-^x .1n 15-23, 15-24] ZEROS OFBESSEL FUNCTIONS 481 Toprovetheformerpartofthetheorem, weobserve that, if A/\=GJJ^)+^JJ (x) '^^''^~AJAx)-\-BxJ:{x)' then0'{x)= ,^,,^ „rw ^)o\ii'^, f' J^'(0tdt, ^^'X[AJ,{x)+BxJJ {x)Y\C,DU andso{x)ismonotonia. Thepositivezeros of^(a;)aretherefore interlaced with thepositive poles, andfrom thisresult theformerpartofthetheorem isobvious. IfthefunctionJ^,{x)isreplaced byarealcylinderfunction a./^,(x)-\-j3Y^,{x), wehave j'gf,{x), x'W^ ix) %V(0tdt=!* Id;^(x)d[x^i^J {x)]+ Idx'dx providedthat—1<y<1;andsothetheoremsconcerning non-repetitionand interlacingofzeros aretrue forA9^^{x) +Bx'WJ (x)andC'(ff^{x) +Dx^ifJ {x) providedthat/3(asinvtt+/3cosvir)ispositive. Again,since %^.{x\ x^J(x)'2vj3(asinvtv+/3cosvtt) ITSmVTT ixd^'S.ix) dlx'^Jix)]=-1 [{x-- v'')'gf;-{x)+0^9^ J-"{x)], .dx'dx thetheorem istrue forzerosexceeding +\/v-,whether vliesbetween —1 and 1ornot. The result of§15"22 isthespecialcase ofDixon's theorem inwhich A=l,B=0,G=v,I)=-l. 15-24. Theinterlacing ofzeros ofcylinder functions oforder v. Let'Wv(x)and^^{x)beanydistinctcylinderfunctions ofthesame order; weshallprovethat theirpositivezeros areinterlaced*. If'^.{x)=a/,{x)+^Y,{x\%\{x)=7J",(x)+h\\{x), 2(aS-/37)then%".(x)%%'(x)-'^%(x)-gf;(x)= TTX Now itisknown that, atconsecutivepositivezeros of^^(x),^J(x)hasopposite signs, andtherefore, from thelastequation, '&y{x) hasopposite signs;that is tosay'^(«) hasanoddinimber ofzeros between each consecutivepairof positive zeros of'^^{x) ;similarly "Wyix)hasanoddnumber ofzerosbetween each consecutivepairofpositivezeros of '&'„{x);and sothezerosmust be interlaced. Ifwetakeoneofthecylinderfunctions tobeafunction ofthe firstkind, wededuce that ailrealcylinderfunctions haveaninfinityofpositivezeros. * Olbricht, Nova Acta Caes.-Leop.-Acad. {Halle), 1888, pp.43—iS,hasgiveu anelaborate dis- cussion ofthisresult withsome instructive diagrams. w.B.F. 31 482 THEORY OFBESSEL FUNCTIONS [CHAP. XV 15'25. LommeVs theorem onthereality ofthezeros ofJv{z). Anextension ofatheorem duetoFourier*, thatthefunction J^iz) hasno zeroswhich arenotreal,hasbeen effected byLommel-f-, Theextended theorem isthat, iftheorder vexceeds —1,then thefunction Jv{z) hasnozeros luhich arenotreal. ToproveLommel's theorem, suppose,ifpossible,thataisazeroofJ^{z) which isnot real. Itfollows from theseries for J^,{z)that aisnotapure imaginary,because then X(-)'"(ia)^ „,^Qm\T {v-\-m+1) would beaseries ofpositiveterms. Let Opbethecomplex numberconjugatetoa,sothatolqisalsoazero of J^{z),because J^,{z)isarealfunction ofz. Since i/>—1,itfollows from§5"11(8)that X tJ^(at)JAaot)dt= „ ,a—«oJ- dJ^(a^w) dJ^{ax)' Joand so,since a-^a,,-, ''1 tJ,{at)J,{a^t)dt=0. Theintegrandontheleft ispositive,andsowehave obtained acontradiction. Hence thenumber a.cannot exist, andthetheorem isproved. Similar arguments |maybeused toshew that,ifAandBarerealand v>- 1,the functionAJv{z)-\-BzJJ{z) has allitszerosreal,except that ithastwopurely imaginary zeroswhen {AjB)+v<0. These results follow from theseries for-^[z-^IB j^(2)}combined withtheformula 1 tJ^{^t)J^{Mdt=0,1: which issatisfied if/3and^0areanyzeros ofAJ^ {z)+BzJ^' (2)such that^^^^q\ 15'26. Theanalogue ofLommel's theorem forfunctions ofthesecond kind. Itisnotpossibletoprove bythemethods of§15"25that§ V„{z) hasno complexzeros intheregion ||inwhich |arg^rj<it.But ithasbeenproved bySchafheitlinl that Fq(~)hasnozeros with apositiverealpart,other than thereal zeros. *LaTheorie Analytique delaChaleur(Paris, 1822), §308; seealsoStearn, Quarterly Journal, XVII.(1880), p.93. tStudieii iiber dieBesseVschen Functionen(Leipzig, 1868), p.69. XSeeA.C.Dixon, Messenger, xxxii. (1903), p.7. §Or,moregenerally, <^^ (2). iiWhen arg^=±tt,1^(2)=e"^""^ l\ (-2)±2/cosvirJ^ (-2),andhence, by§3-(53(1),1^[z) cannot vanish unless vishalf ofanoddinteger. Thistype ofreasoningisduetoMacdonald, Proc.London Math. Soc.xxx.(1899), pp.165— 179. HArchiv derMath, nndPhijs. (3)i.(1901), pp.133—137. Inthispaper Schafheithn alsosub- jectsthecomplex zeros ofYi(z)toasimilar treatment. 15'25-15-27] ZEROS OFBESSEL FUNCTIONS 483 For let/3beacomplexzero of¥0(2), and let/3obetheconjugate complex, sothat/3oisalsoazeroofY,(z). Then, by§§5-11(8)and8-51(1), X dYo(/3,a:) ,_- ^dY,(0x)Y.iM-^^-1- (^„.r-^j-^-.^^^/3 /8^-/3o^ and so,if/3=pe'",wehave Jo TT-p-sm2w andtheexpressiononthe left ispositivewhile theexpressionontherightis negativewhenwisanacuteangle. 15'27. ThetheoremsofHurivitz onthezerosofJ^{z). Theproofwhich wasgiven byFourier thatthezeros ofJ^(z)are allreal wasmademorerigorous andextensive byHurwitz*, whoproved (i)thatwhen V>—l, thezeros ofJ„{z) are allreal, (ii)that,ifsisapositive integeror zeroand vliesbetween —(2s+l)and—(2.s+2), Ji,{z)has4s+2complex zeros, ofwhich 2arepurely imaginary, (iii) that, ifsisapositive integer and Vliesbetween —2sand—(2s+1),Ji,{z) has4scomplex zeros, ofwhich none arepurely imaginary. Toestablish these results, weusethenotation of§9"7. Wetake thefunctiong2,n,u(^)which has, intherespectivecases(i)m positive zeros, (ii)m—2s—1positive zeros, 1negativezeroand2scomplex zeros, (iii)vi—2spositivezerosand '2scomplexzeros. Wenowprove that, iff^it)=2-7-— tt,then thefunction LiK) hasatleast asmany complexzeros asg^m^vi^)-After Hurwitz, wewrite 9m(?,V)=w_^, wheref,t]arerealand^=f+iy,^'=|—i^-Theterms ofhighest degreein 4'm{^> v)3'i"6easily shewn tobe Im(m+l)(v+ m){v+m+1){{i'+m){2m+1)+m- 1}(p+ t?'-)'"-' ; andsinceg-jm,^'isareal function, itfollows that if^isacomplexzero of 9-Mi,v(0>'^^^^^is^';andtherefore thecomplexzerossatisfytheequation c/>m(fV)=0. Again,itisnotdifficult todeduce from therecurrence formulae(§9"7)that «/>m+i(Iv)=(v+2m+2)r/,,^,, (0.7..+,,. (r)+(P+t)^m(^.V)- *3Iath.Ann.xxxin.{lS89), pp.246— 2G6; cf.alsoSegar, Messenger, x\u.(1893), pp.171—181, foradiscussion oftheBessel coelKcients. Theanalysisofthissection clillers insome respects from that ofHurwitz; seeWatson, Pruc, London Math. Soc.(2)xix. (1021), pp.2G()—272. 31—2 484 THEORY OFBESSEL FUNCTIONS [CHAP. XV Hence, forsufficiently largevalues ofm(i.e.those forwhich v+2?nispositive), thecurve ^raihv)—^^i^sinthe finitepartoftheplane,and<f)m{^,v)is negativewhen<^,„+i (|,v)iszero sothecurve^„,+i (^, '>?)= lieswhollyinside oneorother oftheclosed branches whichcomposethecurve^„,(^,77)=0. Hence asm-^00,thecomplexzeros ofgzm,^(0li® i^iboundedregionsof the^-plane, andconsequentlyhavelimit-points. Now, since, by§§9"65, 9'7, canbemadearbitrarilysmall inanybounded domain ofthe^-plane, bytaking msufficiently large,itfollows fromLagrange's expansion*thatthenumber of zerosof/^(^)inanysmall area isatleastequaltothenumber ofzeros ofg^m^^i^) inthatareawhenmissufficiently large;andso/I(^)has25complexzeros. None ofthese zeros isreal, forifoneofthem were real itwould bealimit pointoftwoconjugate complexzeros of^am,^(0>'^^^soitwould count asa double zeroof/^(^); and/^(^)hasnodouble zeros. Again,fromtheseriesfor/^ (^)itisseen that,when vliesbetween -(2s+1) and—(2s+2),forthen/^ (^)hasonenegative zero,and itcannot havemorethan onenegative zero,thengom,y{^)could bemade tochange signmore thanonce as^varied from to—00[since g.>m,v{K)canbemade todiffer from/„(t) by anarbitrarilysmall number], andthis isimpossible. Forsimilar reasonsjfy(^)cannot havemorethan 2scomplexzeros. Ifwereplace ^by^z-,sothatnegativevalues of^correspondtopurely imaginaryvalues ofz,weobtain theresults stated inthecaseofJt,{z). Foradiscussion ofzeros ofBessel functions inassociation with zeros ofpolynomials based onrather differentideas, thereader should consult Lindner, Sitz.derBerliner Math. Ges. XI.(1911), pp.3—5.Itmaybementioned thatHurwitz hasextended hisresults to generalisedBessel functions inabriefpaper, Hamburger Mittheilungen,11.(1890)^ pp.25—31. 15*28.Bourget's hypothesis. Ithasbeenconjectured byBourgetf that,when visapositive integer (zero included), thefunctionsJt,{z), -/„+,«, (2^)have nocommon zeros, other than theorigin,forallpositive integralvalues ofm. Itseems that thistheorem hasnever been proved except (cf §15'22)in thesimplecasesm=1,m=2. Theformula Jv+m (^)=Jv{z)Rm,f(z)-Ju-i {z)Rm-i, f+i(z) *Cf.ModernA7ialysis, §7-32. tA7in. Sci.deVEcole norm.sup.iii.(1866), pp.55—95. 15-28, 15-3] ZEROS OFBESSEL FUNCTIONS 485 shews that, since J„ (-2^)andJ^_^ {z)havenocommon zeros, thecommon zeros oiJv{z)andJ"^4.,„(2^) mustsatisfytheequation i.e.theymust bealgebraicnumbers. Thetruth ofBourget's hypothesiscantherefore beestablished ifitcan beprovedthat ./„{z)hasnozeros which arealgebraicnumbers when visan integer;butattemptstoprovethistheorem* have sofarfailed. When Vishalfofanoddinteger,itiseasytoshew thatJ^{z)andJv+m {z) havenocommon zeros+;forsuch zeros arealgebraic numbers and itisknown thatnoalgebraic number Jcansatisfytheequation cot(^--|^7r-i7r)=- p' since theright-handside isalgebraicinzwhen vishalf ofanoddinteger. Theproof Jgiven byLambert andLegendrethat ir-isirrational maybe appliedto§5"6(6)toprovethat ./„{z)hasnozerowhosesquareisrational when Visrational; and so,from aconsideration ofR,n-i,v+i{z), Bourget's hypothesisistrue inthecasesm=3,vi=4. 15"3. Elementary properties ofthezeros^ ofJ,{x). Itispossibletoacquireaconsiderable amount ofinterestinginformation concerningthesmallest zeros ofJ^{x)andrelated functions, when vispositive, byadiscussion ofthedifferentialequationsatisfied byJ^{x)togetherwith therecurrence formulae;weshallnow establish thetruth ofaselection of theorems"concerningsuch zeros. Thereader will findamoresystematic investigation ijofthese theorems invarious papers bySchafheitliu, notably JournalfiirMatli. cxxii.(1900), pp.299—321;Archiv derMath, undPhys. (3)i.(1901), pp.133—137; BerlinerSitzungsherichte,III.(1904), pp.83—85. Forbrevity,thesmallestpositivezeros of/^(x),JJ{x),JJ'(x),...willbe calledj^,jj,jj', Thesmallestpositivezeros ofY^{x), YJ{x),YJ' (oj),... willsimilarlybecalledy^,yj,yj',.... We firstprovethat (1)/^jV>V,jj>V. Itisobvious fromthepowerseries for/„{x)andJJ{x)thatthese functions *Iconsider that thetheorem isprobably true;itisanabstruse theorem, and Ihave not succeeded inprovingit. tThiswasnoticed byPorter, American Journal of2[<itli. xx.(1S9S), p.203. XCf.Hobson, SquaringtheCircle (Cambridge, 1913), pp.44,51—53. §Some related results aredue toWatson, Froc.London Math. Sac.(2)xvi.(1U17), pp.165— 171. 486 THEORY OFBESSEL FUNCTIONS [CHAP. XV arepositiveforsufficientlysmallpositivevalues ofx;and,fromthedifferential equation d {dJ^,{x)\={v^-a?)J,{x\doc\dx itisevident that, solongas^<yand J^,{x)ispositive, xJ^ (a-)ispositive and increasing,andsoJ^,(x)increases with x. Therefore, solongas<a;<v,bothJ^(x)andxJ^' (x)arepositive increasing functions sothatj^,andjjcannot*belessthan v. Again,from thedifferentialequation vj;'(v)=-j;(v)<0, andso//'(x)hasbecomenegativebefore xhasincreased tothevalue vfrom zero. Hence, when Iv>l, (2) jj'<r. Next, since theexpressionontherightispositivesolongasa;<z^+2.Now, if//were less than^/[v(v-f2)},theexpressionontherightwould benegative whenxisequal tojj(which, from agraph,isobviouslylessthanj„)>andthis isnotthecase. Therefore (8) j;>^[v{v +2)}. Now, from§15*22 itfollows that Jv<Jv+\<Jv+2> and, ashasjustbeen stated, sothatJvijJ)andJ„+o{jJ)arebothpositive.Ifnowweputcc=jjinthe formula /,+, {x)=- |1 ^^— ^1J,{x)-^J,{x\ itisobvious that (4) ><V{2. {v+1)}. Similarly, byputtingx=j^intheformula (v+3)/,(x)+2(i.+2)|l- il^i±^0:±?)|j^,^^(^)+(^+1)J^^^ (^.)=0, wededuce that andtherefore (5) ^{v(v+2)}<j,.<^{2(v +l)(v+S)]. *Cf.Eieinann, Partielle Differentialgleichungen (Brunswick, 1876), p.269. tWhen 0<;v<l, J/' {x)isnegativeforsufficiently small values of.r. 15-3] ZEROS OFBESSEL FUNCTIONS 487 Inlikemanner, wecandeduce from theformulae andJ\^, {x)=-vW-"^1J,{cc)-{v+l) JJ'{x) that (6) >^\v{v-\)]<jj'<^{ir'-l). Some rather betterinequalitiesthan these areobtainable bytaking more complicatedformulae;thus, from theequation "p+ti(*')="v+l(*'}-^5,v+\(j^)—JV\^)^i,v+i ('^7; Schafheitlin* deduced that -R5..+i(j.)>0, i.e.3>^-lQ{v +2){v+4)j,-+16(i/+1)(z;+2)(i;+4)(z/+5)>0. Sincejj^iscertainlylessthanf(i/+2)(i^4-4),byresultsalready proved, J„ must belessthan thesmallerpositiverootoftheequation 3^'-\Q{v+ 2){v+4)X'+16(z/+1)(j^+2)(1/+4)(i/+5)=0, andhence, afortiori, (7) ><V[|(^ +l)(^+5)}. Similarly,from theequation 4J"'^+4 {x)=J^{x)[R,^^{x)+i^3,^+,{x)-Rs ^(x)- i?,^,+o(x)]-2// (.c)[i?o ^^.1(ic)-i?^„^i(«)}, Schafheitlin deduced that (8) jj>^{v(v+2)}, and,when v>4, (9) i;>V[z.(. +3)i, theseinequalities beingderived from theconsideration thatj^'liesbetween thepositiveroots oftheequation x'-S(u +2yx""+2v{v+1){v+3)(i^+4)=0. Thediscussion ofy^,requires slightly more abstrusereasoning. Weuse theresult that JJ^(x)+IV (a-) isadecreasingfunction ofx;this isobvious from§1373.Hence itfollows thatYv"(x)decreases throughtheinterval(0,jV),aiidsoy^exceedsjV;again, inthisinterval Y^{x)isnegative,and itfollows from§3'63(1)thatY^(j^,)is positive,sinceJJ(ju)isobviously negative. Hence (10) j;<i/,<j,. Thisinequality (with yVreplaced byi^-fh)wasestablishedbySchafheitlinf with theaidofrather elaborateanalysis. *Berliner Sitzungsberlchte,in.(1904), p.83. tJournal furMath, cxxii.(1900), pp.317—321. 488 THEORY OFBESSEL FUNCTIONS [CHAP. XV 15*31. Stationaryvalues ofcylinder functions*. Ithasalreadybeen seenthatthecylinderfunction J"^{x)cosa—Y^(x)sina, or^t,{x),hasaninfinite number ofpositive zeros, andsothere areaninfinite number ofpositivevalues ofxforwhich itisstationary. Such values ofx which exceed theorder v(supposed positive)willbecalled/u,i, /Uo, /u.3,...,where yLti<yU.2<Ms< •••. Weshallnowstudysome ofthesimpler propertiesofthesequence "g^.W, -^.W, "^.W,.... The firsttheorem which weshall establish isthat Toprove this,observe thatthefunction A{x)defined as x'^^J-'ix) T- 7)^X-— v" hasthenegativederivate -2x'^;'(x)/(x^--v'y, andsoA(/u.j) >A(/Xo)>A(/u,3)>.... SinceA(fXn)=^V" (/*«), thetruth ofthetheorem isnow evident. Amoreinterestingresult issuggested byHankel'sasymptoticformula (§7-21) %%(x)=(^ycos{x+a-^v7r-l7r)+0[^^. This indicates thepossibilityofproving inequalitiesconsistent with whenfinislarge. Itcaninfactbeshewn that (I)Thevalues assumedby(x-— v^)^ |^^(ic) |when xtakes thevaluesfi^,/m, fi-i,-'.formanincreasing sequenceivhose inernhers arelessthan»J(2/'7r). (II)Thevalues assumedbyxi\^t,(x)\ whenxtakes thevalues/Xr,f^r+i, /jLr+2>-..formadecreasing sequence whose members aregreaterthan\/{2j'ir) providedthat (i)v>^ V3, (ii) yu,">V-[^v-+4+V(48i.-^+13)}/(4i/='-3). Consider thefunction A{x)W,^{x)+25{x)%{x)'W:{x)+G{x)-^/^{x)=e(x), whereA(x),B{x),C(x)aretobesuitablychosen. Wehave ©'(x)={A'(x)-2(x'--V-)B{x)lx^] '^:~{x) +2[B'{x)+A(x)-B (x)/x-(x'-v')G(x)/x'} ^,(x)-^Z(x) +[C(x)+2B(x)-2C(x)/x} ^:^{x) =i){xYW:-'{x\ whereD{x)=G'{x)+25{x)-2G{x)lx, *Cf.Proc.London Math. Soc.(2)xvi.(1917), pp.170—171. 15-31, 15-32] ZEROS OFBESSEL FUNCTIONS 489 providedthatA(cc)ischosenarbitrarilyandthatB(x)andC(x)arethen defined bytheequations 2B(x)=x-A'(x)l(x-- v'), 0{x)=x"-[B'(x)+A(x)-B{x)lx]j{x^- v"). (I)IfA{x)={x--v-)Kthen 2D{x){x"-v'-f=Of(3a,^+\4>x-v'+^v')>0, andso{x)isanincreasingfunction ofxwhich istherefore lessthan lim0(ic)=2/7r. Since©{ixn)={i^n'— v'-)^^V(/"«)weseethatwhen 7iassumes thevalues 1,2,..,,then thenumbers(/u.,;-- v^)^ |'^^(fin)|formanincreasing sequence lessthanv'(2/7r). (II)IfA(x)=X,then 2D{x)(«--v-y=-X'{(4z.--3)X'-8j/-(v-+1)x'+v'{^v~-1)} <0, providedthat 4y->3andxexceeds thegreatestrootoftheequation {^v--3)ic"-Sv-{v-+l)X'-+v'(4z/--1)=0. Inthiscase{x)isadecreasing function andwecanapply arguments, similar tothose used intheorem(I),todeduce thetruth oftheorem(11). 15-32.Schafheitlins investigation ofthezerosofJo(x). Bymeans oftheintegralswhich havebeengivenin§6"12,ithasbeen shewn bySchafheitlin* thattheonly positivezeros ofJo(x)lieintheintervals (mTr+^ir,nnr+^7r)andtheonly positivezeros ofYq{x)lieintheintervals {niTT+^TT,rnir+f-rr),wherem=0,1,2,— We shall firstgiveSchafheitlin'sinvestigationforJo{x), withslight modifications, andthen w^eshallprovesimilar results forcylinderfunctions ofthetype J,,(x)cosa—F„(x)sina (wherevliesbetween —hand|),bythemethods usedbySchafheitlin. Schafheitlin'sinvestigationswere confined tothevalues and^irofa. From aninsjDectionoftheformula of§6"12(7), 77.sm6vcos u itisobvious that,when rnv<x<vnr+ftt, sgn(sin(x+^0)}=sgn(—1)'", andsosgnJo(x)=sgn(—1)"'. Consequently Jq(x)hasnozeros intheintervals(nnr,imr+ftt). *Journal JiirMath. cxiv. (1894), pp.31—44. 490 THEORY OFBESSEL FUNCTIONS [CHAP. XV Toprovethat J^{x)hasnozeros intheintervals {mir+^vr,imr+tt),write X={m4-1)TT—0, T. .2(-)'"+M-^-sin(A^-(f)) ,,,„ andthen J,{x)^-^—^— •^jS—^^"d6. ^ ^TT jsm^vcos6 The lastintegi-andisnegativeorpositive accordingas <^<2(^or2<f><e< l-rr. Since <Itt,thesecond ofthese intervals isthelonger;andthefunction g—2a;cot9 sin6Vcos isanincreasingfunction* of6when x>f7rand6isanacuteangle. Hence toeach value ofdbetween and2c^therecorrespondsavalue between2</>and^ttforwhich sin(|^— <^)hasthesame numerical value, but hasthepositive sign,andthecofactor ofsin(i^—0)isgreaterforthesecond setofvalues of6than forthe first set.Theintegralunder consideration is consequently positive,andsoJo(x) cannot have azero inanyoftheintervals (niTr+^TT,mir+Tr). Therefore theonly positivezeros ofJo(^)'T-reinthe intervals (^mr+ftt,??i7r+|7r). 15'33. TheoremsofSchafheitlins type,when—^<v^^. Weshallnowextend Schaf heitlin's results tofunctions ofthetype ^^(x)=Jy{x)cosCL—Y^, {x)sina, where ^a<ttand—|<v^\. Weshall firstprovethecrude result thattheonly positivezeros of^„{x) lieintheintervals {mir+fTT+^i^TT— Of,1MT+TT—a) wherem=0,1,2,—This result follows atoncefromtheformulae of§6"12, which shew that c^(^^-^"^^^^f-'^cos-^^sinCr+g-r^+i^) ,„,,, for,when mir—a<x<irnr+ftt+|i/7r—a, wehavesgn[sin{x+a—vd -\-\6)'\—sgn(—!)'", and so,forsuch values ofx,^^(x)isnotzero.Consequentlytheonlyzeros of^y{x)lieinthespecified intervals, andthere areanoddnumber ofzeros ineach interval, with thepossible exceptionofthe first ifa>ftt+^vir. Nextweobtain themorepreciseresult that theonlypositivezerosof^^{x) lieintheintervals (mTT+fTT+^vTT— a,niTT+1TT+5i/TT—a) *Itslogarithmic derivate is {2x- sin^cos^) cosee-^ +itan 6. 15-33, 15-34] ZEROS OFBESSEL FUNCTIONS 491 wherem=0,1,2,...,except that, ifaissufficientlynear tott,theremaybe zeros* intheinterval (Itt+^vir—a,tt—a). Weshallprovethisresultbyprovingthat^^^(x)hasafixedsignthroughout each oftheintervals f (iHTT+|7r+^VTT—a,mv+tt—a). Write x={m+l)iT-a.-{l-'2.v)(^, where ^isananglebetween and^tt. With thisvalue ofx, (_)^^i2-^'^-p-cos-^^sinKl-2.)(i^-c/))| _,,,,,..• Toeach value of9between and2</)therecorrespondsavalue between 2^ andIttforwhich sin{(1—2i^)(|^— (/>)]hasthesame numerical value, but hasthepositive sign. Again e--*'«'ot«cos''-^6'/sin-''+i^ isanincreasingfunction of6providedthat sin'^"Ix>maxr (2z^+1)sin^cos^+(27-1)cos^ andthiscondition issatisfied whenx>\since y^|. Hence, ifa->|and mir+|7r>(-\v7r—a<x<nnr+tt—a, wehavesgn ^j,{x)=sgn(—l)'""^^, andthisprovesthemoreprecisetheorem. 15'34. TJieorems ofSchafheitlins type,when\<v<^. Wenext consider thefunction 'Wv{sc)=Jy(x)cosa—Yy(x)sina, where ^a<tt,asbefore, inwhich itisnowsupposedthat|<v<§. Weshall firstprovethecrude result thattheonly positivezeros of^^y{x) lieintheintervals (??i7r—a,liiTT—^TT+-\t-TT—a) wherelm=0,1,2, This result follows atoncefrom theformulae of§6'12, which shew that forwhen mir—^7r+ ^vTr—a<x<{in+1)tt—a, *Bytaking asanalternative functionJ\p\ (x)andapplying thetheorem of§15-24, weseethat there cannot bemore thanonesiTch zero. tIf;c<^and ?)j=0,thereasoningfailswhen'fir+\vir-a<h. tila>{hv-\)tt,theinterval forwhichm=is,ofcourse, tobeomitted. 492 THEORY OFBESSEL FUNCTIONS [CHAP. XV wehave sgn[sin{x+a-v6+l6)]=sgn(-1)*", whence thetheorem stated isobvious. Nextweobtain themorepreciseresult that theonlypositivezeros of9^^(x) lieintheintervals {miT—^TT-V{vir—ayinir—^TT^-^vir— oi), wherem=0,1,2,...,except that, ifaissufficientlynear toit,theremaybe azero intheinterval(0,IvTr—^ir—a),andtheremaybeoneintheinterval (it—a,l7r+ \v7r—a). Weusethesame notation andreasoningasin§1533;onlynow, if g-2xcote cog.-i6'/sin-''+i6=/(^), f(6)isnotnecessarilyanincreasingfunction of6;but itissufficiejittoprove that,when 0<ylr<2ct),then Toobtain this result, observe that •^^og-^Jr~^,=2.^'{cosec2 {2(j>+x/,)+cosec^(2c^- yjr)}n\\r Tl'zfh—Air)dy}r °/(2(^-x/r) —sin40^+2^+1 "1 .+^/r)sin(20-V')J' _cos(20+^//-)cos (2(^—\^)sin(20- But[cosec2 (2<^+1|^)+cosec^{2(p-^)]sin(20+^^)sin(20- ij/) isanincreasingfunction of\|^,andtherefore, afortiori, [cosec2 (20+\//)+cosec2 (20- ^)'\cos(20+^)cos(20- -y/r) isanincreasing function, since thisfunction exceeds theformer byanincreasing function because 40isanacuteangle;andso^log'-.— y-isahvays positiveifitispositive when i|/'=0,i.e.if Ax>{{v-i)tan220+(2i/+1)}sin40, andthis isthecasewhen .v>!»'+§. Hence, when|<v<^,theonlyzeros of^^(x),which exceed fi;+1,liein theintervals (mTT—^TT+\v7r—a,nnr—{tt+^vtt— a). Themethod seemsinapplicableforlargervalues ofvonaccount ofthe oscillatorycharacter ofsin{a;+a—v6+hO)as6increases from to^tt;a method which iseffective fortheselargervalues willnowbeexplained. 15"35.Schafheitlinsinvestigations ofthezerosofcylinder functions of unrestrictedly large order. Weshallnowprove that, if i^>|,those zerosofthecylinder function J^{x)cosa—Y^{x)sina which exceed(2i/+1){2v+3)/7rlieintheintervals {niTT—a-\-\v'Tr +\'ir,nnr—a+|i'7r+ftt) ivheremassumesintegervalues. 15-35] ZEROS OFBESSEL FUNCTIONS 493 Themethod used toobtain this result isdue toSchafheitlin*; buthe considered thecase offunctions ofthefirstkindandofintegralorderonly,and hisreasoningismadelengthyandobscure bytheuseofarguments equivalent totheuseofthesecond mean-value theorem when theexplicituseofthat theorem isobviouslydesirable. Asinthepreceding analysis,write 9ov(^)=JV(^)COSa—Y^(./)sin a, sothat "r(.+f)r(i)J^^dd\cos-'-^i e\'^^' Now cot'-''+^ 9 .e-2-«cot8 illcreases as6increases from to$2andthen decreases as6increases from 62to^tt,where ^o=arctanj-. Itwillbeobserved that 62isnearly equalto^irwhen xislargecompared with V. Nowsupposethat £cliesbetween ^iTr—a+hTT{v—h)and 7H7r—a+^tt{v—h)+ftt, andthen choose d^sothat a;-f-a—(f+I)^1=?/i7r. Itiseasytoverifythat 2v-I TT^2v+2TT 2vTS'2'^ '^2^7^- 2' sothat 61isapositive anglelessthan0..,providedthat arctan <X 4i/+6" Wesuppose nowthat a;>(2i' +l)(2j^+3)/7r, sothat ^1iscertainlylessthan 6.2. Then, bythesecond mean- value theorem, there exists anumber^0,between and^suchthat , i".cos-^^sin(^' +a-z^^+i^) ^^^^^,, =fcot-+>^ e-=xcot.,]•^'A|cosj^rJ^a-^^^-^[ ={cot-+^ e,.6-cotM {cos(m7r+^0_cosjx+a-ve,-m ^ ^ \cos"-^^6'i cos''+^(^o J *Journal filrMath, cxxir.(1900), pp.299—321. 494 THEORY OFBESSEL FUNCTIONS[CHAP. XV Nowquafunction of6, cos{a:+a-v6 -|^)/cos''+^ 6 isstationary when s,m{x-\-a—v6+\6)=^0, and forsuch values of the fraction isequalto+l/cos""^6. cos(«+a-i^6?o- 2^o)Hence-tt-?, cannot exceed numericallythegreatestvalue ofl/cos""*6intheinterval(0,^i), andtherefore (cos(m^+^0_cos{x+a-ve-W\^° I008"+* ^1 COS-'+^^o Jo^ ^ Therefore, since thesignofsin{x+a—v9-\-\6)isthesignof(—1)'"when 6 liesbetween 6^and^tt,weseethat, forthevalues ofxunder consideration, sgn W^,(a;)=sgn(-1)'«. Hence, when xexceeds {2v+1){2v+S)/7r, '^„(a;) hasnozeros inintervals of thetype (lUTT—a+^rTT— ^TT,mir—a+^v7r+ ^tt), andsotheonlyzeros of^^(x)which exceed(2v+1)(2i/+3)/7rlieinintervals ofthetype (niTT—a+hi'ir+|7r,mir—a+hvir+fvr), andthisreduces toSchafheitlin's result* when a=and visaninteger. Thereader willobserve that thistheoremgivesnoinformationconcerning thesmaller zeros of^„(^) when vislarge;itwillbeapparentin§15"8that there arealargenumber ofzeros lessthan {2v+1){2v+d)/7r,and that intei-estinginformation canbeobtainedconcerningthembyusing Debye's integrals. 15"36. Backer's tJteoremf onthezerosofW^ix). Aresult ofaslightlydifferent character from thosejustestablished was discovered byBocher from aconsideration oftheintegralformula§11-41(16). Thetheorem inquestionisthat^^{x)hasaninfinite number ofpositive zeros, andthedistance between consecutive zeros doesnotexceed2;owherej^isthe smallestpositivezeroofJq{x)- Toestablish this result, write j^=0,z=2oin§11-41(17),andthen f'^#0 (tir)d<^=TT-^^o{Z)J,(jo)=0. .'0 Hence'<0'o(sr)cannot beone-signedas^increases from tott,i.e.asot increases fromZ— j,^toZ+j^;and soWoi'^) must vanish foratleast one valueJoforintheinterval {Z—Jq,Z+j^).SinceZisanarbitrary positive number(greaterthanJo),Bocher's theorem isnow evident. *Schafheitlin gives (2c+3)(2v+5)/7r asthelower limit ofthevalues ofxforwhich thezeros lieinthespecified intervals. tBulletin American Math. Soe. v.(1899), pp.385—388. XCf.ModernAnalysis, §3-G3. 15-36, 15-4] ZEROS OFBESSEL FUNCTIONS 495 [Note. Byaform ofGreen's theorem, [dv ,fdu iu^^as=IV;r~as, JdvJcv where w,«aretwo solutions of—^-\-^-\-u =<)with continuous second differential ox''dy'- coefficients inside theclosed curves,andl^l'hvindicates differentiationalongthenormal. Bytakingv=J^{s]{x^-\-y'^)\ andthecurve tobe.t-+^-=/u^Weber* deduced thatu must vanish atleast twice onanycircle ofradiusy^. Bochor inferred from thisresult that sinceWn(r)cosn6satisfies therequisite conditions exceptattheorigin,ifacircle ofradiusj^isdrawn with centre ontheaxis ofxand subtending ananglelessthantt/?*attheorigin, /In (?')must vanish somewhere onthe circle. Hence thepositivezeros of(?«(?)aresuch thatconsecutive zeros areatadistance apartlessthan 2/n,andthedistance from theoriginofthesmallest ofthem does not exceedj^|l-fcosec^|. These results areofinterest onaccount oftheextremesimplicityofthemethods used toprove them.] 15*4. OnthenuniberofzerosofJ^{z)inanassigned strip ofthez-plane. Weshall nextgivetheexpressionforJv{z)asaWeierstrassianproduct, andthendevelop expressions involving quotientsofBessel functions inthe form ofpartialfractions;butasapreliminaryitisconvenient toprovethe following theorem, whichgivessome indication astothesituation ofthose zeros of .7^(2) which areoflargemodulus. Inthisinvestigationitisnotsupposed that Visrestricted tobearealnumber, thoughitisconvenient tosuppose that Visnotanegative integer. When visrealtheresults of§15"2 tosome extent taketheplaceofthetheorem which willnowbeproved. LetCbetherectangularcontour whose vertices are ±iB+17ri7(z'),±iB+unr+^i^TT+^tt, where i?isa(large) positivenumber. Weshallshew thatwhen 7/^isasufficiently large integerthenumber of zeros ofz""J^,(z)inside Cisprecisely equal ftoni. Sinde z~^J^(z)isanintegralfunction ofz,thenumber ofitszeros inside C'is 1 Idr\og{w-''JAw)} ^^^_lf J_^^^^ZTrlJc dw ZTTlJcJ^iW) *Math. Ann. i.(18(59), p.10. tWhen t'isarealnegative number (andforcertain complex values ofv)theremaybepairs ofzeros ontheima^^inaryaxis;insueVicircumstances thecontour Cliastobeindented, andeach pairofzeros istobereckoned asasinglezero. 496 THEORY OFBESSEL FUNCTIONS[CHAP. XV Wenowconsider thefour sides ofCinturn. Itisfirst tobeobserved that onallthesides ofC, If^i)(w)=(—)%'<^-i--J-)(1+^1.(w)},yirwj.N/J iT,®(w)= (''-^Ye-'-('«'-i--^-){1+^,_^(^y)}, where?7i_^{w)"and772,v(w)are(1/w)when jw |islarge. Now, since theintegrandisanoddfunction*, wehave, as5-^x, ^iB+lTriI(v) ^., idiu=^il{v). Next taketheintegral alongtheupperhorizontal sideofG\this isequalto —;—.—rdlU [1+(e^"0]div27ri jiB+mn+iun+^n J^v{w) 27rjiB+imI{v) {1+V2,u{w) 1 27rm.r+Ki^(.) +ivr+-^log ^^>^.^-^^^/+0(l/i?) asB^'oo . Similarlytheintegral alongthelower sidetends tothesame value, and sothelimit oftheintegral alongthethree sidesnowconsidered ism-^^v+l. Lastly wehave toconsider theintegral alongthefourth side,andtodo thiswefirstinvestigatethedifference ^>-tan(»-i..-i.), which, when\iu\islarge,isequalto 2otW riB+mn+^vn+iiTNow andsortB+mn+^vn+iTr 1 tan(w— ^VTT—Iit)diu=0, J—iB+mn+\viT-{-\iT __1_n^- 27riJ -11 r^'^"-^"\^±l+ 0(1)1dn,]^riB-\-mit-\-iv7,-\-\n (2v+1 /'I 27rij _i i-(2i.+l)+0(l/m). Hence thelimit oftheintegralround thewholerectangleism-\-0(l/m). *Allowance ismade fortheindentations, justspecified,inthefirststepofthefollowing analysis. 15-41] ZEROS OFBESSEL FUNCTIONS 497 IfwetakemsufHciently large,wecanensure thattheexpressionwhich is(l/m)isnumericallylessthan 1;andsince theintegralround therectangle must beaninteger,itisequaltom. That istosay,thenumber ofzeros of2""/^ (z)between theimaginaryaxis andthelineonwhich isexactlym. Note. Theapproximateformulae quotedforthefunctions ofthethird kindshew thatthelargezeros cannot have alarge imaginary port; andsoallthezeros ofJ^(z)lie inside astripwhose sides areparalleltotherealaxisandatdistances from itwhich are bounded when ji' |isbounded. 15•41.Theexpression ofJy,{z)ascminfinite product. Itispossibletoexpress J^{z)asaproductof' simplefactors'of Weierstrassiantype,each factorvanishingatoneofthezeros ofJv{z). In order toexpress Jv{z)inthis form, itisconvenient first toexpressthe logarithmicderivate ofz~^Jv{z)asaseries ofrational fractionsbyMittag- Leffler's theorem*. The zeros ofz~''J^(z)aretaken tobe±jy,i, ±ji._2> ±jf,3,•wheref R(j>',n)>and 1R(j„,i) \^\R {jv,2)\^\R (As) I^•••,thevalues ofj.^, yV^o, j;,3,...beingallunequal (§15"21). Wedraw a(large) rectangle D,whose vertices are±A±iB,whereAandBarepositive,andwesupposethat±ju,„i arethezeros ofhighestrankwhich areinside therectangle. Wenowconsider 27riJJ)w{w—z)Jv{w) Avhere zisanypointinside therectangle,other than azero ofJi,(w), and vis notanegative integer. Theonlypolesoftheintegrandinside therectangleare z,±jv,i, tjy,2,•, ±Jv, III- Theresidue atzisJ^+i{z)jJt,{z) andtheresidues at±j^, „are [Z+Jv,n Jv,n) since -//{z)=—J^^^ (z)whenz=±j^,n,by§3*2. Itfollows that Jv{z) n=l[z—Jv,n Jv,n] n=l[z+Jy,n Jv,n 27ri}.^^A^^hw, J)w{lu—z)'Jv{w) *Acta Soc. Scient. Fennicae, xi.(1880), pp.273—293. Cf.ModernAnalysis, §7-4. tIfiv(±j^,J=foranyvalue ofu,wechoosej^ ^^tohave itsimaginary part positive. W.B.F. 32 498 THEORY OFBESSEL FUNCTIONS [CHAP. XV Wenextshew that,bygivingAandBsuitable sequencesofvalues which increase without limit, Jy+i(w)/J^(w)canbetaken tobebounded onD. Since thisfunction isanoddfunction ofw,itissufficient toconsider the right-handhalfofD. WetakeA=Mir -\-R{\v+\)'ir,whereMisapositive integer;andthen wetakeMtobeatleast solargethatM=m,which ispossible by§15'4,and alsotobesolargethatwecantake thefunctions-ij^i^iv), Vv,-2{w),defined in §15"4, tobelessthan, say,^inabsolute value. Then Z^+i {iv)/J^ (w)isbounded whenever jg2l(W—Ji'ir— Jn-)I is*lessthan|orgreaterthan 2;andwhen theexpressiondoesnot liewithin these limits, I(lu)isbounded andwisnotarbitrarilynear azero ofJ^(w); sothat,from theasymptotic expansionoi^1'2l,J^+i{iu)IJy(w)isbounded on thepartoftherectanglewithin thisstrip. That istosay J^,+i (tu)/J^ (w)isbounded onthewhole oftheperimeterof therectangle DasBandMtend toinfinity. Hence 1 r 2J„+i{w) ]dw(w-z) JAw) 2iri]i)w{w—z)Jy{w) andtherefore t "fV-^/ n=\(•^Jv,n Jv,n) n=\{.^"^Jv,n Jv,n) When weintegrate, wefindthat (('Ju+.it),]» [/z\_^fz\]^if, ,z\ /z 7- zandhence This istheexpressionofJ^{z)inthespecifiedform. Theformula mayalsobewritten inthemodified form Thisformula wasassumed byEuler, ActaAcad.Petrop.v.pars 1,(1781) [1784], p.170, when j/=0,audsubsequently byvarious writers forother values ofr;cf.§§15-5, 15-51. Theanalysisofthissection isdueinsubstance toGrafandGubler, Einleitungindie Theorie derBessel'schenFunktionen,i.(Bern, 1898), pp.123—130,and itwasgiven explicitly byKapteyn, Monatshefte furMath, undPhys.xiv.(1903), pp.281—282. *Because'^>_ 1+i2- tIfwetaketherectangle tohave itsvertices atA±iB, -A'^iB, weseethat thetwo series ontheright converge separately. 15*4:2] ZEROS OFBESSEL FUNCTIONS 499 Theexpansion ontherightof(1)isevidently expansibleinapower series; the coefficients insuch aseries havebeenexpressed asdeterminants byKapteyn,Proe. Section ofSci.,K.Acad, van Wet. teAmsterdam, viir.(1905), pp.547—549,640—642;Archives Neerlandaises, (2)xi.(1906), pp.149—168.Some associated formulae have justbeen published byForsyth, Messenger,L.(1921), pp.129—149. 15"42. TheKneser-Sommerfeld expansion. Anexpansion which, insomerespects, resembles thepartialfraction formula obtained in§15'41isasfollows : .!:(^'-A..)>.» //=(>.7)= iiJAi)i-^'(')''(-^-)-I'(^>-^^(^^>l. mluhich xandXarepositive numbers such tlait ^x^X^1, while zandvareunrestricted(complex) numbers, exceptthat itisconvenient totakeR(z)>0. Theexpansionvrasdiscovered inthecasev=0, asaspecial form ofanexpansion occurringinthetheoryofintegral equations, byKneser, Math. Ann. LXiir.(1907), pp.511— 517. Proofs oftliisandofrelated expansionsforintegral values ofvwerepublished laterIjySommerfeld, Jahresbericht derDeutschen Math.Vereinigung,xxi.(1913), pp.309— 353,butSommerfeld's method ofproof hasbeen criticisedadversely byCarslaw, Proc. London Math. Soc.(2)xiii.(1914), p.239. Itmaybenoticed thattheexpansion hassome connexion with the'Fourier-Bessel' expansions which willbediscussed inChapterxviii. Toobtain aproofoftheexpansion,consider theintegral 1[H,w(Xiu) ^,'^'(w)-H^^jXiv) ^,">(w)J,{xw )^ ZTTt J Z-—W-'Ji,\W) inwhich thepathofintegrationisarectanglewith vertices +Bi,A±Bi,and itissupposedthat the leftsideoftherectangleisindented attheorigin. Theintegral round theindentation tends tozerowith theradius ofthe indentation, whether vbeanintegerornot;andtheintegrals alongthetwo partsoftheimaginaryaxis cancel. Also,when xandA^satisfythespecified inequalities,thefunction {Zr,w{Xw) R/'^ (lu)-H,^'>(Xw) ff,<" (w)}J,{xw)/J, (tv) remains bounded ontheother three sides oftherectangle whenB-*acand whenA-*oothroughthevaluesspecifiedin§15'41. Hence thelimit oftheintegralround therectangleiszero,andsothe limit ofthesum oftheresidues oftheintegrandatthepolesontherightof theimaginaryaxis iszero. Now theresidue atzis S '^^j^[.L(Xz)YAz)-JA^)yAXz)], 32—2 500 THEORY OFBESSEL FUNCTIONS [CHAP. XV while theresidueat_y^,,jis -2tJ^(j,,nX)7,(j\n) Jv(jv,n x)/[J,' (>,„) (^'-j\n)] —~^"^''v(>,n^)'Jv ijv,nx)\y (\T'/A\_T(i\V'(i V-T'o/•\T^ ^\{'^v\Jv,n)^V\Jv,n)"v\Jv,7i}-tu\Jv,nJ) ^v~\Jv,n)\2~ J'v,n) 'Jrj,,nJ.'^ (j.,n)(2^-j\n)' andonsummingtheresidues weatonce obtain thestatedexpansion. Forageneralisationofthisexpansion,obtained byreplacing Jy{xiv)jJy (w)by <^i,(.rw)/^^ (w)inthecontourintegral,seeCarslaw, Proc.London Math. Soc.(2)xvi. (1917), pp.84—93;Carslaw has also constructed some similar series which contain Legendre functions aswell asBessel functions, andthese seriesrepresenttheGreen's functionsappropriatetocertainphysical problems.SeealsoBeltrami, Lomhardo Rendiconti, (2)XIII.(1880), p.336;andLorenz, OetivresScientifiqties,II.(1899), p.506. 15*5.Eldersinvestigation ofthezeros ofJ^{2\/z). Aningeniousmethod ofcalculatingthesmallest zeros ofafunction was devised byEuler*, andapplied byhim todetermine thethree smallest zeros ofJo(2^z). Ifthezerosarrangedinascending orderfofmagnitudebeQj,Wg,Cs,•••> thenby§15-41, j„(2V^)=n (i-f).11=1 \ "n/ Ashasalreadybeen stated(§15"41),thisformula wasassumed byEuler;ifit isdifferentiatedlogarithmically,then d=°1 -;i-logJ'o(2V^)=Sdz°n=lan-Z CC 00 yVd providedthat |^^ |<Oj ;andthelastseries isthenabsolutely convergent. 00 Put2l/a,i"*+^=a-,„+iandchangetheorder ofthesummations;then d az,„= Replace Jq{2\Jz) oneach sideby z z'^ X2+]^2 2212.2^3-'^'"'' *Acta Acad.Petrop.v.pars 1,(1781) [1784], pp.170 etseq.Apaper byStern, Journal fiiv Math. XXXIII. (1846), pp.363—365should alsobeconsulted. tFrom §15*25 itfollows thatthezeros arepositive andunequal. i 15-5] ZEROS OFBESSEL FUNCTIONS 501 multiplyouttheproduct ontheright, andequatecoefficients ofthevarious powersof2intheidentity; wethusobtain thesystem*ofequations 1=CTi ,—1=0-0—0-1, j2=o"3—o"2+i^n, - -rii:=^4- 0-3+i0-2-aV^i> 2580=o"5-^4+jo-a-aVo"2+sfo o-i. ~ HS'Toci—^s—0-5+ i0"41_ 3 5T0tO-.>144U"1' whence 0-1=1,o-o=|,0-3=4,cr,=^i, cTg=yVij, 0-6=i%*#,ii^=_UL 473 Since <Oi<Wo<Oo< ... ,itisevident that andso o-„r^"" <«!<cr„,/o-,„+i. Byextrapolatingfrom thefollowing Table : m 502 THEORY OFBESSEL FUNCTIONS [CHAP. XV 15*51.RayleigKsextensionofEuler sfomnula. Themethodjustdescribed wasusedindependently byRayleigh*tocalcu- latethesmallestpositivezero oft/„{z). Takingtheformula(§15'41) 1andwriting S -r^^^—=o-^**"', M=lJ'v,n wefind, afterRayleigh,that .(1)= = -. fl-..(2)= -. .O-<="="'^ 5r+ll o-'••'= o-t«'= 29(i;+ly{v+2)-' (i;+3)(y+4)(i/4-5) The smallestpositivezeros ofJ^(z)andJj(z)arededuced tobe2-404826 and3-831706. Immediately afterwards Cayley fnoticed thato-^(''>canbecalculatedrapidly when ris apowerof2byaprocess which heattributed toEnckeJ,butwhich ismoreusually known asGraefte's§ method ofsolving anequation. _^ Themethod consists incalculating o-^*'')when risapowerof2bystarting with the given equation andforming from itasequence ofequations each ofwhich hasforitsroots thesquares oftheroots ofitspredecessor; anda-J'^^thenrapidlytends toaratio of equality withllj'^''y^i. Cayley thusfound o-..'^)tobe 4291^5^7640i/<+ 53752./3 +185430i/2 +311387i/+202738 2i6(,.+l)8(,.+2)i(v+3)2(r+4)2(i'+5)(^+6)(v+7)(v+8)* Itwasobserved byGrafandGubler|| thatthevalue ofo-^t'')caneasily bechecked bythe formula ,.j('-)=22'-ii?,/(2;-)!, where B^isthe ?-thBernouUian number;thisformula isanevidentconsequenceofthe equation Extensions ofsome ofthese results tothezeros ofzJi,' {z)+hJ^ (z),where hisaconstant, havebeenmade byLamb, Froc.London Math. 8oc.xv.(1884), p.273. Thesmallest zeroof«/;,(s),forvarious values ofvbetween and1,hasrecently been tabulated byAirey,Phil.Mag. (6)XLi. (1921), pp.200—205, withtheaidoftheKayleigh- Cayleyformulae. *Proc.London Math. Soc. v.(187i), pp.119—1-24.[Scientific Papers,i.(1899), pp.190—195.] tProc.London Math. Soc. v.(1874), pp.123—124.[Collected Papers,ix.(1896), pp.19—20.] :;:Journal furMath. xxii.(1841), pp.193—248. §DieAuflosung derhoheren numerischen Gleichungen (Zurich, 1837). IIEinleitung indieTheorie derBesseVschen Funktionen,i.(Bern, 1898), pp.130—131. 15-51, 15-52] ZEROS OFBESSEL FUNCTIONS 503 [Note. Theproeedui-eofcalculatingthesum oftherthpowersoftheroots ofan equationinorder toobtain thenumerical value ofitslargestrootseems tobedue to Waring, Meditationes Analyticae (Cambridge, 1776), p.311;other writers whowere acquainted withsuch amethod before GraefFe areEuler(cf.§15-5); Dandelin*, Mem. de VAcad. R.desSci.deBruxelles,ill.(1826), p.48;Lobatschevsky*, Algebra,orCalculus ofFinites (Kazan, 1834), §257.] 15*52. ThelargezerosofJq{x). Themost effective method ofcalculatingthelargezeros ofcylinder functions (when theorder visnottoolarge) is,insubstance, duetoStokes-f*, though subsequentwriters have, tosome extent, improvedonhisanalysis. Stokes' method willbesufficientlyillustrated byhisownexample jJo(rf'), whose zeros aretheroots oftheequation with thenotation of§7"3. Itwillberemembered that theasymptotic expansionsofP(a;, 0)andQ{x, 0)are „, ^, ^1.9 1.9.25.49 1 1925 Wbx 3 !{%xf"" Forsufficiently largevalues ofx,P(x,0)ispositive, Q(x,0)isnegative and thequotient Q(x,0)/P(x, 0)isanegative increasing!function ofx. The function cot(^—^tt)isadecreasingfunction which vanishes when x=n7r — ^'7r,and soitisobvious from agraphofcot(*'— |^7r)that there exists apositive integeriVsuch thatwhen n>N,Jo(x)haspreciselyonezero ineach oftheintervals (mr— ^vr,mr+^ir),andthatthedistance ofthezero from theleft-hand endoftheinterval tends tozero as7^-^oo . Again,if«,-, v,-denote the(r+l)thterms ofP{x,0)andQ(x, 0)wemay write m-l m~lP(x,0)=S^^.+du,„ Q(x,0)=SV,+d,v„^, where 6and 0^arecertain functions ofxandmwhich liebetween and 1. *Iowethese tworeferences toProfessor Whittaker. tCamb. Phil. Trans, ix.(1856), pp.182—184. [Math, andPhys. Pajyers, u.(1883), pp.350— 353.] fStokes alsoconsidered Airy's integral (§6-4)andJj(.f),forthepurposeofinvestigatingthe positionofthedarkbands seen inartificial rainbows. §Thereader may verify, by§3-63, that itsderivate is |l_p2_Q2}/p2, where P,Qstand forP{x,0), Q(x,0); and,bytheasymptotic expansions,this isultimately positive. 504 THEORY OFBESSEL FUNCTIONS[CHAP. XV Now consider theequationm-l 2 V,.+d^V.n cot (os—Itt)=^^^ , 1Uy+0Um r= inwhich itistemporarily supposedthat6and ^j,instead ofhavingtheir actual values, areanynumbers which liebetween and 1. Theequation nowunder consideration involves nofunctions morecompli- cated thantrigonometricalfunctions. Ifxweresupposed complex,there would beanumber ofcontours inthea;-planeeach ofwhich enclosed oneof thepointsn-rr—^ttandonwhich |cot(a;— \'jt)\exceeded themodulus ofthe quotientontheright. ByBiirmann's theorem*themodifiedequation would have onerootinside thepartofthecontour which surrounds mr—^tt,and this rootcanbeex- pandedindescending powersofnir— ^tt. Wethus obtain anexpansionfortherootoftheequationintheform inwhich thecoefficients/^(^,^i)areindependentofnbutdejDend on6and 6^ ; and itisreadily perceivedthat the firstmofthecoefficients areactually independentof6and 6^,sothat,when r<iii,wemaywrite Now thesum oftheterms after the7nth isabounded function of6and 0^ asand 6^varybetween and 1;and itisclear thattheupper bound ofthe modulus ofthefunction inquestionis0(?i~-"*~^)asn-* cc .Hence, when 6 and $1aregiventheir actual values whichtheyhave atthezerounder con- sideration, thesum oftheterms after themth isstill0(n~'^'"~^). That istosay,ithasbeenprovedthatthere exists onezero(nearly equal toWTT—^ir),and itsvaluemaybewritten Hence theasymjitotic expansionofthezero is f Itremains tocalculate the firstfewofthecoefficientsf,..If . . Q(x,0) whereyjr^0asx^x,then , ,1 33 3417tanvri-^ "—I- *Cf.ModernAnalysis, §7-31. 15-53] ZEROS OFBESSEL FUNCTIONS 505 25 1073 andtherefore theequationtobesolved assumes theform , ,1 25 1073 ^"-^"^-^^^^8^--384^+5120.^^--- The result ofrevertingtheseries is ,,,1 31 3779 S{mr-lTr) 384(wtt-^Tr)^15360 (nvr-|7r)s This series isadequateforcalculatingallthezeros ofJ^,(x),toatleast five placesofdecimals, exceptthesmallest zero, forwhichn= 1. 15"53. Thelargezeros ofcylinder functions. Itiseasytoseethat thelargezeros ofanycylinder function, Jy(z)cosa—F„(z)sin«, where vandaarenotnecessarily real,maybecalculated byStokes' method from aconsideration oftheequation cot(^ -^VTT—iTT+a)=7j7r-r{z,V) Itseemsunnecessarytoprovetheexistence ofsuch zeros (with large positiverealparts)orthe factthattheymaybecalculated asthoughthe series forP(z,v)andQ(z,v)wereconvergent,because theproofdiffers from theinvestigationoftheprecedingsectiononlyintedious details. Theexpressionforthelargezeros ofacylinderfunction ofanygivenorder wascalculated after themanner ofStokesbyMcMahon*; butthesubsequent memoirs ofKalahnef andMarshall :|:havemade theinvestigationmoresimple andhave carried theapproximationastagefurther withnogreater expendi- ture ofwork inthecalculation. FollowingMarshall wedefine§twofunctions of^,calledMandi/r,bythe equations i/cosi/^=P(z,v),Msiny^=-Q{z, v), ontheunderstandingthat il/-*+1andi/r^-as^-^+x . Itisthen clear that J^,{z)cosa-F^{z)sina= [— jJfcos{z—^vit— \'jr-\-a— y^). *Annals ofMath. ix.(1895), pp.23—2.5;seealsoAirey, Proc. Plujs.Soc. 1911, pp.219—22-1, 225—232. fZeitschriftfiir Math, nndPhijs.liv.(1907), pp.55—86. +Annals ofMath.(2)xi.(1910), pp.153—160. §Cf.Nicholson, Phil. Mat;. (6)xix.(1910), pp.228—249. 506 THEORY OFBESSEL FUNCTIONS [CHAP. XV Again arctan j^^^Y=z—^vtt—^tt— i/r, and,whenwedifferentiate thisequation,anduse|3*63(3),wefindthat dy^_ 2/(7rg) sothat,by§7"51, dyjr '-dzTO= 2m^2?n When theexpressionontherightisexpandedasfarastheterminvolving l/z^,wefindthat 1_^^1_^-1_(M-l)(At-25) _(;a-l)(^''-114yL6 +1073) dz^ 2^z^ 2V 2">z'' (fi-l) (5fji'-1dS5/m'+54703/^-375733) 215^ inthisequation fihasbeen written inplaceofAiv^forbrevity.Itfollows, on integration,that {^l-1){bij?-1535yLt^ +54703/^-375733)+7.2>«2^"^•••' andsotheequationtobesolved is* 1 1 At-1 (y^-l)(/^-25)z-mr-lv'Tr +l'7r-\-a^--^^ g-^f^g.... If/3= (/i,+ly-1)TT—a,theresult ofreversion is ^l-l (/i,-l)(7/^-31) (ya-1) (83/^2_982^+3779)^~/3— 2^/3 3 .2'y8^ 15 .2i"/S' (/i-1)(694V-153855/^2 +1585743/z-6277237) 105.215/3^ Therefore thelargezeros of/^(2)cosa—Fj,(^)sinaaregiven bythe asymptotic expansion 4i.— 1 (4i/--l)(28i^--31) (?i+1^—^)77—a— 8{(/I+li/-1)TT-a}384[{n+^v- \'rr)- a]' *This equation (inthecase v=l)wasgiven byGauss inhisnotebook with thedate Oct. 16, 1797, butnoclue isgiven concerning themethod bywhich heobtained it.[Cf.Math. Ann. Lvii.(1902), p.19.] 15-54, 15-6] ZEROS OFBESSEL FUNCTIONS 507 [Note. The factthatJ^ (-)+Y^(z)hasasimple asymptotic expansion shortens the analysisinamanner which wasnotnoticed byMarshall;heused theequations 1- andhesolved thelatter byassumingadescendingseries forJ/.] 15*54. Zeros offunctions related tocylinder functions. Themethod ofStokesis,ofcourse, applicabletofunctions other than thosejust investigated. ThusMcMahon* hascalculated thelargezeros of 9Bv{z)andof-—"^when thecylinderfunction isaBessel function of the firstorsecond kind. Thegeneralformula forthelargezeros of'Wv{z)is R/^+37/x-^+82 /ii-9 ^' 8/3i 384y3i^ where/3i=(w+iz^+^)7r—a,while thecorrespondingformula forthelarge .d[z-'-^~{z)]. zeros 01—-— -,——^^isdz _/^+7_7yti-+15V+95_^' 8/3i "384/3i^ Thezeros of J,{£)\\ikz)-\\iz)J^{kz\ where kisconstant, andof J:{z)Y;{kz)-Y^{z)j;{hz) havebeen treated inasimilar manner byMcMahon. Kalahnet hasconstructed tables of thezeros oftheformer function when khasthevalues1*2, 1"5,2*0and vis0,^,1,f,2,#; while ithasbeenproved byCarslaw, ConductionofHeat (London, 1922), p.128,thatthese zeros are allrealwhen vandkare real. The zeros of//{£)J\,{kz)—Y^{z)J^(kz)have beenexamined bySasaki, Tohoku Math. Journal,v.(1914), pp.45—47. 15"6. Themode ofvariation ofthezerosofacylinder functionwhen its order isvaried. Theequationinz hasaninfinite number ofroots, thevalues ofwhichdependon z^;since J^(z) isananalyticfunction ofboth zandv,solongasz^O,itfollows thateach root oftheequationis(withincertain limits) ananalyticfunction ofv.A similar statement holdsgoodwhen thefunction ofthe firstkind isreplaced byanycylinderfunction ofthetypeJ";,(2')cosa—7,,(^)sina,where aisany constant. Ifjdenotes anyparticularzeroofJ^(z),therateofchangeofJ,asvvaries, isgiven bytheordinaryformula ofpartialdifferentiation (1) ^'0)1+'cV.(z) =0. ov *Annals ofMath. ix.(1895), pp.25—29. tZeitschriftfilr Math, undPhys.liv.(1907), pp.oo— 8(5. 508 THEORY OFBESSEL FUNCTIONS [chap. XV SinceJ^(j)=0,itfollows that <//(j)=—J^+i(j)t^O,solongasjisnot zero,andhence, from§5'11(15),whenR(v)>0, dj_2i/ (2) ^^no?.dvjJ\+,{j)Jo"'"' t Thisformula shews that tuhen vispositive,thepositivezerosofJ^ix)in- crease asVisincreased. Equation (2)wasstated withoutproofbySchlafli, Math. Ann. x.(1876), p.137;audthe deduction from itwasestablished inadifferent manner byGegenbauer*, Mem. delaSoc. B.desSci.deLiege, (3)ii.(1900),no.3,inthecase ofthesmallest zeroofJy{x). Weproceedtoextend theresultsalreadyobtained tothepositivezeros of ^^(z)=J^{z)cosa—Y^ {z)sina, where visanunrestricted realvariable, andaisconstant(i.e.independentofv). Theextended theorem isasfollows : Any 'positive zero, c,of^^(z)isdefinableasacontinuousincreasing function oftherealvariable v. Toprovethistheorem w-eobserve that cisafunction ofvsuch that arctan isconstant, sothat dc dv and therefoj-e^arctan -^j-' \dz\J^{z))_+ z=cJAc) s-arctan cv TTCdv+J,.{z)^^-Y^{z)a/,.(^)'=0, dv=0. Hence, by§13-73(2), wehave (3)dc -r=2c^0(2csinht)e-^^Ht.dv j Since theintegrandispositive,thisformula shews that cisanincreasing function ofv. Alessgeneral theorem, namely that, ifcisazerowhich isgreaterthan theorder v(supposed jjositive), then cisanincreasingfunction ofv,hasbeen proved bySchafheitlinf with theaidofveryelaborateanalysis. Itwillbeobserved from thedefinition ofY^(z)that ctends tozeroonly when Vtends toant/negative value which satisfies theequation sin(a—vtt)=0. *Thereader should note that theanalysisinthelatter partofGegenbauer's memoir is vitiated byhisuseofEudski's erroneous results(§15-1). tBerliner Sitzungsberichte,v.(1906), pp.82—93;Jahresbericht derDeutschen Math. Vereini- (jimg,XVI.(1907), pp.272— 279. 15-6] ZEROS OFBESSEL FUNCTIONS 509 Itshould benoticed that (3)shews that,when vistaken tobeacomplex number and cisa(complex) number, with ajiosifiverealpart,then cisan analyticfunction ofv;and so,asvvaries, thezems of'^^(z)varycontinuously, andtheycanonlycome into existence ordisappear when cfails tobean analyticfunction ofv,i.e.when c=0. Itfollows thatthepositivezeros of9^^(2)arederived from those of'^^(z) byaprocessofcontinuous variation asvvaries, exceptthatonepositivezero disappears whenever vpasses throughoneofthespecified negativevalues. Ifwenowchoose asothat* O^ccKtt, weseethat, asvvaries from| toanyvalueexceeding (ck/tt)—1,nozerosdisappear duringtheprocessof variation ofv,and sointhecase ofzeros which aresolargethattheformula ofStokes'type (§15"58)isavailable, theformula 7177+^VTT—iir-a-—-Jr- ... givesthenthpositive zero,when thepositivezeros areregardedasarranged inorder ofmagnitude. If,however, vhasvaried sothat itfinallyliesbetween(o/tt)—A;and (o/tt)—k—1,where kisapositive integer,kzeros havedisappeared, andso theformulajustquoted givesthe(n—k)th. positivezero. ThistjijeofargumentisduetoMacdonald, Proc. Londoii Math. Soc.xxix.(1898), jjp..575—584;itwasapplied byhimtothediscussion ofthezeros ofBessel functions ofthe firstkind oforderexceeding-1. Ifwedraw thecurve^t^xiy)=0,itevidentlyconsists ofanumber of branchesstartingfrompointsonthenegativehalf ofthe«-axis andmoving upwardstowards theright,bothxandyincreasingwithout limit oneach branch. Ifwetakeanypointwithpositivecoordinates(v^t,yo)anddraw from ita line totherightandalinedownwards terminated bythe^-axis, itisevident that thecurve "^-j(y)=meets each ofthelines inthesamenumber of points.Itfollows thatthenumber ofzerosofVJ'^(y,^), quafunction ofv,which exceedi\tisequaltothenumber ofpositivezeros of?J?^^(y)quafunction ofy which arelessthany^,.This isageneralisationofatheorem duetoMacdonaldf, whotook Vn=andthecylinderfunction tobeafunction ofthe first kind. Fig.33illustrates thegeneral shapeofthecurves Jx(y)=0,thelength ofthesides ofthesquares being5units.Amuchlargerandmore elaborate diagramofthesame character hasbeen constructed byGasser|, whohasalso constructed thecorresponding diagz-amforYx{y)=0.Thediagramfor 'Wxiy)= isofthesamegeneralcharacter asthat forJ^(y)=0,exceptthat *This doesnotlead toanyreal lossofgenerality. tSeealetter fromMacdonald toCarslaw, Froc. London Math. Soc.(2)xiii.(1914), p.239. XBern Mittheilungen, 1904, p.13o. tl 510 THEORY OFBESSEL FUNCTIONS [chap. XV theportionsofthecurves below theaxis ofxconsistmerelyofanumber of isolatedpointsonthelinesonwhich 1xisanoddinteger. 11 1 15-61, 15-7] ZEROS OFBESSEL FUNCTIONS 511 thecircle r=a,and ifthestraiglit edgesofthemembrane arefixed, thedis- pUicementinanormal vibration isproportionalto J^(rp/c)sinvdcos(pt+e), where cisthevelocityofpropagationofvibrations. Ifthecircular boundary ofthemembrane isfixed, thevalues ofap/carethezeros ofJ^(x),while if theboundaryisfree tomovetransversely theyarethezeros ofJJ(x). The eff'ect ofintroducingconstraints intheform ofclampswhichgradually diminish theeffectiveangleofthesector istoincrease vandtoshorten the periodsofvibration, sothatpisanincreasingfunction ofr,andtherefore (since aand careunaltered) ap/cisanincreasingfunction ofv.That isto say,thezeros of-/,,(x)andJJ(w)increase with v. Byusing argumentsofthis character, Rayleighhasgiven proofsofa number oftheorems which areprovedelsewhere inthischapter byanalj'tical methods. 15-7. ThezerosofK,.(z). Thezeros ofthefunction K^(z),where i^isagiven positive number(zero included), andzliesinthedomain inwhich )arg^ |<|7r,havebeen studied qualitatively byMacdonald *. From thegeneralisationofBessel'sintegral, givenin§6"22, itisobvious thatK^(z)hasnopositive zeros; and ithasbeenshewn furtherbyMacdonald thatKy(z) hasnozeros forwhich |arg2^ |^^tt.Thismaybeprovedatonce from aconsideration oftheintegral givenin§13'7l;for,ifz=7'e^'^were such azero(r>0,—^tt<a<^tt),then z=re~^°-would beanother zero ;butthe integralshews that jr , ,^rr ^ ,xIf^ {V ?'-COS 2a]„/?'- \dvK.(r.'^)/C(r.-'^)=- j^exp|-,--^\k. (-)- >0, which iscontrarytohypothesis. Ifaisequalto+^tt,wehave andsoK^(z) hasnopurely imaginaryzeros. Nextwestudythezeros forwhichR(z)isnegative,thephaseofzlying eitheHbetween ^ttand ttorbetween —|7rand—tt. Itmaybeshewn thatthetotalnumber ofzeros inthispairofquadrants istheevenintegerfnearest tov—^,unless v—\\s,aninteger,inwhich case thenumber isv—^. Inthe firstplace,there arenozeros onthelinesSiTgz= ±tt,unless v—h *Proc. London Math. Hoc.xxx.(1899), pp.165—179. +This isnotthenumber given byMacdonald. 512 THEORY OFBESSEL FUNCTIONS [chap. XV isaninteger;forK^,(re*'"')=e^""^K^{r)+iriI„(r),and, ifboth therealand theimaginary partsofthisexpressionaretovanish, Avemust have cosVTT .K^(r)=0,sinvtt .K^(r)+vr/^(?•)=0. Since theWronskian ofthepairoffunctions ontheleftoftheequationsis (7r/7-)cos Z'TT,theycannot vanishsimultaneouslyunless cosi'7r=0. Now consider thechangeinphaseofz"K^{z)aszdescribes acontour consistingofarcsoflargeandsmall circles terminatedbythelinesarg^=+tt, togetherwith thepartsofthese lines terminatedbythe circular arcs. (CfFig.15of§7-4.) Ifthecircles becalledVand7,theirequations being \z\=Rand\z\=h, itisevident thatthenumber ofzeros ofK^,{z)inthepairofquadrants under consideration isequaltothenumber ofzeros ofz"K^,{z)inside thecontour, andthis isequaltol/(27r)times thechangeinphaseofz^K^{z)asztraverses thecontour. Now thechangeinphaseis arg [z"A%{z)] +s.vg[z-K,{z)]arg [z^K^{z)] Sexp-rri +arg[z''I{,(z)]Rexf)(-Tri) 6exp(-7ri) _Eexpiri AsR^cc andS-^0, the firsttwoterms* tend to27r{u—l)and respectively,because when |^ |islargeorsmall onthecontour, z"K^{z)-•^-'-ig-V^TT),2"7C{z)~2"-!r{v) respectively f. The lasttwoterms become lim 2arctanTTcosvir .I^,(r) K^(r)+TTsinVTT .7^(?•)_R NowK^(?)isapositive decreasingfunction ofrwhile /^(r)isapositive increasing function, and sothe lastdenominator hasonezero ifsinvttis nf^gative,andnozero ifsinvttispositive. Iftherefore wetake theinverse function tovanish when?-->-0, itslimit when 1—»-X) isarctan(cot i/tt),thevalueassignedtotheinverse function being numericallylessthantworight anglesandhavingthesamesignasthe signofcosVTT. Hence thetotalnumber ofzerosofK^{z)inthepairofquadrants :]:inwhich R(z)isnegative and |argz\<-rris v—-k+—arctan(cot vtt),TT *This isevident from theconsideration thattheasymptotic expansion of§7"23isvalidwhen Iargz I^TT. tThesecond ofthese approximate formulae requires modificatiou when ^=0. XThetwozeros ofA'o{z)arenotveryfarfrom thepoints-l-29=fc0-44;(. 15-8] ZEROS OFBESSEL FUNCTIONS 513 andthereader willfind iteasytoverifythat thisnumber istheeveninteger which isnearest tov—^. When v—^isaninteger, Ky(z)isapolynomialinzmultiplied bya function withnozeros inthe finitepartoftheplane,and sothenumber of zeros forwhichB(z)<0 isexactlyv—^. Next consider theportionoftheplaneforwhich vr<arg^^27r. Ifwewrite z—^e^"*,wehave K^(z)=-^7re-i-' [F,(0+ ^•(1+2^2-) J,(^)], and soK^(z)hasasequenceofzeroslyingnear thenegative partofthe imaginaryaxis. The zeros oflarge modulus whichbelongtothissequence aregiven approximately bytheroots oftheequation tan(f-hvTT-Itt)=-{(1+ 262""') ; itmaybeverified thattheyareultimatelyontherightorleftoftheimaginary axis inthe^-plane accordingascos- z/tt islessthan orgreaterthanI;i.e. accordingasvdiffers from thenearestinteger bymore orlessthanJ.The sequencedoesnotexistwhen e'^'"'^=—1,i.e.when vishalfofanoddinteger. There isacorresponding sequenceofzeros near thelineargz=—ftt. 15*8. ZerosofBesselfunctions ofunrestrictedly largeor^der. Theprevious investigations,basedmainlyonintegralsofPoisson'stype^ have resulted inthedetermination ofpropertiesofzeros ofBessel functions,, when theorder visnotunduly large.Thisis,ofcourse, consistent with the factthat Hankel'sasymptotic expansions,discussed inChapter vii,are significant onlywhen v^isfairlysmall incomparison with theargumentof theBessel function. The factthatDebye's integralsof§8"31 affordrepresentationsoffunctions oflargeordersuggeststhat theseintegrals mayformaneffective means of discussingthezeros ofBessel functions oflarge order; and this, infact,proves tobethecase*. Moreover, themajorityoftheresults which willbeobtained arevalid forfunctions ofanypositive order, though theygaininimportance with theincrease oftheorder. Weshalladoptthenotation of§8'31, sothatf gvi(tan/S— /3)/00+ni— ifi H,<"{vsec/3)= ^ e-"^dw, where—t=sinhw—lu+itan/3(cosh'W—1),andthecontour intheplaneof thecomplexvariable wischosen sothattispositiveon it. *Watson, Froc. RoyalSue. xciv. a,(1918), pp.lyU— 206. tWeshall usethesymbols xand vsec/3indifferently when x>v. W.B. F. 33 514 THEORY OFBESSEL FUNCTIONS[CHAP. XV /;Ifw=u+iv,where uand vare real,uandvboth increasesteadilyasw describes thecontour, sothat e-'^du, e'"^dv J-e arebothpositive. Hence,ifweregard /3asvariable, anddefine arg€'"''dw J-X-?j3 tobeapositiveacuteanglewhen/8=0,and tovarycontinuouslywith/3,it willremain apositiveacuteangleforallvalues ofy8between and^tt;and, moreover, by§8"32,itcannot exceedJtt,since dwjdu^\/3. Thispositiveacuteanglewillbecalled;t^,andthen^willbedefinedby theequation -^=i;(tan /S-yS)+x-^TT. Itisthen evident that ^,w(i,sec/3)=ilWe^*, where J¥lispositive (not zero); and J-^(a;)=iHilcos^, r,(«)=ittsin^. If ^y{x)=Jy(x)cosa-Yy(x)sina, itisclear that theonlyzeros of^^(x),greaterthanv,arederived from the values of^which make^+otequaltoanoddnumber ofright angles. Itiseasytoshew that^increases with x,when vremains constant. For wehave ^=arctanV^,-j-=t,/ %t/,/ x>^^ J^(x) dx JJ"(x)+Yj"(x) Hence, asxincreases, ^increasessteadily, and so,toeach ofthevalues of^forwhich^=(m+I)77--a, correspondscmeandonlyonepositivezeroof^^(x). Nextweshallprovethat%isalsoanincreasingfunction ofx.This isa theorem ofamuchdeeper character, since theresult of§18'74 isrequired toproveit;wethence have dx_(^_d(tan /3-/3)_ 2/(7rA-) _^/jx"-v') dxdx dx J^(x)+Fy-(x) X' . From Hankel'sasymptotic expansionitisclear that limX=lini[x—hvTr—lTT—^/{x-—v~)+varccos(vjx)+|tt+(1/^)] A . (•^•'(^) 1 1andso arctan i—„;.><y<xtt, inwhich theexpressiononthe left isapositiveacuteangle. 15-8] ZEROS OFBESSEL FUNCTIONS 515 Toformanestimate ofthevalue of -y^when vislarge,wewrite J^(v)=—Y^(v)tan7^ ; hence, from§8*42,wehave lirn7,=^TT, and,when vislarge, tan7^11-r(§)+ V'^i2ior(i).a^V sothat,when vislarge, 7^isanincreasingfunction ofv. ThefollowingTablegivesthesexagesimal measure(tothenearest half minute)oftheangle whose circular measure is7^;itexhibits thecloseness of 7^toitslimit, evenwhen visquitesmall: V 516 THEORY OFBESSEL FUNCTIONS[CHAP. XV 15"81. Thesmallest zerosofJy{x)andY„(oc). Ithasbeen seen(§15'3)that/^(x)and¥„(x)havenozeros intheinterval (0,v),when pispositive,and itisfairlyobvious from theasymptoticformulae obtained in|842thattheyhavenozeros oftheform v+o(v^)when vislarge. Theasymptoticformulae which werequotedin§8'43shew that,with the notation of§15'8, where theinversetangentdenotes anegativeacuteangle. Hence, atthesmallest zero ofJ^,(x), tan\v(tan/3-/8)-|7r +{l/\/v)\=- PGi^tan^^,!)- As ySincreases from to^tt,theexpressiononthe leftincreases from 1+(1/^v),while theexpressionontherightdecreases from0'2679 to0;the smallest rootoccurs foravalue of/3forwhich v(tan^—^)liesbetween ftt andfTT,sothat V(tan/3-^)=^vtan^^+0 (v-i). Hence, ifwesolve theequation tan(^-f7r)=-Q(^,J)/P(^,i), thevalue of^soobtained isthevalue of^ytan^'yS atthezerowithanerror which is{v~^). Thevalue offisapproximately 2"383447, andhence the smallestpositivezeroofJ^(x)is v+vix1-855757 +0(1). Inlikemanner, bysolvingtheequation tan(|-i7r)=-Q(f,i)/P(^, a ofwhich thesmallest root isapproximately f=0'847719, wefind that the smallest zeroofY^(x)is z;+7.4X0-931577 +0(1). Theformula forthesmallest zero ofJy(x) hasbeen given byAirey,Phil.Mag. (6) XXXIV. (1917), p.193. Airey's formula wasderived byusing Debye's asymptotic expansion of§8-42 for/„{x)whenxhasavalue such that x-v=0{vi)^o{v^). Forsuch values ofthevariables,ithasnotbeenprovedthatDebye's expansionisvalid, andalthough Airey's methodgivesthetwodominant terms ofthesmallest zerooft7„(x) correctly,thenumerical result which Airey givesforthesmallest zeroof//(x)isnotthe same asthatof§15-83. Thereason whyAirey's method givescorrect results isthatJ^{v+^) isexpansibleinpowers of(solongasfiso{v\andinthisexpansionitispermissibleto substitute Debye'sformulae forJv{v), //(v),JJ' (v), Aformula forthesmallest zeroofJ^^ix) wasgiven byAirey. This zeromayheany- where between andthesmallest zeroofJ^(x),accordingtothevalue ofv. 15-81, 15-82] ZEROS OFBESSEL FUNCTIONS 517 Itdoesnotseem tobepossibletomake furtherprogress bythemethods used inthis section. Weshallnowmake adigressiontoexplain themethods ofSturm (which have beenappliedtoBessel's equation byvarious mathematicians), andweshallthen givean investigation which leads tothefascinatingresult thatthetwoexpressions which,inthis section, wereprovedtobe0(1) areinreality {v~^),sothatapproximationsareobtained forthesmallest zeros ofJ^(x)andFy(.f)inwhich theerrors are0{v~^),i.e.theerrors becomenegligible when vislarge. [Note. Anelementaryresult concerningthesmallest zero ofJ^(x)hasbeenobtained from theformula of§5'43byGegenbauer,Wie7ier Sitzungsberichte, cxi. (2a), (1902), p.571; if^=I'+6where<e<1,then thesmallest zeroofJ^v+e{^)islessthan twice thesmallest zeroofJ^,(x),because, forthelatter value of.vtheintegrandcannot beone-signed.] 15'82.Applications ofSturm's 'methods. Various writers have discussedpropertiesofBessel functions bymeans of thegeneral methods invented bySturm* fortheinvestigationofanylinear differentialequationofthesecond order. The results hitherto obtained in thismanner areofsome interest, though theyarenotofaparticularly deep character, andmost ofthem havealreadybeenprovedinthischapter by other methods. Thetheorem which isatthebase oftheinvestigationsinquestionisthat, givenadifferentialequationofthesecond order initsnormal form d"u^ inwhich theinvariant /ispositive,then thegreaterthevalue of/,themore rapidlydothesolutions oftheequationoscillate asxincreases. Asanexampleofanapplicationofthisresult, wemaytake atheorem duetoSturm {ibid. pp.174—175)andBourget, Ann. sci.deVi^cole norm.sup.in.(1866), p.72,that,if v^- 1-bepositive and cbeanyzeroof"^^ (,^•)which exceedss/{v^- ^),then thezeroof'g'^{x) which isnestgreater than cdoesnotexceed c+ ,,„"^ „— j-,. This result follows atoncefrom theconsideration ofthefacts that thefunction ^'"^('(•*')isannihilated bytheoperator (§4-3) dx- \x'^ andthat,when x^-c, 1—-—^^>~''—? . X' 6-2 Aslioiitly more abstruse result isduetoPoi-ter, American JournalofMath. xx.(1898), pp.196—198, totheefiect that,ifi/->jand ifthezeros of^^{x),greater thans^{v--^), inascendingoi-der ofmagnitudeareCi,Co,C3,...thene^+i-Cn decreases asnincreases. Thishasalready beenprovedini^15-8byanother method. Other theorems oflikenature aredue toBocher,Bulletin American Math. Soc. in. (1897), pp.205—213; vii.(1901), pp.333—340; andtoGasser, BernMittheilungen, 1904, pp.92—135. *Journal deMath. i.(1836), pp.106—ISO;anaccount ofrecent researches ondifferential equations bySturm's methods isgiveninalecture byBocher, Proc. Int.Congress ofMath. i. (Cambridge, 1912), pp.163—195. 518 THEORY OFBESSEL FUNCTIONS[CHAP. XV 15*83. Applications ofSturm's methods tofunctions oflargeorder. Weproceedtoestablish anumber ofresultsconcerning cylinderfunctions oflargeorder which arebased onthefollowing theorem ofSturm'stype: LetWj(x)andU2{x)hesolutionsoftheequations such that,whenx=a, Ui(a)=U2(a), u((a)=u^(a), and letIiand Inbecontinuous intheinterval a^x^h, andalso letu/(x)and U2(x)becontinuous inthesame interval. Then, if1^^/othroughouttheinterval*, \u2{x) \exceeds\u^(x)1solongas Xliesbetween aand thefirstzeroof tt^(x)intheinterval, sothat thefirstzero ofui(x)intheinterval isontheleftofthefirstzeroofuoix). Further, ifw/(a)hasthesamesignasu^(a),thefirstmaximumpoint of IUt^{x) Iintheinterval isontheleftofthefirstmaximumpoint of \u^ix) |,and, moreover max iWi{x)i<max1^2(^)1- Toprovethetheoremf,observe that, solongasu^ix)andu^{x)areboth positive, d'Uo aU-i .J jrV^rv and so,whenweintegrate, du2 dui ^dx""^dxX Since theexpressionnowunder consideration vanishes atthelower limit, wehave du2 dui^_ Hence wehave andtherefore that istosay, (2)ax U2X a Ui(x)'^ u^(a) *Tosimplifythepresentationoftheproof ofthetheorem, itisconvenient tochange thesigns ofu-y(x)and (/._>(.r),ifnecessary, sothat Wi(x)ispositive immediately ontheright of j:=a;the signs indicating moduli maythen beomitted throughout theenunciation. tThetheorem ispractically duetoSturm, Journal deMath. i.(1836), pp.12-5— 127,145—147. 15-83] ZEROS OFBESSEL FUNCTIONS 519 Itfollows thatjustbeforeUi(x)vanishes forthe firsttimeUii^)isstill positive, and ithasremainedpositivewhile xhasincreased from the value a. The firstpartofthetheorem isthereforeproved. Again,ifUi'{a)ispositive,aswellasUi(a),thenUi(x)musthaveamaximum before itvanishes, andatthispoint, /Xj,wehavefrom(1) soihatii2(fJ-i)ispositive and u.^{x)must bepositiveintheinterval(0, /Hj). Therefore the firstmaximumpoint /loofu,2,{x)must beontherightof/^i. Finally wehave max Ml{x)=Ui(/ii)^Uo(/ij)^u.2{fx-,)=max k,,i^), andthetheorem iscompletely proved. When twofunctions, Ui(x)and u^(x),arerelated inthemannerpostulated inthistheorem, itisconvenient tosaythatUi(x)is7noreoscillatory* than Wj(x)andthat Uo(x)islessoscillatorythan Uj(x). We shallnowapplythetheoremjustprovedtoobtain results fconcern- ingJ^{x)andY^(x)when vislargeand ic—i;is(v^).Ourprocedurewill betoconstructpairsoffunctions which arerespectively slightlylessand slightlymoreoscillatorythan thefunctions inquestion. Inthe firstplacewereduce Bessel'sequationtoitsnormal formbywriting X=ve^;wethenhave (3)<l&^^-^-^9r%{ye')=^, Afunction which isobviously slightlylessoscillatorythan'^^(ye^)for smallpositivevalues ofQisobtainable bysolvingtheequation m^-^'''(*) since e~^—\'^Wwhen 6'^Q. Thegeneralsolution of(4)isu=0, andtheconstantsimpliedinthiscylinderfunction have tobeadjustedso thatuand itsdifferential coefficient areequalto"^^(ve^)and itsdifferential coefficient at^=0. *Thereason fortheuseofthese terms isobvious from aconsideration ofthespecialcase in whichZjand I2arepositive constants. tThese results supersede theinequalities obtained byWatson, Proc.London Math. Soc.(2) XVI.(1917), pp.166—169. 520 THEORY OFBESSEL FUNCTIONS [CHAP. XV Itfollows that afunction which is(slightly)lessoscillatorythan ^v{x), when x'^v,'\'& (2e)i' (f)a-)*^.(v)J-,(^")+ra)(i.)3 <^;(^)/j3 Wenowendeavour toconstruct afunction which is(slightly)more oscil- latorythan^y(x),inorder thatwemayhave^^(x)trappedbetween two functions which aremoreeasily investigatedthan^^(x). Theformula forthe lessoscillatory function, combined with theresult stated in§8*43, suggeststhatweshould construct afunction ofthetype* whereo/r(6)isafunction of6tobedetermined. Itmightbeanticipated from§8'43thatthesuitable form foryjr(6)would be|tan^/3,where sec/3=e^ ; but itappearsthat thisfunction leads toadifferentialequation whose solution issuch that itsdegreeofoscillationdependsontherelative values ofvand 6, andwearenotable toobtain anyinformationthereby. Theinvariant o/theequationdetermined by isknown tobe(§4'31) 2tT^T" 4WW)\+36\^Td-)\+'i^^^^J' and itisrequisitethat thisshouldslightlyexceed p^(e-^— !)•Itisconsequently natural totestthevalue ofyjr(6)which isgiven bytheequations ,|r'(^)=V(e^«-l), ir{0)=0, bydetermining whether, forthisvalue oft/t(6), 2i/r'(6) 4{yjr'(d) ]"^36(^/r{0)\^''• When wereplacee*bysec^,wefindthat •>|r'(6)=tan13, ^Ir{6)=,tan/3-/3, andhence wehave totestthetruth oftheinequality sin^yS (5) 3(tan;8-/3)^ ^''' J-03(tan/3-yS)-cos2/3V(H-itan'/3) sin^/3 cos2/SV(l +itan-yS)_ isnegative when tan-/3<\/24—3,and itispositiveforgreatervalues oftan^/S. *Themultipleofthecylinder function istaken sothat theproductsatisfies adifferential equationinitsnormal form; cf.§4'31 (17). 15-83] ZEROS OFBESSEL FUNCTIONS 521 Hence, since(5)istruewhen /3=0,itistruewhen0^/3^ ySo,where ^p isacertainanglebetween arctan\/{\/24—3}and Itt.Thesexagesimal measure ofj3,is59°39'24"-27. Proceedingasintheformer case,wefindthatthefunction V{3(1-^cot/3);.[r(f)(i^)i^^{v)J_j [v(tan y3-^)] +r(i)(^)5 '^Z(^)Ji[v(tan /3- /3)]] isslightlymoreoscillatorythan/^(i/sec /3),solongas* ^^^/Sq. Wecannowobtain anextremely importantresultconcerningthesmallest zeroof^^^,(w) which isgreaterthan v;forlet^X.^*bethesmallest value of^ which makes r(f)(1v)i'^.(v)./_.(I)+r(^)(1v)^-#;(^)/^(|) vanish. Then 6o^A-oftheequations 26=\,yv^V(tsin 13-/3)=I\/ givex=V+^\/v-^+ (v~-^). Since, bySturm's theorem, thezero of'^^(x)liesbetween twoexpressionsof thisform,weseethat thevalueofthezeroof'Wv(^)which isnextgreaterthan v isexpressibleintheform When ^y(x)isequaltoJ^(x)itiseasytoverifyfrom aTable ofBessel functions oforders ±^that K=1-926529 + (i/-S), andsothesmallest zeroofJ^,(x),when vislarge,is v+v'^x1-855757 +0(z^-i). Inlikemanner, thesmallest zeroofY^(x)is V+v'^X0-9S1577 + (v-i). The firstmaximum ofJ^(x)maybeobtained inasimilar manner, bydiffer- entiating fthetwoexpressionsconstructed asapproximations. Theresult isthat if^fi^,^isthesmallest value of|which makes r(f)(ir.)ij.(^)j-j(f)-r(i)(i^)s/;(i.)j-_3(l) vanish;]:, then the firstmaximum ofJ^,(x)isatthepoint v+^fi^'fi+Oiv-i), i.e.atthepoint v+v^x0-808618 +(i^-i). The^rst maximum ofthefunctionV^,(x)cannot betreated inthismanner because its firstmaximum isontherightofitsfirstzero;this follows atoncefrom§15"3,because Yy(x)increases from-qotoas.rincreases from tothe first zero. Foraninvestigation ofthemaximum value ofJy{.v) quafunction ofvthereader should consult apaper byMeissel, Asir.Nacli. cxxviii.(1891),cols.435—438. *This restriction istrivial because wehave toconsider values of^forwhichv^"^isbounded; i.e.small values of/3. tThepermissibiHty ofthisfollows from thesecond part ofSturm's theorem just fj;iven. XThis vaRie offisapproximately 0-685548. CHAPTEH XYI NEUMANN SERIES ANDLOMMEL'S FUNCTIONS OFTWOVARIABLES 16*1. Thedefinition ofNeumann series. TheobjectofthischapterandofChapterxvii istheinvestigationof varioustypesofexpansionsofanalytic functions ofcomplexvariables inseries whosegeneralterms contain oneormore Bessel functions orrelated functions. Theseexpansionsaretosome extentanalogoustothewellknownexpansions ofananalyticfunctionbythetheorems ofTaylor andLaurent. Theexpansions analogoustoFourier'sexpansionofafunction ofarealvariable areofamuch more recondite character, andtheywillbediscussed inChaptersxviiiandxix. Anyseries ofthetype iscalled aNeumann series, althoughinfactNeumann considered*onlythe special typeofseries forwhich visaninteger;theinvestigationofthemore generalseries isduetoGegenbauerf. Todistinguishthese series from thetypesdiscussed in§16'14, thedescription 'Neu- mann series ofthe firstkind'hasbeensuggested byNielsen, Math. Ann. LV.(1902), p.493. Thereader willremember thatvariousexpansionsoffunctions asNeumann series havealreadybeen discussed inChapterv.Itwillbesufficient toquote herethefollowingformulae : (^.)., s<'-^^">T'"^"^ j>«..(^).n=o n\ J,(z+t)=SJ,.,n{t)Jm{z),m=-00 where -sr^=Z"" -{-z--2Zzcos</>. Weshall first discuss thepossibilityofexpandinganarbitraryfunction intoaNeumann series;thenweshallinvestigatethesingularitiesofthe analyticfunction definedbyaNeumann series withgivencoefficients;and finally weshall discuss theexpansionsofvariousparticularfunctions. Foraverygeneral discussion ofgeneralisationsofallkinds ofseries ofBessel functions, thereader mayconsult memoirs byNielsen, Journal furMath, cxxxii.(1907), pp.138— 146; Leipziger Berichte,lxi.(1909), pp.33—61. *Theorie derBesseVschen Functionen (Leipzig, 1867), pp.33—35. tWiener Sitzungsberichte,lxxiv.(2),(1877), pp.125—127.' 16-1, 16-11] NEUMANN SERIES 523 Various expansionsoftypes which resemble Neumann's (other than those giveninthis chapter)areduetoH.A.Webb, Phil. Trans, oftheRoyalSac. cciv. (1905), p.487and Nielsen, Atiidella R.Accad. deiLincei, (5)xv.(1906), pp.490—497. 16'11.Neumanns expansion* ofanarbitrary functioninaseries ofBessel coefficients. Letf{z)beafunction ofzwhich isanalyticinside andonacircle of radiusRwith centre attheorigin.IfCdenotes thecontour formedbythis circle and ifzisanypointinside it,itfollows fromCauchy'stheorem that fit) Now, by§9-l,dt. z '—=SenOn{t).Jn{z); I—Z,„=o and thisexpansion converges uniformlyonthecontour. Itfollows atonce that (1) /(^)= 2a„J„(4 where (2) an= {^^^^Jit)On{t)dt^ and this isNeumann's expansion. IftheMaclaurinexpansion <^if{z)is f{z)= ShnZ^\ n=0 weseethat 524 THEORY OFBESSEL FUNCTIONS[CHAP. XVI 16*12.Neumanns*analogue ofLaurent's theor 6771. Letf{z)beafunction ofzwhich isanalytic andone-valued inthering- shaped regiondefined bytheinequalities LetCand cbethecontours formedbythecircles \z\=R,\z\=r,' bothcontoursbeingtaken counter-clockwise;then, if^beapointoftheregion between thecircles, wehave ^^ ^27riJct-z 27riJe z-t =X^Jn{z)l f(t)On{t)dt+X ~On(z)t f{t)Jn{t)dt. Consequently f{z)isexpansibleintheform 00 00 (1) f(2)= SanJn(z)+ San'0n{2), where (2) an=9—ff(t)On(t)dt, a,/=^ ffit)Jn(t)dt. IftheLaurentexpansion off{z)intheannulus is 00 00 I-' f(z)^ SbnZn+ t^, wehave, asin§16'11, («) C=«'f'2»-'.»<''-'",-'>'i,_.„, („>1), ,m "(—) m= 16*13. Gegenhauer's generalisation ofNeumann sexpansion. Byusingthepolynomial An,v{t) defined in§9"2, Gegenbauerfhas generalisedtheformulagivenin§16"11. Iff{z)isanalyticinside andonthecircle\2\=R,and ifGdenotes the contour formedbythis circle, wehave ,,,If z''f(t)dt = 2^Jc{ Jo^"^"^'^^«>^(0}/(0dt, andso 00 (1) Z^f{z)= 2anJ^+n{z), M= *Theorie derBesseVschen Functionen (Leipzig, 1867), pp.36—39. tWiener Sitzungsberichte, lxxiv.(2),(1877), pp.124—130. SeeWiener Denkschriften,xlviii. (1884), pp.293—316forsomespecial cases oftheexpansion. 16-12-16-14] NEUMANN SERIES 525 where (2) an= -^.\j{t)AnAt)dt, provided onlythat visnotanegative integer. If,asin§16"11,theMaclaurinexpansion oif{z)is f{z)== thnZ^\ then (3) ,,.=(„+,o1"2—»nid:4^)t ';«=0 m\^n—zm• Neumann's expansionof§16"12 maybegeneralisedinasimilar manner. 16'14. TJieNeumann-Gegenbauer expansion ofa/miction asaseriesof squaresorproducts. From theexpansionof§9"o,namely t—Zn=0 which isvalidwhen \z\<\t\,wecanatonce infer that, iff(z)isanalytic when1^1^?',then theexpansion 00 (1) Z>-+''f(z)= %anJ^+in(2)J,+Uz) isvalidwhen \z\<r,andthecoefficients aregiven bytheformula (2)an=^^.j^f(t)Bn.,,,.{t)dt, Cbeingthecontour formed bythe circle\z\=r.Thisexpansionisdue to Gegenbauer*;anexpansion closelyconnected withthis,namelythat (3) f(z)=ia,:J,Hz), where (4)a,:=-^^.^f{t)nn{t)dt, andQ.n{t)isNeumann's secondpolynomial (§9"4),isy-Alidprovidedthatf(z) isanevenanalytic function;thisexpansionwasobtainedbyNeumannf. Gegenbauei''sformula hasbeeninvestigated morerecently byNielsen, Nouv. Aim. de Matkr(A)II.(1902), pp.407—410. Atypeofseriesslightlydifferent from thosepreviouslyconsidered is derived from theformula of§5'22(7)intheform ^-'=2^r(1.+1)s^^^/i.+,(4 *Wiener Sitzungsberichte, lxxv.(2),(1«77), pp.218—222. tMath. A7171. III.(1871), p.599. 526 THEORY OFBESSEL FUNCTIONS[CHAP. XVI which shews that (5) iftn^"-^'^=ia„(iz)i(''+'^>Jh.+u) (^Xn=0 w=0 where Expansionsofthistypehave been thetopicofadetailedinvestigation by Nielsen *. 16'2. Pincherles theorem and itsgeneralisations. LetSanJv^n{z) beanyNeumann series, and letthefunction defined by this series and itsanalyticcontinuations becalled/(^). Letalso Thefunction definedbyf{z)yand itsanalyticcontinuations willbecalled theassociated powerseriesoff(z). TheNeumann seriesconverges throughoutthedomain inwhich lim^i{an J'„+„(^)1|<1, n->-<x> andthisdomain isidentical withthedomain inAvhich Km"/ Jl-*-00Vaniljr^<1,T{p+n+l) byHorn's asymptoticformula(§8-1). Itfollows that aNeumann series hasacircle ofconvergence, justlikea power series, andthecircles ofconvergenceofaNeumann series andofthe associated powerseries areidentical. Thetheorem thattheconvergenceofaNeumann series resembles that of apowerseries isdue toPincherlef; but itispossibletogomuch further, and,infact, itcanbeproved that/ (2^)hasnosingularitieswhich arenotalso singularities off(z)x^. Toprovethistheorem, wewrite 2Sctna^Y"^' * Nijt Tidsskrift,ix.(b),(1898), pp.77—79. tBologna Memorie, (4)iii.(1881), pp.151—180; seealso Nielsen, Math. Ann. lv.(1902), pp.493—496. 16-2, 16-3] NEUMANN SERIES 527 andthen, inside thecircle ofconvergence *, f{z)= j^^cos{^(1-t^]. (/>(zt^)-^^^--^^. From thetheoryofanalyticcontinuation itfollows that, if (f)(z)isanalytic foranyvalue of2',soalso isf(z),providedthatthepathofintegrationissuitably chosen;andsoallthesingularities oif{z) must besingularitiesof(f>(z). Now theseriesdefining cf)(z)maybewritten intheform ^^«»^ T(v+n+l) Vtt„ro2''+«r{v+n+l)r{v +n+|-)' andatheorem duetoHadamardfstates that, if F,(Z)^ ib„Z>\ F,(Z)= 5CnZ'\ F,(Z)= ibnCnZ\«=0 »=0 M=0 then allthesingularitiesofF^(z)areexpressibleintheform ^j,where/3is somesingularityofF^(z)and7issomesingularityofF.2(z). Since theonlyfinitesingularityofthehypergeometricfunction isatthepoint^=1,itfollows that allthesingularitiesof(z)aresingularities off(z)y; andtherefore allthesingularitiesoff{z)aresingularities off(z)x; andthis isthetheorem which wastobeproved. Thereader should have nodifficultyinenunciatingandprovingsimilar theorems:!:connected with theothertypesofexpansionswhich aredealt with inthischapter. 16*3. Variousspecial Neumann series. Thenumber ofNeumann series, inwhich thecoefficients areofsimple forms, whose sumsrepresentfunctions withimportant analytical propertiesis notlarge;weshallnowgive investigationsofsome such series which areof specialinterest. Byusingtheexpansion (^-2f- cos2^+1)-*= ir-^"--P„ (cos 2^), n= *Itisassumed thatR{v+h)ispositive;ifnot, theseveral series under discussion have tobetruncated bytheomission oftheterms forwhichR(v+h+A)isnegative, butthegeneral argumentisunaffected. tActa Miith. XXII.(1899), pp.55—64; Hadamard, LaSerie deTaylor (Paris, 1901), p.69. XForsuchtheorems concerning theexpansion of§16-14, seeNielsen, Math. Ann. lit.(1899), p.230 etseq. 528 THEORY OFBESSEL FUNCTIONS[CHAP. XVI Pincherle* hasobserved that i7.,. (.)P„(cos2^)=1 f""' ;,!?'If- Vl^it,. n=Q ^TTl J \/{P—2t-COS2^+1) where thecontour lieswhollyoutside thecircle j< |=1. Ifnowwewrite ^{t—1/t)=w,sothatthecontour inthew-planeisa (large)closed curvesurroundingtheorigin, wefindthat 00 1 /•( 2J,,+,(^)P,(cos2^)=^-^(0+) e^'^dw ^^^.n+.v/ nv /^^.j V{K+l)0^Hsin2^)}' andsoweobtain theformula X1/•(0+) 1r{iK-+) (1)SJ.,n+i {Z)Pn(cos26)=-:- e-^'"'''du =-r-e-f^sinesnw ^j^^ M= ^"^J ^TTj where themodulus oftheellipticfunction issin6. Theinteresting expansion hasbeengiven byJolliffef,whoprovedthattheseries ontherightsatisfied thesame differentialequationasJy-{^z). Thisexpansioniseasilyderived as aspecialcase of§11"6(1),butthefollowingdirectproofisnotwithout interest : ByNeumann's formula(§5"43)wehave| and, ifweexpand Jzv{z\jt)intotheseries T i+\-^^' s(2i^+2m+l)r(2i; +m+l ) J,.{zv«)- -^^^^ ;^r(2. +i) X2^1(-TO,2i;+7?i+1;2v+\;t)J2,.+mt+i i^), wefindonintegrationthat ^2{2v+2m+l)J.,. m= Z *Bologna Memorie, (4)vin.(1887), pp.125—143. Pincherle used ellipticfunctions ofmodulus cosec inhisresult. tMessenger, xlv.(1916), p.16.Thecorresponding expansionofzhJv-^ (\z)Jv{^z)wasobtained byNielsen, Nyt Tidsskrift,ix.b,(1898), p.80. XliR{v+^)<0, weuseloop integralsinstead ofdefinite integrals. 16-31] NEUMANN SERIES 529 where T(2u+m+l)n TT .m-L7>74-n /i -^7o^At''-Hl-trK,F,(-m,2v +>n+l;2j.+1;t)(U I{-Zi'+1)Jo bympartial integrations. Itfollows that a„,isthecoefficient ofh'^intheexpansionof 1 TT^2.(^_]^t.(1_^)}-^-i {1_^+/,^(1_t)]-^dt inascending powersofIt;andthisexpansionisabsolutely convergent when IA I<1. Itwewrite t—-^-,— ,1—hu wefindthat iaji'^=^ff-i {1-h(l- OJ-"-^ (1-t)-i(1+/?e)-i dt =-I^i-'-i(1-m)-* (1-h-u)-^du. TTJo Itisnowevident thatftoji+i=andthat 11.3...(2/i-l) fi^„,,, ,,,«-=- •- 2.4...(2n) Jo''^'-''^~'^" _l^1.3...(271-1) r{v+n+i)r(^) ~7r' 2.4... (2n) r(z/+n+l)' andthisformula atoncegivesJolliffe's form oftheexpansion. 16'31. TheNeumann seriessummedhyLommel. The effects oftransforming Neumann seriesbymeans ofrecurrence formulae have been studiedsystematically byLommel*; andhehassuc- ceededbythismeans inobtainingthesums ofvarious series ofthe^tyjje inwhich a^isapolynomialin ?i. Tafee thefunctions 00 M,,,n {z)=2(z.+-In+1)/:,, {V+2n+1)./,+„,,^j {z), i»= \/(= where/^,i,(t')isafunction tobedeterminedpresently. *Studien ilher dieBessel'scheii Functioiten (Leipzig, 1868), pp.46—49. \v.B.F. ,34 530 THEORY OFBESSEL FUNCTIONS [CHAP. XVI Bytherecurrence formula wehave 2°° (1)-64y,,n {Z)=Sf2y,{v+2m+1) {J'.+s^ {Z)+J^+2n+2 (2)} =f-2m {V+1)J",(^)+IM.^yn {z\ •providedthat/^(v) satisfiestheequation ofmixeddifferences* /;„(v+2n+3)+f,,n (v+2n+l)=2(v +2n+2)/,,^^^, (v+2n+2). Asolution ofthisequationis Weadoptthisvalueof/w(i') andthen itisfoundbythesamemethod that 2 (2) 'v,m{Z)=-f-2m-i {v)J^+i (z)+2S4u,m-i (z). Hence itfollows that S^.,m (z)=W2m {V+1)'/.(Z)-\Z%„_, {v)J,+, (z)+Z'S^^,^ ,n-l(z\ m andsoa,,rn (z)=z^^S{i^^-^/;, (v+l)./.{z) 71= -l^^-^'7;«-i ('')^.+1 (Z)]+Z^^+^-a., -^(Z). Therefore, since wehave001f^ S^u.-l (z)=2J„+2n+i (^)=o/•^•'(0dt, M=0 ^Jo andsimilarly,from theexpressionforMy^m(z), V/or,.Tav+n +m+^)J ,. (4) _1^(.+2«+2)r(^^^„.,„^^^^.«« (.) *RecentapplicationsofNeumann series tothesolution ofequations ofmixed differences are duetoBateman, Proc. Int.Congress ofMath. i.(Cambridge, 1912), pp.291—294. 16-32] NEUMANN SERIES 531 Thepotentialitiesoftheother recurrence formula were alsoinvestigated byLommel, buttheresults arenotsointeresting. Asexamplesof hisexpansionsthereader maynotice that 22.i{n+ra-iy.j^^rn) (,)^(2..- 1)!^.(4n=m{n-m)\v+tn^^ ^ Jv\n These results were given byLommel, though hisformulae contain numerical errors. 16'32. TlteNeumann seriessummedhyKapteyn. Thesum oftheseries 00 SnJn{z)Jn{a) isexpressibleastheintegral 1^r'M^^l^ J(a-v)dv;Joz-v thesums ofthealternate terms oftheseries havebeenexpressed byKapteyn* intheform ofintegralsfrom which thisintegral maybededuced, and conversely Kapteyn'sformulae maybededuced from thisintegral. Weproceedtoestablish this result byasimplifiedform ofKapteyn's methods. The seriesmaybewritten inthelormf n=0 {STTl) J J „=o xex])Hzit-jj+^oi(u—-]}dadt where thecontours maybetaken tobethecircles\u\=l,\t\=A>1. Now, let ^=2^/'"'"' ^.7^1"^P{*« {''-I)}^^- Then, iijfi= h(t— 1/t),wehave —+ml=J-.ll(^+]^exp||a (^u- Jjjdu=^{/„(«)-Ji{oi)lt].da. 'liri *Nieuw Archief voorWiskunde, (2)vii.(1907), pp.20—25;Proc. Section ofSci.,K.Akacl. vanWet. teAmsterdam, vii.(1905), pp.494—500;Kapteyn hassubsequently summed other series, ibid. XIV.(1912), pp.962—969. tTheinterchange ofsummation andintegrationispermissiblesolong as |<« |>1,where/,a areany iDoints outhecontours. 34—2 532 THEORY OFBESSEL FUNCTIONS [CHAP. XVI Therefore, onintegration, I=Ce-'«"+^I"e-"^<»-^' IJo{v)-JI(v)/t} dv,2.0 whereCisindependentofa.Bytakinga=0,weseethatC=lft. Hence wehave \zt^^+H'^l %yA^)Jn{a)=^^.j—, exp\l{z-a)\t--^)\ +2IexpU(^-a+v)(^--U[tJ,{v)-J"i{v)\dv = ^r[J,(2-a+v)+J,(2-a+v)]Jo (v)dv,dt themajorit}^oftheterms havingazeroresidue at^=0. Consequently SnJn(z)Jn(aj= .sI——-—— -'o(v)dv, «=i ^ 2—ai-V that istosay (1) S7iJ„ (^)/„(a)=^f"JA'^Aj^(„_^)dv. Ifweselect theoddandevenpartsofthefunctions ofzoneach sideof thisequation, wefindthat (2) i(2n+l)J,n+,('^)J2n+Aa) ^}q[Z-V Z\-V] which isoneofKapteyn'sformulae;and zr» (Ji{z-v) J,{z+v)] ^^ ^, 4Jo Iz-v z+v1'^ ^ zf"(Jo{z+v) jAz-v)\ J.,.,=T\-^ '^^^'\J,{a-v)dv,4.J0 (z-\-v z-v] whenweintegrate byparts. Hence itfollows that (3)i2n/,„(^)/,,(a)=^\\ltni^^L±J^+'Ili'JZl)\j^^t-v)dv,n=\ 4JoJo(-S' +'V Z—V)^ which istheother ofKapteyn'sresults. 16-4] NEUMANN SERIES 533 Thereader should havenodifficultyinproving bysimilar methods that, whenR(v)>0, (4)l(v+n)J^^n {z)-/.+« («)=^"^-^ I"'^^—^J^(«-^0dv «= ^.'0z—V 2Jo z—vdv. 16*4. TheWebh-Kapteyn theory ofNeumann series. Neumann series have been studied from thestandpointofthetheoryof functions ofrealvariablesbyH.A.Webb*. Histheoryhasbeendeveloped byKapteynfandsubsequently byBateman +.Thetheoryisnot soim- portantasitappearstobeatfirstsight, because, asthereader willpresently realise,ithas todealwith functions which must notonlybehave ina prescribedmanner asthevariable tends to±x ,butmust alsosatisfyan intricateintegral equation.Infact,thefunctions which areamenable tothe theoryseem tobeincluded inthefunctions towhich thecomplex theoryis applicable,andsimplefunctions have been constructed towhich thereal variabletheoryisinapplicable. The result onwhich thetheoryisbased isthat(§13'42) dt (0 {m^n), I'Jzm+i yt)^211+1 {t),]-ii/A^, Io\ / \t(l/(4w+2) (m=n), sothat, ifanoddfunctionf{x)admits ofanexpansionofthetype 00 JyX)=—,<^2tH-l'J271+1\^)} w= and ifterm-by-term integrationispermissible, wehave CLtC^2)i4-lr^" df \j\t)J.n+i{t)^= 4/1-h2 Wearetherefore ledtoconsider thepossibilitythat (1) f{x)=S(4n-f-2)J,,^, {X)I"-^V(0dt;H=0 .' andweshall establish thetruth ofthisexpansionunder thefollowing conditions : (I)Theintegral rf(t)dt Jo exists and isabsolutely convergent. *Messenger, xxxiii.(1904), p.55. tMessenger, xxxv.(1906), pp.122—125. XMessenger, xxxvi.(1907), pp.31—37. 534 THEORY OFBESSEL FUNCTIONS [CHAP.XVI (II)Thefunction f{t)hasacontinuousdifferential coefficient forall 'positivevaluesofthevariable which donotexceed x. (III) Thefunction f(t)satisfiestheequation (2) 2/'it)=r'^{f{v+t)+f{v- 1)]dv Jo V luhen tdoesnotexceed x. Wenowproceedtosumtheseries oc (•« 7" (f\8=1.(4/1+2)J,n+^ (^)I^^^^^ fit) dt, 71= JOf andwefirstinterchangetheorder ofsummation andintegration.Itisevident that 00 ^JfM+l \^) ['Jin \i)+"2rH-2(01n=0 converges uniformlywithrespecttotforpositive (unbounded)vahies of t, since |J^n(01^1^^^^ !^2(1+1 (^)!isconvergent. Hence, sincef{t) possesses anabsolutely convergent integral, wemayeffect theinterchange,andthen, by§16-32, S=r fit){2(4n+2)J,,^, ix)J^,it)]^Jo U=o Jf =2.^^'\Li"^^ "7+7^f^"^"-'^"^ = IfVo(^- t^)IJ'^{fit+V)+fit- V)}dtdv +fVoix-v)rfiv-1)'^dtdv. Jo Jot Wenowtransform thelastintegral byusing §12"2,andthenwehave* n''j,ix-v)fiv-t)'^dtdvJo.t =rIVoiu-t)fix-u)'^^dtdu JoJ t = \t/iiu)fix—u)du Jo =fix)— jQiu)f'(x—u)du.Jo *The firsttransformation iseffected bywriting v=x+t-u. 16-4] NEUMANN SERIES 535 Hence (3)i(4/. -F2)X,+, {x)\'' "^^/lOdt = f{x)-\y,{!n-v)W'{v)-\f^^'^[f{t+v)+f{t-v)\dt Now write sothat^(w)isacontinuous function ofv,sinceJi{t)/thasanabsolutely- convergent integral. IfthenwearetohaveS=f{x) when xhasanyvalue insuchaninterval as(0,X),wemust have rj,{x-v)F{v)dv=0, Jo throughoutthisinterval;and, differentiatingwithrespecttox, F{x)=\ J^{x-v)F{v)dv.Jo Since jJj{x—v)\^ l/\/2, itfollows byinduction from thisequation,since \F(x)\^^j')F(v)\dv, A.x"" that\F{x)\'i^j^^, whereAistheupperbound of jF{x)\intheinterval andnisanypositive integer. Ifwemaken^- -^,itisclear thatF{x)=0,andsothenecessityofequation (2)isestablished. Thesufficiencyofequation (2)forthetruth oftheexpansion*isevident from(3). IthasbeenpointedoutbyKapteynthat thefunction sin(i*?cosec a)is one forwhichequation (2)isnotsatisfied; andBateman hasconsequently endeavoured todeterminegeneralcriteria forfunctions whichsatisfy equation (2) ;but IjOsimplecriteria have, asyet,been discovered. [Note.If/(a-)isnotanoddfunction, weexpandthetwooddfunctions i{/(-^-)-/'(--0}, 4-*-{/(.^)+/(-.^)} separately;andthen itiseasytoprove, byrearrangingthesecond expansion,that f{x)=2 ClnJni-O-), 1f^where «o= gI/W"^i (I•*'I)^-^j f°°dv «»=?iI ./(•'•)'A(-^Orri > (»>0) , J—» I*^ I providedthattheappropriate integral equations aresatisfied.] *Thesufficiency (butnotthenecessity) oftheequation wasproved byKapteyn. 536 THEORY OFBESSEL FUNCTIONS [CHAP. XVI 16'41. Cailler'stheory ofreducedfunctions. TheWebb-Kapteyn theoryofNeumann series which hasjustbeen ex- poundedhasseveralpointsofcontact with atheorydue toCailler*. This theoryisbased onBorel'sintegral connectingapairoffunctions. Thus, if CO «= then thefunction/(2^)^defined bytheseries 00 f{z)R= 2Cn.n\z'\ »= supposed convergentforsufficientlysmall values of\z\,mayberepresented bytheintegral f{z)j,=re-^f(tz)dt. J Thefunction y(2")^maybetermed thereducedfunction {lareduite) o{f(z). IftheNeumann series whichrepresents f{z)is 30 f(z)= 2anJn{z),n=0 thenwehave, formally. 00 rx f{z)R=%an\e-f.In{tz)dtn=0 JQ Nowput andweseethatv(n-^^)„ro" Z 1+^-./2^=K Hence, iftheNeumann series forf{z)is2anJn (z),then thegeneratingM= function of2a,i^"is w=0 providedthat thisfunction isanalyticnear theorigin. Moregenerally, iff{z)hasabranch-pointneartheoriginofsuch anature that 00 (1) /(^)= 2anJ^+n(2), then ="1+c- /zr\ Mem. delaSoc.dePhys.deGemve, xxxiv. (1902—1905), pp.295—368. <^ 16-41, 16-5] LOMMEL's functions 537 Inlikemanner, if (3) f{z)= ia„^''+»J,+«(^), )j= then ^^.to2''+- r(i.+rn+1)"""^v(i-n^^v(i-n^• [Note.If e"^sin&3= 2a„y„ (0),n=l ^ 26^(1+^2)tbeu 2«„i«=-——^ ,T>^,,,,• Thisresult, wbich isimmediately deducible from Cailler'stheory, wassetasaproblem intheMathematicalTripos, 1896.] 16'5.LommeV sfunctions oftwovariables. Two functions, which areofconsiderableimportanceinthetheoryof Diffraction andwhich aredefinedbysimpleseries ofNeumann'stype,have been discussed exhaustively byLommel* inhisgreat memoirs onDiffraction ataCircular ApertureandDiffraction ataStraight Edge. Thefunctions ofintegralorder n,denoted bythesymbols Un{w,z) and Vn(w,z),aredefined bytheequations (1) UA^U,Z)= X(-)'M- Jn^2m(z),m=Q \ZJ (2) Vn(^^,^)=S(-)--/_„-,,« {Z). Itiseasytoseefrom§2*22(3)that (3) Un{w,z)-V,n^,(w,z)= 2(")'"(^ Jn+U^) »i=-00 \Z/ Iw z- nir= ^^H2+27.-T /AS'rr / XTr / n fW Z'mr\ (4) Un+^ (w,z)-F_„+, (w,z)=»^n (^-+2^- "2-J. The lastequation maybederived from thepreceding equation byreplacing nbyn+1. There isnodifficultyinextending (1)todefine functions ofnon-integral orderj_forunrestricted values ofvwewrite• (5) UAw.z)= s(-r-J".+...(4m=0 \Z/ *Abh. dermath. phys. Classe der k.b.Akad. derWiss.(Miinchen),xv.(1886), pp.229—328, 529—664.The firstmemoir deals with functions ofintegral order;andthedefinition ofV,^(w,z) initdiffers from thatadopted subsequently bythefactor(-1)".Much ofLommel's work isrepro- duced byJ.Walker, TheAnalytical Theory ofLight (Cambridge, 1904). Theoccurrence ofLommel's functions inadifferentphysical problem hasbeen noticed byPockliugton, Nature, lxxi.(1905), pp.607—608. 538 THEORY OFBESSEL FUNCTIONS [CHAP. XVI Theexpressionontherightisanintegralfunction ofz,and(whenthe factorWisremoved) anintegralfunction ofw. Thecorresponding generalisationof(2)givesaseries whichconverges onlywhen visaninteger. Andconsequentlyitisconvenient todefine V„{w,z)forunrestricted values ofvbymeans ofthenaturalgeneralisationof (3),namely (6) F,(w,z)=cos (^^+^+—^+f/_,+, {lu, z). Itisevident that (7)U^{w,z)+U,^,{w,z)={^)^J,{z), (8) F,{w,z)+n+,3{w,z)=(^y/_. (z). Asspecial formulae, wededuce from§2*22that (9) U,(z,z)=V,{z,z)=l[J^{z)+cosz], (10) U,{z,z)=-V,{z,z)=l^xnz; andhence, by(7)and(8), (11) U^^{Z,Z)= F„(^,^)=l(-)4cOS^- S'{-T€^„,J^rn{z)\, \ j»=J (12) U,n+, {z,z)=-F^+i {z,z)=l{-T jsinz-^'t(-)'« e^„,^,/^m+i (^)l ; providedthat ?i^1in(11),and /<^in(12). Itisalsotobeobserved that, asageneralisationofthese formulae, (13) Vn{w,z)={-YUn{z'jiu,z). Thefunctions 2md .J„j(s), 2sin(m+^)6.J^j^I (^)) OT=ocos TO=0 which arecloselyassociated withLommel's functions, havebeen studied byKapteyn, Proc. SectionofSci.,K.Akad. van Wet. teAmsterdam,vii.(1905), pp.375—376,andby Hargreaves,Phil.Mag. (6)xxxvi.(1918), pp.191—199,respectively. 16*51. Thedifferential equations forLommeV sfunctions oftiuovanables. Itisevident bydifferentiating §165(1)that (1)^UA'w,z)=--U,+,{w,z), andhence 9" z- 1 andconsequently 16-51, 16-52] LOMMEL's functions 539 Itisnowevident thatU^(w,z)isaparticular integraloftheequation ^^^ dz-^zdz^w''-\z)^''^^'*- Since thecomplementaryfunction ofthisequationis Z' z- jdcosr—+i?sin^r— , whereAandBareindependentofz,itisclear from§16'5(6)thatV_^+2(w, z) isalsoaparticular integral. Therefore F^(w, 2-)isaparticular integralof (3) ^,.\^^^=(^J^,^^(,y dz- zdz IV- \w/ Theseequations areduetoLommel, Milnchener Abh. xv.(1886), pp.561—563. 16'52. Recurrence formulae forLommeV sfunctions oftwovariables. Wehavejustobtained onerecurrence formula forU^{iv,z), namely 9 z (1) ^U^{w,z)=f^^+i(w,z).dz w Toobtain other formulae, weobserve that ~L\(w,z)=^{-y-(v +2m){w/zY^^---^ J,+,,n {z)lz andso=^i:(-)- (w/zy^^--^-^ {J.+^n-, (^)+•/.+2m+i (z)],^m=0 (2)'2^CT,(w,z)=U,_,(w,z)+{zli6f U,+,(w,z). Again, bydifferentiating §16"5(6) wededuce that (3)• ^J.{w,z)=-^V^_,{w,z), (4)21^F,{tu,z)=F,+i{w,z)+{zlwf V,_,{w,z). Ifnowwetakew=cz,where cisconstant, wededuce that {b)r2^t^,{cz,z)=cU^^, {cz,z)-(1/c) U,+, {cz,z), (6)21^F,{cz,z)=cV,^, {cz,z)-(1/c) F,_, {cz, z). Hence weget 4^3U,{cz,z)=c-f^,_2 {cz,z)-2U, {cz,z)+(l/c^) U,+, {cz,z) =c"J.-^ {z)+C-^/, {z)-(c+IjcfU,{cz, z). 540 THEORY OFBESSEL FUNCTIONS [CHAP. XVI Hence itfollows that Z7„{cz, z),andsimilarly V_v+2 {cz, z),areparticular integralsoftheequation a) ^^+ (^"^ c)"^=''''^"-^ ^^^"^'^''"'^'' ^^^• Theparticularcase inwhich ^r= isofsome interest;wehave (8) U,{w,0)= 2 =or(y+-2m+l)' andsoU^(tu,0)andF_^+2 ('^.0)areexpressibleinterms ofLommel's functions ofonevariablebytheequations (9) (10)U.{w,0)J^''^'^^^^^^^ F_^+2 (iv,0)=r(v-i)' Ofthese results, (1)—(8)weregiveninLommel's memoir. Thefollowing formulae, validwhen nisapositive integer (zero included). should benoticed : (11) V',n{w,0)=(-r cos|w—S'»-i(-)"'(Jw)^' ^=o"~(2»"«)!^. (12) U^+,(w,0)=(-r 2^=0(2m+l)! . (13) ?7_„(w, 0)=cos(|w +^W7r). Hence itfollows that (14) Fo(^^,0)=l,Vn+^(w,0)=0, (15) (16)F_.(.,o)=(-)nj;-M^, F_^_, (i^,0)={-rti-y^ilwYsm+1 ^0(2m+1)! 16*53.Integral representations ofLommel'sfunctions. Theformulae (1) (2)w"'^ U^(w,z)=-^^ J^,_i(^).cos{|w;(1-^0}•^''dt,^. C^.+i(w,^)=^;zi ^.-1 {zt).sin{iw(1- f")].^''(;^,3 /n which arevalidwhenR{v)>0,maybeverified immediately byexpanding theintegrandsinpowersofwandthenusingtheresult of§12"11(1)in 16-53] lommel's functions 541 performing term-by-term integrations.Forother values ofv,theymaybe replaced bytheequations (3) U.{w,z)=- 2,-^.-isi^2.7r J,'^"'^^~'^^'''''^^^"^^^~ ^'^^"^~^^^^^' (4) ^.4.,{w,z)=- ^.^^_^ ^.^^^^ I^J,_,(-^0-sin{li^(l-^01-(-0''^«> inwhich thephaseof—^increases from—tttottas^describes thecontour. Itisclear that,whenR{v)>0, w" [^ (5) U,(w,z)±iU,+,(w,2)=-^1 ^.-1 (2t)exp[±^{w{l- 1^)}.t"dt.Z'ft.' Bymodifyingthisformula wecanobtainintegral representationsofF^(w, z) valid forpositivevalues ofwand z.Letusconsider W ./,_i {zt)exp{±liw(1- r-)].t"dt. Theintegral convergesatthelower limitwhenR{v)>0 andattheupper limitwhen R{v)<'^,\iwandzarerestricted tobepositive. Toevaluate thelastintegral, swingthecontour round until itcoincides with therayargt= ^-^ir,thisambiguityinsignbeing determinedbythe ambiguityinsignintheintegral;such amodification inthecontour is permissible byJordan's lemma. When weexpandthenewintegralinascending powersofz,asin§13"3, wefindthat W ["^ zJo./,_i (zt)exp{±^iw{l- t-)}.t"dt 2"1,„=ow!l {v+m)Jo1V2 / "~ 2",^0 ^lVw)' that istosay (6)^ [V._, {zt)exp {±liiv(1- t-)].t''dt=exp(±f±^^+ "f-')• When wecombine theresults contained inthisformula, weseethat, ifiv>0, z>0, and0<R(v)<^,then W W Z- I'TT (7)^J^/._:(^0cos[lw(l- f-)].t^dt=cos (^-+ 2^y 2 (8)^^[^./,_,(.0sin{!..(1- t?)].t^dt= sin (^^+ -"^-^ 542 THEORY OFBESSEL FUNCTIONS [CHAP. XVI Itfollows atoncefrom (1)and(2)combined with§16"5(6)that (9) V,_,(w,z)=-'^\ /,_! (zt)cos{|w(1-f)}.t"dt,^.'1 (10) F,_, (iv,z)=- ^.^ /._, (zt)sin{1w(1- t^}.t"dt. ^.'1 Sinceconvergenceattheoriginisnowunnecessary,thetheoryofanalytic continuation enables ustoremove therestriction It(v)>0. Changingthenotation, weseethat (11) n(w,z)=--- IJ,_. {zt)COS[Iw(1- ^^)}— t"--' z"-^ (12) n_,{w,z)=- -l^^_/,_. {zt)sin{1..(1- t^)]-^^, providedthat m;and 2^arepositiveandR(v)> |. Thefollowing specialformulae areworth mention : (13) Ujn^^^ j' J^^_^ (^^)COS{|0(1- t^)}.f^-dt n= I«/^i-2(^0sin{i^(1- ^^)|.t'""-' dt, .'6 (14)Um+iiz^ =IV^,(^0cos{i0(1- <2)}.r-^'^^^ cZ^ = \J,n-i (zt)sin{^z(1- «2)}.t^"dt. . Again,from(6),weseethat r=^^ /X /^iwf-\ ,,,^"-1 /iz-_v'iri\ (15)J^./.-,(.0exp(+^j.«-*=—exp(±j^+^), and, inparticular, r=^ .,,,cos/w«2x 1sin /^^\ ^ ' ./ sinV2/wcos\2w/ The lastresults should becomparedwith§13'3; seealsoHardy,Trans. Camb. Phil. Soc.XXI. (1912), pp.10,11. Theformulae ofthissection (withtheexceptionofthecontourintegrals)arealltobe found inoneorother ofLommel's twomemoirs. 16'54.Lommel'sreciprocation formulae. Itisevident from§16'5(13)that functions ofthetype U^(z^/w, z)are closelyconnected with functions ofthetype U^{w,z)providedthat visan integer. 16-54, 16-55] LOMMEL SFUNCTIONS 543 Toappreciatethesignificanceofsuch relations observe that d dtcos(Iwt-).U^[— ,2t]+sin(Iwt-).U^+^(^,zt =sin(|w^").w —wtU^{—, zt\—wtU^^2[-,zt;z' iU =-zJ,{zt)sini^wt').(wt/zy-'. Onintegration wefindthat (1) to J(I./,(zt)sin(Iwr-)t'-"dt =—cos-kw .U^"^ 2 and, similarly,wsmi».f/.«(|,.)+ f,',g.O w"(2)-^ J,{zt)cos{lwt')t'-''dt =sin|w.UJ—,zj—cos|w. f/'^+i(—,zj+U^^^f— , Hence itfollows that (3) and (4)ivu ,-1 -^^ J,_, {zt)cos{fw(1- t')].t"dt z" J =-Uo^^f— ,^j+sin|w.C7'i_^f- , j+cos|w.?72_^(—, ], wVn-/!_, {zt)sin{|-w(1- t')\.fdt =f/i_^ (— ,-2 )-COSiw. f/'i_p(— , )+sin Itv .Uo^^ I— , |, \'W J'\w J" ' \iu J andtheseintegralsdiffer from thecorresponding integralsofthepreceding sectiononlyinthesignoftheorder oftheBessel function. Thereader will findsome additional formulae concerning Lommel's functions ina paper bySchafheitlin, BerlinerSitzungsberichte,viii.(1909), pp.62—67. 16*55. Pseudo-additionformtdae forfunctions oforders^and §. Someverycurious formulae havebeen obtained byLomrael, which connect functions ofthetype ?7^{iv,z)with functions ofthesametypeinwhich the second variable iszero,providedthat visequalto^or|. When wewrite v=^in§16"5o(5),weget f/'i{w,z)iill?,(w,z)=(^^YIexp{+i{Iw-zt-^wt-)]dt V-^TT/ Jo +(^- jIexp{+i{Iw-\-zt-Iwf^)]dt. 544 THEORY OFBESSEL FUNCTIONS[CHAP. XVI Now write _(w4-zy ^_(«^-zf andwefindthat U^(w,z)±it^3{w,z)=gT'zj^^y [ exp {±o-i(1-p)}d^ +e±^'^f-Vf exp{±hi(1-p)}rff, intherespective integrals, and\Ja,\/Saretoheinterpreted bytheconventions* w+z ,^w—z V/°"=' //o \'V^= V(2?/;)'^ V(2iy)' Hence wehave U{w,z)±iU^_{w,z)=igT^^ (u^(2a,0)+iU^ (2(7, 0)} +^e^'^{U,{28,0)±iU^{26A))} -e'^^'l-] Iexp{+0-1(1-^OWf +e±^'M- 1 exp I±8i(1-f-)}d^ \7r/Jo When wetakeo-|'andS^-asnewvariables inthelasttwointegrals respectively, theseintegralsareseen tocancel;andsowehave thetwo results combined intheformula (1) U,(w,z)±iU^ {lu,z)=le^i' [U,(2(7,0)+iU, (2(7, 0)} +^e^^-{Ui{28,0)±iU,{2h,0)], and, asacorollary, (2) U,{z,z)±iJJ^_ (z,z)=le^fe {U,(4>z,0)±iU,(4^, 0)}. These formulae are(iuetoLommel, Miinchener Ahh. xv.(1886), pp.601—605;theyare reproduced byWalker, TheAnalytical Theory ofLight (Cambridge, 1904), pp.401—402. 16"56. Fresnel'sintegrals. Itiseasytoseefrom§16"53 (1)and(2)that,whenR{v)>0, L\^,(w,0)= 2^;=^^ j^t^"-'sin{^w(1- t^)\dt, w f^ ^1)o.-i-p/ Xi'"''COS(Iwt-)dt=U^{w, 0)coslw+U,+i(w,0)sin|w,^ I{^)Jo w" /"' (2) ^i^;z:^r7-^*""'^in(lwi^)dt=U,{w,Q)sin|w-f7,+i(w,0)cos^w. *These arenotthesame astheconventions usedbyLommel,sothat 16-56] lommel's functions 545 Ifwetake v=^andmodifythenotation bywriting ^w=z=^iru'-,we seethat (3)JJcos(^TTf-)dt= ^j~^(;^J^costdt=^j'/_i (t)dt =[U^(2z,0)cos2+U^(2z,0)sin2]/V2 =2+[^^i(2^,0)sin^+F|{'2z,0)cos2]/V2, and (4)JJsin(ITTt')dt= ^I"(^J'sintdt= ^j''J^(t)dt = [f^j(2^r,0)sinz-U^ (2z,0)cosz]/^/2 =i- [T^j(2^,0)cosz-V.^ (2z,0)sinz]{^/2. Wethus obtainascendingseries andasymptotic expansionsforFresnel's integrals* ru ru Icos(^7rt')dt,Isin{^7rt-)dt. Jo J{) Theascending series, originally given byKnockenhauer, Ann. derPhysik undChemie, (2)XLi.(1837), p.104,arereadilyderived from thef-series, namely \7rJ {1.3 .5• 1 .3 .5 .7 .9 while theasymptotic expansions, duetoCauchy, Comptes Rendus, xv.(1842), pp. 5.')4,573, arederived withequal easefrom theT-series, namely Tables ofFresnel'sintegrals were constructed byGilbert, Mem. coihronnees de I'Acad. R.desSci.deBruxelles, xxxi.(1863), pp.1—52,andLindstedt, Ann. derPhysik undChemie, (3)XVII. (18S2), p.720;andbyLomiuel inhissecond memoir. Lommel hasgivenvariousrepresentationsofFresnel'sintegrals byseries which arespecialcases oftheformulaef (9)^ \~JAt)dt=2i,L+,n+,{z) J n= 00 \ZJv+iiyZ) n=o(v+l){i'+-^}...{v+ 2n+1)' (10)fJAtXU^ I(l+l>iii^gI„-(-'^^"- l)^,,.(.).J2 «= Z *Mem. deVAcud. desSci. v,(1818), p.3.S9. [Oeuvres,i.(186G), p.176.] tItissupposed in(9)that 7?(c)>-1. w.B.F.3,-, 546 THEORY OFBESSEL FUNCTIONS [chap. XVI These arereadilyverified bydifferentiation. Other formulae alsodue to Lommel are (11)rJ^{t)dt=2icosl.Jo (12)rJi{t)dt=2^sin^zJos(-rJn+idz)n= +25sin iz ti-yjn^iilz) M= —2^cos^z Thesemayalsobeverified bydifferentiation.2(-)"^.+i(i^) M= ti-y-Jn+iiiz) LM= 16"57.Hardy's integrals forLommel' sfunctions. The factthattheintegrals [^ f^h\dtr f.h\tdt Jo'"n"'-^'r±^' io''""r-viit^ areexpressibleinterms ofelementaryfunctions* when aand harepositive suggestedtoHardy ftheconsideration oftheintegrals r=°f^h\dtr f.^\id* Jo'''r^ijiTt- Jo^^"r-'^"Jr±i- andhefound them tobeexpressibleinterms ofLommel's functions oftwo variables oforders zeroandunity respectively.Thisdiscoveryisimportant because themajorityoftheintegrals representingsuch functions contain Bessel functions under theintegral sign. Ifl/tbewritten inplaceoft,itisseen that (1,IJCOS [at+ ^)^,=±I'cos[U+^)jf^,. ,„, i''.I^h\tdt [' f,b\dtC. (2) 1^sm(a«+^)j^=±j^sm («(+-)-- (^sm/, ,a\tdt and since, by§6*13(8), -i^sin(at+j)^=Jo{2^{ab)},TT. V I/I itissufficient toconfine ourattention tothecase inwhich b<a. Wenowwrite c=^/ib/a), x=2^/(ab),(9=|(1- c')/c, *Hardy, Quarterly Journal, xxxii. (1901), p.374.When thelower signistaken itissupposed that theintegrals have theirprincipal values. +Messenger, ixxvm.(1909), pp.129—132. 16-57] lommel's functions 547 andthen thesubstitutions t=ce"andcoshu=rshew that b\dt f°^cos(wcoshu)du '^'''r^'^^yi+^^-.Loo ce-+l/{ce-) cos(^cosh?t)^^ ^^-r,r+ --.//-„Sdu a+bf^cos(a't).rdr Now consider 1r e'^^rdr 27^iJr(0''+T')^/(T'-l)' where Fisacontourconsistingoftherealaxisandalargesemicircle above it,therealaxisbeingindented att=+1. Theonly poleoftheintegrandinside thecontour isatid,andso 1f e''-*'"TcZT e"^^ 27rijr{&"+t')V(t^-1)'2i^{¥+T)' Astheradius ofthelargesemicircle tends toinfinity,theintegralround ittends tozerob}'Jordan's lemma, andhence pcos(xt).rdr 1psin(scr) .rdr _7re~*^* ii{0'+T-)V(t^-1)''2j_i(>+T-^)V(l-T^)~ 2\/(^'M^l)* Thuswehave 6\ c^i_vre"'**"^' a+6/"""sin(ir cos<^).cos0c?</) I ^^cos[at+^j^_^^„ 2 2*- i ^-+cos'^cf) COS c^ 4c -^"t^fl^-;—^r-7= ^i r,(ccos6—c^cos3(6+c'cos 5<f>- ...],P-+cos- (/)1+c^^ ^ andsowefindthat? (3) j^cos[at+])y~^,= -^-rr ^^_^c—J„„_, (^). Similarlyitisfound that* (4)\y'''[''^-^\)l%-''^--'^}j''^ and f«= / h\dt* (5)P cos a«+~—-=l-rrsin(a+6)-ttS(-)'"-' c^—^X,^., (a-), .' V tjl-t,„=! (6)P sin at+- =-^ttcos(ft+6)-tt:S(-)'"-' c^J^n (^).Jo \ tJI—I-^^i *The details oftheanalysis willbefouud inHardy's paper. 35—2 548 THEORY OFBESSEL FUNCTIONS [CHAP. XVI The lasttworesults maybewritten intheform (8) U,{w,x) +V,{w,x)=-~Pj^^'"^V2^+2"Jrr72' providedthat<w<x. 16*58. Integrals ofGilbert'stypeforLommeVsfunctions. Anobvious method ofrepresenting L\(w, z)and F„(?^', z)byintegralsis tosubstitute theBessel-Schlafliintegralof§6*2foreach Bessel function in theappropriateseries.Wethusget ^-(»-^)= 2^-„!.<-)"" «»)'""'/!!'-p('- S)A.• When thecontour issochosen that itlieswhollyoutside thecircle on which I^ I=^Ity I,wemaychangetheorder ofsummation andintegration andget (1) U.{w,z)=—.^_^ ^-^^^_^exp(^--)-. Now theresidues oftheintegrandat+\iware , [iw iz-_vTri] andso Makingaslight changeinthenotation, wededuce that (2)^^(^'^)=2^-j_ r+-F/«7^^"P(2-2^)T' and, inthisintegral,thepoints+iwlieoutside thecontour. Ingeneralitisimpossibletomodifythecontour in(2)intothenegative halfoftherealaxistaken twice, inconsequenceoftheessentialsingularityof theintegrandattheorigin. Theexceptionoccurs when z=0,because then theessentialsingularity disappears,and 1f^^+Ht/wye^*dt (3) '^'(».0)= 2^-/.27rij_„ 1+t-lw-t' andhence (4) V,(w,0)=, .,-du, TT J I-i-U providedthatR{v)>0 and a.isanacuteangle (positiveornegative)such that )«+argw|<Jtt. 16-58, 16-59] LOMMEL's FUNCTIONS 549 IfVisequalto^orf,theintegral ontherightin(4)iscalled Gilbert'sintegral*. Formula(4)wasobtained byLonimelt from theformula of§16"53(11)byatransforma- tionofinfiniteintegrals. From(4)itisclear that,when vandwarepositive, V^{w, 0)hasthesamesign as,and isnumericallylessthan SniVTT I" „_l_ 1,,„,, 1 TTjo r{i-u).{hwy Asimilar butlessexact inequality wasobtained byLommel. Thereader willalsoobserve thatVv{i(', 0)/sin vnisapositive decreasing function oflo whenwispositive. 16"59. Asymptotic expansions ofLommeVsfunctions oftwovariahles. From Gilbert'sintegralsitiseasytodeduceasymptotic expansionsof F^(w;, 0)andU^(io,0)forlargevalues of\w\; thus, from§16'5(8),we have wherepisanypositive integer. Wechoose ptobesolargethatR{v+2p)> andthen,by§16'58(4),wehave {-y F,+,^ {w,0)=-—-y-— Irxexpta '\ =0(w-''-'P), when I?^ Iislarge and, asinthesimilaranalysisof§7'2, |argw |<tt. Hence (-r (1) VAw,0)ZoT{l-V-2m). (Iwy^'-^ forthevalues ofwunder consideration. Whenu+2p and tvarebothpositive, (—)^F^+2^(w, 0)hasthesamesign as,and isnumericallylessthan sinVTTf^ ^„„ ,, ; (-)P TTJo r(l-v-2p).{lwy+-'P' sothat theremainder afterpterms in(1)hasthesamesign as,and is numericallylessthan the(p+l)thterm. Itmaybeprovedinlikemanner from§16"53 (11)that (2) .F,[w,z)^X(-)- {z/ivy+"-- J"_,_.,^ (3) when\io\islargewhile vandzarefixed;but itisnoteasytoobtain asimple expression whichgivesthemagnitude andsignoftheremainder. *3Iem. couronnees deVAcad. R.desSet.deBruxellea, xxxi.(18(53), pp.1—52. tMUnchener Ahh. xv.(1886), pp.582—585. 550 THEORY OFBESSEL FUNCTIONS[OHAP. XVI Itisevident from§16'5(6)thatthecorrespondingformulae forU^{w, z) are (_)m (3) t^,(w;, 0)-'cos(iw-ii'7r)+ 2 =0r(i'-1-2m)(iw)^-''+'»' (4) Uy(w,z)~cos{\w+^z^/w-Ii/tt)+S(-)"^ {z/iuy-"'-''+' J,.o.o,„ (z). These results weregiven byLommel*, buthedidnotinvestigatethem in anydetail. Theasymptotic expansionofVy(ex,x),when i/isor1and cisfixed, while Xislargeandpositive,hasbeeninvestigated byMayallf. Thedominant term forgeneral (real) values ofvgreaterthan—|is readilyderived from|16'53 (12)which shews that Vy{ex,x)^--^^j (—^cos(xt+Iv-jr-Itt)sin{|ex(1-f^}— . Now, ifc>l, thefunctions^ca;(l— ^^)+(a?i+^vir—^'Tr)varymonotonically astincreases from 1tooc,andhence itmaybeverified bypartial integra- tions that /2\^c^""" (5) Vy{ex, a;)~ (—— -cos(a;+^r;7r-^it),virxj G—1 thenextterm intheasymptotic expansion being {x~^). If,however, e<l, then^cx(1- t"^)+(xt•\-^ptt—lir),quafunction oft, hasamaximum at1/c;andhence, bytheprincipleofstationary phase (§S'2), itfollows that (6) Vy(ex,x)-^-1^^cos\-^x(c+-]+^vir[ Finally, whenc=l, themaximum-pointisatoneend oftherangeof integration,andsotheexpressionontherightin(6)must behalved. We consequently have (7) Vy(x, x)'^^cos(x +^v'7r). Thisequation,like(5)and(6),hasbeen established onthehypothesis thatv>—^;thethreeequations maynowbeprovedforallrealvalues ofv byusingtherecurrence formula§16'5(8). *MUnchener Abh. xv.(1886), pp.540,572—573. tProc. Camb. Phil. Soc. ix.(1898), pp.259—269. CHAPTER XVII KAPTEYN SEKIES 17*1.Definition ofKcipteynseries. Anyseries ofthetype CO 11= inwhich*vandthecoefficientscinareconstants,iscalled aKapteynseries. Such series owetheirname tothefactthattheywere firstsystematically investigated, quafunctions ofthecomplexvariable z,byKapteyn finan important memoirpublishedin1898. InthismemoirKapteynexamined the questionofthepossibilityofexpandinganarbitrary analyticfunction into such aseries, andgenerallyheendeavoured toputthetheoryofsuch series intoapositionsimilar tothatwhich wasthenoccupied byNeumann series. AlthoughthepropertiesofKapteynseries are, ingeneral,ofamore recondite character thanpropertiesofNeumann series, yetKapteynseries areofmorepractical importance; theyfirstmade theirappearanceinthe solution ofKepler's problemwhich wasdiscovered byLagrange^andredis- covered halfacenturylaterbyBessel§;andrelated series areofgeneral occurrence inaclass ofproblems concerning ellipticmotion under theinverse square law,ofwhichKepler's problem maybetaken astypical.Morerecently, inthehands ofSchott|| theyhaveprovedtobeoffrequentoccurrence inthe moderntheoryofElectromagneticRadiation. Theastronomicalproblems,inwhich allthevariables concerned are real, areofamuch moresimple analyticalcharacter than theproblems investigated byKapteyn; andinorder todevelopthetheoryofKapteynseries inasimple manner, itseems advisable tobeginwith adescriptionoftheseries which occur inconnexion withellipticmotion. 17*2.Kepler's problem andallied problemsdiscussed byBessel. Thenotation which willbeused inthis section inthediscussion ofthe motion inanellipseofaparticleunder theaction ofacentre offorce atthe focus, attractingtheparticle accordingtotheinversesquare law,isasfollows : Thesemi-major axis,semi-minor axis,andtheeccentricity oftheellipse aredenotedbya,h,and e.Theaxes oftheellipsearetaken ascoordinate *Itwill, forthemostjjart, beassumed that viszero, tAnn. set.deVEcole norm.sup. (3)x.(1893), pp.91—120. XHist, deVAcad. R.desScLdeBerlin, xxv.(1769) [1770], pp.204—233. [Oeuvres,iii.(18(;9), pp.113—138.] §Berliner Abh.1816—7 [1819], pp.49—55. ilElectromagnetic Radiation (Cambridge, 1912). 552 THEORY OFBESSEL FUNCTIONS [CHAP. XVII axes, thedirection oftheaxis ofxbeingfrom thecentre oftheellipsetothe centre offorce. The centre offorce istaken asoriginofpolar coordinates, theradius vector totheparticle being r,andthetrueanomaly, namelythe anglebetween theradius vector andtheaxisofx,beingw.The eccentric anomaly, namelytheeccentricangleoftheparticleontheellipse,isdenoted byE.Thetimewhich haselapsedfromaninstant when theparticlewasat thepositiveendofthemajoraxis iscalled t. Themean anomalyMisdefined astheangle throughwhich theradius vector would turn intime tiftheradius vector rotateduniformlyinsuch a wayastoperform completerevolutions inthetime itactuallytakes toperform completerevolutions. Thegeometrical propertiesoftheellipse supplytheequations* (1) r=r—^—=a(1—ecosA), ^'1+ecosw^ ' fromwhich theequations (2) taniw=/JU^J tan^E, \/%l-6^) .smE .„Vd-e^sinw (3) smw=^ i=i—,sinA=—=^^1—ecosE 1+ecoszu arededucible;andanintegratedform oftheequationsofmotion(theana- lytical expressionofKepler'sSecond Law) suppliestheequation (4) M=E-€smE. Kepler's problemisthat ofexpressingthevarious coordinates r,w,E, which determine thepositionoftheparticle +,interms ofthetimet,that is, effectively,interms ofM.Itisofcoursesupposedthatthevariables arereal and, since themotion iselliptic (orparabolic,asalimiting case), <e^ 1. The solution oftheproblemwhich was effectedbyLagrangewas ofan approximate character, because hecalculatedonlythe firstfewterms inthe expansionsofEand r. Themorecompletesolutiongiven byBesseldependsonthefactthat(4) definesEas a,continuousincreasingfunction ofMsuch that the effect of increasing Mby27ristoincreaseEby27r. Itfollows thatanyfunction of-E*with limited total fluctuation isafunction ofMwith limited total fluctuation, andsosuch functions ofEareexpansible inFourier series, quafunctions ofM. *The construction ofthese equationswillbefound inanytext-book onAstronomy or Dynamics ofaParticle. See, e.g.Plummer, Dynamical Astronomy (Cambridge, 1918),Ch. iii. tKepler himself wasconcerned with theexpression ofEinterms ofM. 17-2] KAPTEYN SERIES 553 InparticularesinEisanoddperiodicfunction ofM,and so,forallreal values ofE,itisexpansibleintotheFourier sine-series where An=2esin^=SAnsinnM, r esin^siniiMdM TTJ 2esin£'cos«i¥' 11772f'^..dUsmE) ,,,+—cosnM-^ ,,..— '-dM nir' dm 2r,^dE-dM—cosnM TTT— mrJ„ dMdM cosnM.dE n-TT .'o -- ./,,(ne).n- Hence itfollows that (5) E=M+ i.^Jr,{ne)8mnM, andthisresultgivesthecomplete analyticalsolution ofKepler's problemcon- cerningtheeccentricanomaly. Theseries ontherightisaKapteynseries which converges rapidly whene<1,and itisstillconvergent when e=1 ;cf§§8*4,8'42. Theradius vector issimilarly expansibleasacosine series, thus -=^0+-^»icosnM, (^71=1 where Bo=~ l^{l-€cosE)dM (1-ecosEfdE TTJQ while, when n^0, 2 Bn=-\{l-€CosE)COSnMdM 2(l-ecos^)sin?iil/ mr nirJC^(6C0S^),„SinniU TTTdMdM 2e nir JsinEsin(nE—nesinE)dE sothat (6)2e r-=1+4^2-T°°2e aS—Jn{ne)cosnM. «-in 554 THEORY OFBESSEL FUNCTIONS [CHAP. XVII Theexpansionofthetrueanomalyisderived from theconsideration that tu-Misanoddperiodicfunction ofif,andso w—M=%GnsinnM, w=l where C.n,=- {w—M)s,mnMdMTTJ 2(lu-M)COSnAri'' 2f" ^j(dw ^\,,, —+—cosnil/ .hxr-1d^^ niTJonir JQ \dl\i ) 2f"" ,^dw,„ =—cosnM. -,-r,dEnirJ dE _2V(l-e-)f""cos{nE-nesinE) mr JQ1—ecosE Thisexpressionisnotsuch asimpletranscendent asthecoefficients A^ andBn.Themost effective method ofevaluatingitisduetoBessel*, who used theexpansion V(l-e-) ^^^.^^^^^2/2cos2^+'2f-^cos3E+..., 1—6cosE-^ ^ -^ where /'=. Onmakingthesubstitution, wefindatonce that 2r"^ Gn=-\Jn{ne)+2/'«[Jn-m (ne)+Jn-r-m {ne)] 17*21.Expansionsassociated with theKepler-Bessel expansions. Alargeclass ofexpressionsassociated withtheradius vector, trueanomaly andeccentricanomaly,areexpansibleinseries ofmuch thesametypeasthose justdiscussed. Such series havebeeninvestigatedinasystematicmannerby Herzf, andweshallnowstate afewofthemoreimportantofthem; theyare allobtainablebyFourier's rule,and itseemsunnecessarytowrite outin detail theanalysis,which thereader willeasilyconstruct forhimself First,wehave a(l-e-)- r rcosiv=x—ae e sothat (1)'-^-^^^ =-ae+I^ J,'(ne)cosnM,a- „=i?i andnext /-.x rsiniu h .^\/{l-e-) ^2^, . . ,-. (2) =-sm^=— S-Jn(we)smnM,a a 6 n=in *Berliner Abh.1824 [1826], p.42. tAstr.Nach. cvii.(1884), col.17—28.Various expansions hadalsobeengiven byPlana, Mem. deltaR.Accad. delleSci.diTorbio,(2)x.(1849), pp.249—332. Inconnexion with theirconvergence, seeCauchy, Coviptes Rendus,xviii.(1844), pp.625—643. \_Oeuvres, (1)vni.(1893), pp.168—188.] 17-21, 17-22].KAPTEYN SERIES 555 while (3) cosE=-—=-1€+S-Jnine)cosnM. Next, ifIIIisanypositive integer*, "^^1 (4) cosmE=m2- {'/„_,„ (ne)-J„^,„ (ne)] cosiiM, n=\n '^1 (o) sinmE=7>iS- •{-/„_„, (we)+J„+,„ (?ie)|sinnM. Theexpansionofa/risparticularly simple, namely, (6)-=1+22./„(ne)cosnif . ? »=i Theexpansionsofcoswandsinware 1_^200 (7) cos2<;=-eH 22J^ine)cosnM, (8) sin?^=V(l-e-') 22Jn'(n€)smnlM. n--=\ Theexpansionsofcosw/?--,sini^/r^areofasimple form,namely a-"^ (9) -cosw= 22n/,/ (we)cosJiil/, ^'' «=i (10) ^sinw=^^^22n./„(?ie)sin«ilf. [Note.Itispointed outbyPlummer, Dynamical Astronomy (Cambridge, 1918), p.39, thatthese arereadilyderived from theCartesianequationsofmotion intheform (P.X a^GOswd'^y a^sinw 2 ~^^^ ' ,7\f>T"Tig^^^> .' dJJ-^ r''dAJ combined with(1)and(2).] \-^17-22. Sumsofspecial Kapteynseries. Thereader willobserve that, inthecaseoftheexpansionsofeven functions ofM,theresultssimplify whenwetake theparticletobeatoneoftheends ofthemajor axis,because then thethree anomalies areallequaltoortott, while theradius vector isequaltoa(1—e)ortoa(1+e).From theresults ofthelastsection wethus obtain thefollowing formulae, which weregiven by Herz inthepaper already quoted: /;=1 " w=l n *Itisseenfrom(3)that,whenmisequal to1,theexpansion (4)has tobemodified bythe insertion ofaconstant term. These twoformulae were given byJacobi, Astr.Nach. xxvni.(1849J, col. (39.[Ges.Math. Werke, vii.(1891), p.149.] 556 THEORY OFBESSEL FUNCTIONS[CHAP.XVH Moregenerally wefindbydifferentiating §17'21(6)that (4) (5) Since2d' 1 2dM-'^\-ecosE d'"" 1 dM"^ 1-ecos^=Sn^"^Jnine),M=0 n=l =2(-)"-iw-'»j:«(?ie). M=ir n=l di -TYf= T E^te^.theexpressions onthe leftin(4)and(5)can becalculated foranypositive integralvalue ofm,with sufficient labour. Again,ifweregardeandMastheindependent variables, itiseasily seen that _^f1 I_cos£* BE de{sin^(1-ecosE)]~ sin^^(1-ecos^a^ 1 f E.•r^9^ •—rTri Etv> i-cosA+eSinA^r- sinE{1—€cosA^)2 [Be esinA" sothat,by§17-21(6)(1-ecosjE')^ A1 dlM 1-ecosA" (Zic.9f 1]°° de[smE(1-ecosE)] „=i^ andtherefore, ifweintegratewith e=asthelower limit, 1 1" • r* (6)-.—irj^=--.— ^,,=-2Snsin ?iil/ ./„(nx) ^sinA(1-6cos^)smM n=i Jo Ifwedifferentiate withrespecttoM,wefindthat _. cosA" cosilf e ^^sin^a;(1-ecosEf~ su^M"*" (1-ecosEf =2Sn^cosnM . jJn{nx)dx. w=l .'0 The lasttwoexpansionsdonotappeartohavebeenpublished previously. Expressions resembling those ontherightof(6)and(7)haveoccurred intheresearches ofSchott, ElectromagneticRadiation (Cambridge, 1912) />as«m. Thus, ascases of(4)and(5),Schott proved {jhid. p.110)that (8)J^.'.^,„(2ne)= ^iL±^,,In^ J^^.(2..-)d.=^^^^,. The lastofthesemaybeobtained bytakingMequaltoand izin(7). 17-23] KAPTEYN SERIES 557 17'23. MeisseVsexpansions ofKapteyns type. Twoextremely interesting series, namely n=in^+r 1^+r (r^+r)(:i^+r) ^^-^ „=i{M-If+1^P+r"* (1-^+f)(3^+a +^!l^ + havebeen stated byMeissel*whodeduced variousconsequencesfromthem; itistobesupposedatpresent fthat <e^1,and^isreal. Thesimplestmethod ofproceduretoadoptinestablishingthese ex- pansionsistotake theFourierseries]: «cos2'«if_7rcosh(7r-2ilf)f1 Z\n'+f~ 2f^F7r^ 2p' (whichisvalidwhen ^if^tt),replaceMhyE—€sinE,andintegratefrom toTT.Itisthusfound that ^^X^(2ne) ^1r (7rcosh(7r-2£' +26sin£')^ M^7^ ^.rl '^^+r -^ui ^sinh7^^ H _1fi- [ttcosh(2^+2eCOS^i)^_j^. TT-ST ^sinhTT^ ^"- 2fi'^ (ttcosh2^0.cosh(2e^cos0) 1 , TT.'o[ Isinh7r| f^ Now thelastexpressionisanevenintegralfunction ofe,andhence itis expansibleintheform§ z— -; ^-j—^ad m=i C-^w)! .'o smhTTf ^1r(l+rBr(l-t|) ^ Z,r{7n +i+i^)r(m +i-i^y^ m- byaformula duetoCauchy||; andthetruth ofMeissel's firstformula isnow evident. Thesecond formula follows inlikemanner from theFourier series "cos(2n-1)i¥_TTsinh(Itt-M)^ „=;(2/i-ir+p" 4|coshl7rr^• *A)>tr. Nach. cxxx.(1892),col..863—368. tTheextension tocomplex variables ismade in§17'31. JSeeLegendre, Exercices deC'ulc. Int. ii.(Paris, 1817), p.166. §Itiseasytoseethatthetermindependentofevanishes. IIMem. siir lesintegrales dejinies (Paris, 1825), p.40. Cf.ModemAiialy.iis, p.263. 558 THEORY OFBESSEL FUNCTIONS[CHAP. XVII Now, since theseries obtained from(1)and(2)bydifferentiations with respectto^^areuniformly convergent throughout anybounded domain ofrealvalues of^,wemaydiiferentiate anynumber oftimes andthenmake Wethusdeduce that arepolynomials*ine;theformer isanevenpolynomialofdegree Im,and thelatter isanoddpolynomialofdegree 2m—1. Thevalues oftheformer polynomial weregiven byMeissel inthecasesm=l,2,3,4,5; thevalues form=\,2,3are 2'28'232'*'72* Thevalues ofthelatter polynomialfor ??i.=l,2,3are 2'2~18' 2~81"^450" Meissel alsogavethevalues ofthelatter polynomialform= 4.,5. Conversely,itisevident thateveryevenpolynomialofdegree 2m is expressibleintheform oc 71= andthateveryoddpolynomial,ofdegree2m— 1,isexpressibleintheform 00 2hnJ2n-i{{2n-\)e], n=l where ««and6„areevenpolynomialsinl/nand1/(2/?—1)respectively,of degree2m. 17*3. Simple Kapteynseries withcomplexvariables. Itwasstated in§17'1 that, ingeneral, Kapteynseries areofamore recondite character thanNeumann series, andweshallnowexplainoneof thecharacteristic differences between thetwotypesofseries. InthecaseofNeumann series itis,ingeneral, possibletoexpandeach of theBessel functions intheform ofapowerseries inthevariable, andthen to rearrangetheresultingdouble series asapowerseries whose domain ofcon- vergenceisthat oftheoriginal Neumann series. *Itistobenoted thatthecoefficients ofe-'"and e-»"-i intherespective polynomials arenot zero;thev are (-y-'^.(-r-' 2 .(in !)•-2.1- .3^ ..(2m- I):'' 17-3] KAPTEYN SERIES 559 Thecorresponding propertyofKapteynseries isquitedifferent ;forthe Kapteynseries '^anJu+n{(v +n)z] isconvergentandrepresentsananalyticfunction(cf. §87)throughoutthe domain inwhich zexp \/(l—2-) l+V(l-^')<lim "^'i/ar, while thedouble series obtainedbyexpandingeach Bessel function inpowers ofzisabsolutely convergent onlythroughoutthedomain inwhich l^l.expV(l-I^P)1• andthelatter domain issmaller than theformer;thus,when thelimit is1, the firstdomain istheinterior ofthecurve shewn inFig.24of§8'7,in which thelongestdiameterjoins thepoints+1,while theshortestjoins thepoints ±ix0'6627434;while thesecond domain* isonlytheinterior of thecircle\z\=0-G627434. Hence, whenwearedealingwithKapteyn series, ifweusethemethod of expansionintodouble serieswesucceed, atbest, inprovingtheorems onlyfor aportionofthedomain oftheirvalidity; andtheprooffortheremainder of thedomain either hastotaketheform ofanappealtothetheoryofanalytic continuation orelse ithastobeeffectedbyacompletelydifferent method. Asanexampleofthemethods which have tobeemployed, weshallgive Kapteyn'sf proofofthetheorem that (1) -^^1+2iJ,,(nz), providedthat zliesintheopendomain inwhich zexp \/(l-z")<1. :'l+^(l-z"-) Thisdomain occurs sofrequentlyinthefollowing analysisthat itiscon- venient todescribe itasthedomain K;itistheinterior ofthecurve shewn inFig.24of§8-7. Formula(1) is,ofcourse, suggested byformula (2)of§17"22. Toestablish thetruth oftheexpansion, wewrite _1+2iJ„{7iz)=S(z), andthen ithastobeproved thatS{z)=1/(1- z). 1 /•(0+)Since J,,(nz)=,^—./ Ztti J'exip{^z{t-l/t)\ t T *Foraninvestigation ofthemagnitudeofthisdomain, seePuiseux, Journal deMath. xiv. (1849), pp.33—39, 242—246. +Nieiiw Archief voorWiskunde, xx.(1893), pp.123—126; Ann. sci.deVKcole norm.sup. (3) X.(1893), pp.96—102. 560 THEORY OFBESSEL FUNCTIONS [CHAP. XVII weseethat,ifwecanfindacircleTwithcentre attheoriginofsucharadius thatonittheinequality exp{1^(^-1/0} (2)"'^ ;^'<1 istrue, then (^\ ^(z\=J^ il+^~^exp 1^^(^-1/0 1dt ^^ ^'27ri./,r+)1-r'exp[^z(t-1/^}t' Toinvestigate (2),werecall theanalysisof§8'7. If2=pe'",t=e""*"**, wherep,u,a,6areallreal(pandubeing positive),then(2)issatisfied for allvalues of^if pv^sinh- u+sin-a)—«<<; andwhen uischosen sothat the lastexpressiononthe lefthas itsleast value, thisvalue is(|8'7) ,zexp .v/(l—z^) '°8|14-V-^') which isnegativewhen zliesinthedomain K.Hence, when zliesinthe domain K,wecanfindapositivevalue ofusuch thattheinequality (2)is satisfied when |i |=e". Again,ifwewrite\jtinplaceoftin(3)wefindthat ^^ ^^ 27rzJ(^+) 1-^exp{-1^(^-1/0}^' where 7isthecircle\t\=e~". When wecombine (3)and(4)wefindthat 9<^(,\-l^{*+exp{1^(^-1/0}dt ^''^'^-27ri\r+,y-)t-eK^{lz{t-l/t)}t' and so2S{z)isthesumoftheresidues oftheintegrandatitspoleswliich lie inside theannidus houndedbyVand7. Wenextprovethat there isonlyonepoleinside theannulus*, and,having proved this,wenotice that thispoleisobviously^=1. Forthenumber ofpolesisequalto 1[dlog[1-t-^exp{\z{t-I/O]]^^ 2'iri J(^r+.y-)dt ^J_[c^log[1-r^exp{i^(^- 1/01]^^ 27riJ(r+)dt ^_l_irflog[l-^ exp1-1^(^-1/0}] 27rl Jir-i-\ dt -mJ<dt {r+) ^log[l-r^expli^(f-l/0}]^^ (r+)dt +_L Ic?log[-<exp{-1^(^-1/0}]^^^ 27ri J(r+)dt Thecorresponding part ofKapteyn's investigation doesnotseem tobequite soconvincing astheinvestigation given inthetext. 17-31] KAPTEYN SERIES 561 Now the first oftheseintegralsvanishes;for,ifwewrite t"'exp{^{t-l/t)]=U, then I?7|<1onV,andsotheexpressionunder consideration maybewritten intheform J(V+) i»=o ]dt 71"^J(V+) {n=0 andtheintegralofeachterm oftheuniformly convergentseries involved is zero. Hence thenumber ofzeros of1-^^exp{^z(t- l/t)}intheannulus is equalto 1 1^/ 1 i-2^+^clt=l. Iiri Jr+ Itfollows that2S{z)isequaltotheresidue of ^+exp{1^(^-1/0} ^-exp [\z{t-ijt)] ati=1;andthisresidue iseasilycalculated tobe2/(1—z). Ithastherefore beenshewn that8{z)isequalto1/(1—z)throughoutthe domain K,i.e.throughouttheivholeoftheopendomain inwhich theseries defining S(z)isconvergent. [Note.Itispossibletoprove thatS{z)convergestothesum1/(1—2)ontheboundary ofK,exceptats=1,buttheproof requires anappealtobemade totheorems ofanAbelian type;cf.§17'8.] 17'31. TlieextensionofMeisseVsexpansionstothecaseofcomplexvariables. Weshallnowshewhow toobtain theexpansions ^^Zin''+^' p+r''^(p+n(2^+n (i'+r'^)(^'+r)(3--^+r> +..., ^^„Z,(2/1- 1)2+r^ 12+^2+(p+^2^3.+ ^.) which arevalidwhen zliesinthedomainKand^isacomplexvariable which isunrestrictedapartfrom theobvious condition that^imust notbean integerin(1)noranoddintegerin(2).These results aretheobvious extensions ofMeissel's formulae of§17-23. [Note. Theexpansions when^isapureimaginary have tobeestabhshed byalimiting process bymaking (a|)[)roachtheimaginaryaxis ;since thefunctions involved in(1)and (2)arealleven functions of^,nogeneralityislostbyassuming thatli{Qispositive.] w.B.F. 36 562 THEORY OFBESSEL FUNCTIONS [chap. XVII Inorder toestablish these formulae, itisfirstconvenient toeffect the generalisationtocomplexvariables oftheexpansionofthereciprocalofthe radius vectorgiven by§17'21(6).That istosay,wetaketheexpansion 00 1+2SJn{nz)cosil^,n=\ which wedenote bythesymbol S{z, ^),andproceedtosum itbyKapteyn's method(explainedin§17'3),onthehypothesesthat</>isarealvariable and that zliesinthedomain K.Wedefine acomplexvariablei|rbytheequation (^=y^—zQ\n'^. Thesingularitiesofi/r,quafunction of<^,aregiven bycosi/r=\Jz,that is ^=arcsecz— isj{z^—1)• None ofthese values of(/>isreal* ifzliesinthedomainK \and, as^in- creases from toGOthroughreal values, t/tdescribes anundulatingcurve which canbereconciled with thereal axis inthe-\/r-planewithoutpassing overanysingular points. Itfollows thatif,forbrevity, wewrite t^=nexp{i^(«-l/0}, then ^"^^'^P^27rtj(r+)l-2(/cos(^+ U' t' with thenotation of§17"3.Bythemethods ofthat section wehave 9cr^^-_L/l-^'' dt andso2S(z, (/>)isequaltothesumoftheresidues oftheintegrandatthoseof itsjjoleswhich lieinside theannulus hounded hyTandy. Weshallnowshew that tJiere areonlytwopolesinside theanmdus, and, having proved this,wethen notice thatthesepolesareobviouslyt=e±"^. ByCauchy's theorem, thenumber ofpolesisequalto 1r dlog(l-2t/cos(^+ ?70 •^TTt'(r+,v-)dt c^log(l-2£rcos<^+ JJ') (r+)dt 1r (^log[^^exp{-^ (^-1/011dt inJdt + liri{r+) 2dt 00 Sf7"COS(??.+!)</) n.=0dt dU dtdt+2 =9 *Itiseasytoshew thatsuch vahies of(psatisfy theequation sothat 1e±^ I<1.• 17-31] KAPTEYN SERIES 563 theintegralofeachterm oftheuniformly convergentseriesvanishing, just asin§17"3. Now theresidues of 1-U' 1 att=e***arebothequalto1/(1—zcosyjr);andtherefore wehaveprovedthat (3)^ ^=1+21Jn(nz)cosncf>, inthecircumstancespostulated;andtheseries ontherightisaperiodic function ofwhichconverges uniformlyintheunboundedrangeofreal values of(p. Hence, when11(0>^>wemaymultiply bye~^*andintegrate thus: Too 00 TOO ra> g—^<t> e-^*d(b+21Jn(nz) e~^^cosnSd6= \ ^ d6. Jo «=i Jo Ji)l-zcos^fr^ That istosay, (4) re-^(*-.sin^)^^=^+2i?^^^^, where thepathofintegrationistheundulatorycurve inthei/r-planewhich correspondstothereal axis inthe^-plane;and,byCauchy's theorem, this undulatorycurvemaybereconciled with thereal axis. Now,when thepathofintegrationisthereal axis, theintegralontheleft in(4)isanintegral function ofz;andthisfunction maybeexpandedinthe form S—V e-^'^sin'^^lrdylr. m=om\Jq Bychangingthesignofzthroughouttheworkweinfer thetwoformulae which arenowestablished onthehypothesesthatzliesinthedomainKand that^0>0.' Bydividingthepathsofintegrationintotheintervals(0,tt), (tt, 2-77),... andwriting Jtt+6,^7r+d,...for\/rintherespective intervals, weinfer thab Ie-<*sin-"^->irc/-»ir =-T-j— ;—-1cosht^.cos^^^^c?^ Jo smh^TT^Jo 36—2 564 THEORY OFBESSEL FUNCTIONS [CHAP.XVII andthat roc\ fin e-^* sin2'«-i 'Jrdylr=—p-j—-coshtd .cos^'"-! Odd Jo cosh|7r^Jo r{r+ii{?'+3^}...{^^+(2m-i)^r Bysubstitution in(5)and(6)andwriting 2^for^in(5)weatonce infer the truth of(1)and(2)whenR{0> 0:andthemode ofextendingtheresults to allother values of'(hasalreadybeenexplained. Therequired generalisations ofMeissel'sexpansionsarethereforecompletelyestablished. 17'32. Theexpansion ofz^intoaKapteynseries. With theaidofMeissel'sgeneralisedformula itiseasytoobtain the expansionofanyintegral powerofzintheform ofaKapteynseries. Itis convenient toconsider evenpowers andoddpowers separately. Inthecaseofanevenpower, z^^,wetaketheequation given by§17'31(1) intheform ... 1f2r{n+l+iOr(n +l-i^)- J.mi^mz) ^^ 2'7riJr^"-ir{l+iOT{l-iO ,«=im'+^^^^ =J-fi^(.-n+^•D^(. +l-^•o^,„_,,_,^. 27riJ„,t 1r(m+1+t'or(m+1-io^ ^' where thecontour ofintegrationisthecircle |^\=n+^.Since both series converge uniformlyonthe circle, when zliesinthedomain K,term-by-term integrationsarepermissible. Consider nowthevalue of j^f(p+n(2^+n...0i^4-n When in<n,there arenopolesoutside thecontour, and sothecontour may bedeformed intoaninfinitely great circle, andtheexpressionisseen tobe equaltounity;butwhenm>n,thepoles ±imareoutside thecircle andthe expressionisequaltounity minus thesum oftheresidues oftheintegrand atthese twopoles,i.e.to (m+n)I "~?/i'^«+'.(m-«-l)!" Theexpressionontheleftof(])isthereforeequalto 9Vr/9,„^\ 9V(m+n)\.L,J27n2) m=i »i=ji+inv"^^^.{m—n—l)\ Nextweevaluate 1 27rt^(/^+l+^•O^0^ +l-^•O d^ il^nHr{m+l+i^)r(m-h1-iO^^'-'"'+' 17-32] KAPTEYN SERIES 565 Whenm^n,theoriginistheonly poleoftheintegrand, and, ifwetakethe contour tobeaninfinitely great circle, theexpressionisseen tobeequalto1. But,whenm>n,there arenopolesinside thecircle {^\—n+\,andthe expressioniszero. Hence wehave Ifwereplacenbyn-1andsubtract theresult soobtained from(2),we findthat " vi=n W'"--. (m-n)! m=n+i'^n-''+\ (m-n-1)!' andso m=n Ifn=1,equation (3)isatoncededucible fromequation (2),without the intervening analysis. When wehave todealwithanoddpower, z^^~'^,wetake theequation given by§17"31 (2)intheform (4)27ri rrZir+(2m-l)^"^^ 00^2m—1yam—2m— 1 ""Ji{p+?^H3^+n--K2^-iy+n'^^' andwededuce inasimilar manner that (o)- ^-.^J,„,_, \{2m-l)z\. ,,^-^^ (m-ir.(m-^-l)!— =^+2^+ ...-t-^-'^-^ Hence Theformulae(3)and(6)maybecombined intothesingleformula which isobviouslyvalidthroughoutthedomainKwhen nhasanyofthe values 1,2,3,— This formula wasdiscovered byKapteyn*;theproofofitwhich has justbeengiven, though somewhat artificial, seems rather lesssothanKapteyn's proof. *Ann. sci.deI'Ecole norm.suj). (3)x.(1893), p.103. 566 THEORY OFBESSEL FUNCTIONS [CHAP. XVII 17-33. Theinvestigation oftheKapteynseHes forz'iithemethodof induction. Weshallnowgiveanalternative method* ofinvestigaiigtheexpansion ofz^asaKapteyn series, which hastheadvantageofusignoresult more abstruse than theequations (1)-^=1+2iJ-^{mz), t4-.=1+2^(-)" "(^^)' V—z,„_1I-tz„,_1 which wereprovedforrealvariables in§17*22 and forc<jiplexvariables in §17"3;itis,ofcourse, supposedthat, ifzisreal,then-1 ^^<1,and, ifzis complex,then zliesinthedomain K. Theinduction which willbeuseddependsonthefact itwhen thesum, 00 f(z),oftheKapteynseries SUmJmi'niz)isknown, then iisumF(z) ofthe m=l series 1——— ^canbeobtained bytwoquadratures,theformer series m=i w^ converges uniformly.Toestablish this result, observe thabyterm-by-term differentiations. z^ az' az ,„=i =(1-;-) a,„J,n{mz), sothat(z^yF{z)=(l-z')f{z); itfollows atonce thatF{z) canbedetermined interms of iz)byquadratures. Now, from(1),wehave iJ^„(2,nz)=-1^^.($) andSO,if Fiz)= X'^'"^/^mz)^ 111=^1 4»i^ then(^^)V(^)=^^,dzl Therefore, inthedomain K, whereAandBareconstants ofintegration.Ifwemat --*0,weseethat A=B=0. Consequently (2) 2-=2i'^»"*(^^"-^) m=i m" *Watson, Messenger, xlvi.(1',)17), pp.1.50—157 17-33] KAPTEYN SERIES 567 Inlikemanner, wedeace from(1)that :.4„^.,K2m +i)^}=-i^^, andhence that ^^"^ mto (2m+1)^• Theexpansionsofz*^nennis1or2aretherefore constructed. Nowassume that, foisomeparticular value of?i,z'^isexpansibleinthe form z^=n''2K.^n'J'mi'mz),m=1 andconsider thefunction^ (z)defined bytheequation <l>(zh(n-f2)"S^—6,„,„ J,n(mz).m=1^^ Bytheprocessofditirentiationalready used,wehave z' az- az ,re=i =(«+2)={.'* +4^} j:-(«+2)Mi-.')." =(n+2)-^z''+\ Onintegration wededu-; that (ji{z)=z''+-+^' log^+B'. Itisobvious thatA'='=from aconsideration ofthebehaviour of<f){z) neartheorigin. Hence theexpansioofz'^'^^ is 30 m=l where 6m,n+2=——:,—b^,ti- ltfollows atoncebinduction that 'n,m- ,7^i;^=rr (|m-«+1)*"•" ,_.,»'2^'V{m +n)J^{2mz)^^^^° ^^^ ''" „Zi{2m)^-'.nf-r(m-n+1)* That istosay andthis isequation (cof§17-32.Theexpansionof2-"-' isobtained inthe samewayfrom theesansion ofz;theanalysisinthiscase islefttothe reader. 568 THEORY OFBESSEL FUNCTIONS [CHAP.XVII Wetherefore obtain theexpansion /.^/Isn_..yr(w+m).Jn+^m \{n+2m) z\^ ^2^^~"''"m=o (n+2m)"+i.m! which istheexpansionobtained byother methods in§17'32; andtheex- pansionisvalidthroughoutthedomain K. Since theseries ^= (?i+2mf+i.m! isabsolutely convergent (being comparablewithS1/wi^),theexpansion (4) converges uniformly throughout Kand itsboundary. Theexpansionisthere- forevalid (from considerations ofcontinuity)ontheboundaryofK,and in particularatthepoints2^=+1,aswell asthroughoutthedomain K. 17"34. Theexpansion ofl/{t—z)inaKapteynseries. From theexpansionofz''^,obtained inthetwopreceding sections, wecan deduce, afterKapteyn*,theexpansionofl/(^—z)when zliesinthedomain Kand tliesoutside acertain domain whose extent willbedefined later in this section. Assumingthat |^ |> j^^ |,wehave _11^^_1^2"w2«r{n-\-m)Jn+2m {{n+2m) z} Now, ifgexpV(l- z'') \^y 1+^(1-22)I''' therepeatedseries isexpressibleasanabsolutely convergent double series if thedouble series ^«2'*?i-r{n+m)F"+2^»i „ti„r-o(«+2m)"+^m!|«|«+i isconvergent. Buttheterms inthis series arelessthan theterms ofthe double series IX 2'»F''+=''^_ 2FexpF- «=im-om!|«r^" \t\{\t\-2V)' providedthatIi [>2F. Hence, when t\>2^exp \/(l—z^) 1+V(1-^-^) ;' rearrangementoftherepeatedseries forl/(^—z)ispermissible, and,whenwe arrangeitasaKapteyn series, weobtain theformula (1) r^=0„(0+2i<Bn (t)Jn(nz), t—z „=i *Ann. sci.deI'Ecole norm.sup. (3)x.(1893), pp.113—120. 17-34, 17-35] KAPTEYN SERIES 569 where* (2) <Bo{t)=llt, (o^ era/A-1v"('^-2m)^(n-m-l): From thelastformula wemaydeduce averyremarkable theorem discovered byKapteyn;wehave .dV- 1<4«(^n-vi-l)l1<4«(M-2m)-.(??.- w-1)! andtherefore, by§Ol(2), sothat,by§9-12(1), (4) ^„ {t)=w(1- ^2)0„(/^O+sin^\n'rr+«cos'^\mr when ?i=1,2,3,.... Kapteynspolynomial ^„. (^)i«tJierefofe expressibleintermsofNeumann's polynomial 0^(nt). Itisnowpossibletoextend thedomain ofvalidityoftheexpansion (1); for,by§8-7combined with§9-17, itfollows that theseries ontherightof (1)isauniformly convergentseries ofanalyticfunctions ofzand twhen 2 and tlieindomains such that (5) n(z)<n(t), n(z)<n{i), where n(z)^\l^^^jMlzA^) | Theexpansion (1)istherefore validthroughoutthedomains inwhich both oftheinequalities (5)aresatisfied. [Note. This resultgives asomewhat more extensive domain ofvalues oftthanwas contemplated byKapteyn;heignored thetlicoremprovedin§9-17,andobserved that (since thecoefficients intheseries for(©„(0 arepositive) wiien |^ |^1, I©»(0|<®«(1^|)=$©„(1)=1, by(4) ;sothatKapteyn proved that(1)isvalidwhen 0(2)<fl(l), |i:|:$:l.] 17"35. Alternativeproofs oftheexpansioii ofl/(t—z)intoaKapteynseries. Now thatexplicit expressions havebeen obtained forthecoefficients intheexpansion / l~Z „=i itispossibletoverifythisexpansioninvariousways. Thus,if©„ {t)bedefinedas n{\- 1-)On(n«)+sin2\mT+1cos^^rm-, thereader willfind itaninteresting analysistotakethescries 74-f^+2{\-f^)^ n0^{nt)J„{m\ *Cf.Kapteyn, Nicuw Archie/ voorWiskunde, xx.(1893), p.122. 570 THEORY OFBESSEL FUNCTIONS [CHAP.XVH substitute suitableintegralsfortheBessel coeflBcients andNeumannpolynomials, and reduce theresult to1/(^—2)after themanner of§9'14. Oragain,ifwedifferentiate theexpansion twice withrespecttozwefindthat f222 \+ (^^^2}=(1-2')J^^nn^®n it)Jn{m),\{t-zf {t-zf andthen, dividing by1—z^^andmaking useof§17"3(1),wefindthat {t^-\){t-zf if-\f{t-zf {f'-\f{t-zf whence thedifferential equationfor©„ {t)iseasily constructed intheform andhence itfollows that • ®n{t)=nil-f)On{nt) +sm^hin-irtco^'^ Hit+1-"^ {A^Jnint) +Br^Yn{nt)], where A^andB^areindependentoft;but itdoesnotseem easytoprovethatAn=Bn=(). 17'4. Theexpansion ofanarbitrary analytic functionintoaKapteyn series. Weshallnowprovethefollowing expansion-theorem: Letf{z)heafunctionivhich isanalytic throughouttheregioninivhich £1{z)^a,luhere a^1. Then, atallpointszinside theregion, -A (1) f{z)=Oo+2i:aJn{nz\ xvhere (2)^=^.\(Bn{t)f{f)dt, and thepath ofintegrationistliecurve onivhich il(t)=a. This result isobvious when wesubstitute theuniformly convergent expansion n=l forl/(t—z)intheequation J^'^-^iri] t-z' since D,{t)=aonthecontour, while bothD,{z)<l andQ,(z)<Cl (t)when z isinside thecontour. Thistheorem isduetoKapteyn. 17-4, 17-5] KAPTEYN SERIES 571 Itiseasytodeduce that, iftheMaclaurin seriesfory(^)is n=0 then ,.1<^''»(n-2m)-.(7}-7u-l)l an-2m 17'5.Kapteynseries inwhich visnotzero. ThetheoryofKapteynseries ofthetype inwhich visnotzero oraninteger,canbemade todependontheex- pansionofz".The result of§17-33suggeststhat itmaybepossibleto provethat 00 throughoutthedomain K. Itiseasyenoughtoestablish thisexpansion* when\z\<0-6627434;but nodirectproofofthevalidityoftheexpansion throughouttheremainder of thedomainKisknown, andtheexpansionhastobeinferred bythetheory ofanalyticcontinuation. Toobtain theexpansion throughouttheinterior ofthespecified circle, expandtheseries ontherightinpowersofz.The coefficient ofz"'^-'' is V^( t^+wt) (-)'""'" ('^+2m)"+-'• „f^o(iH-"2m)''+i .ml' 2"-*^'-{r-m)l V{v+r+m+1) r(v)^(-^r-m(-J,+2m)2'-i r{v+ m)T(v-{- 2r+1) 2>'+-rr{v-h2r+1),„=om !(r-m)! T{v)r{v+r+m+1)' When r^l, the lastseries isapolynomialinvofdegree3r—1which is known tovanishidenticallywhenever visaninteger.Ittherefore vanishes identicall;yforallvalues of v.Theexpansion (1)istherefore established (inside thecircle) byacomparisonofthecoefficient ofz"oneach sideofthe equation. From thisresult, wecanprove that,under theconditionsspecifiedin§17-4, (2) r^=2Mn,u(t)J.+n {(v+n)z\, where 71—2TOt12,r,^^{\v^nY'-^-'^^Km\t *Thiswasdonewhen|s|<;0-659 byNielsen, Ann. sci.deVEcole norm.)f\tp. (3)xviii.(1901), pp.42—46. 572 THEORY OFBESSEL FUNCTIONS [CHAP.XVII Itisnotdifficult* toexpress S4n,v{t)interms ofGegenbauer's polynomial An,v{nt-\- hvt), defined in§9"2. Andthereader willeasily provethatif/(^)satisfies theconditionsspecified in§17-4,then (4) z^f{z)=Sa„,,/,+,, {(v+71)z], n= where inwhich thecontour ofintegration surrounds theorigin;andhence _1 <^''(v+n-2m)-r(v+n-m)an- (6) «.-2m wherecio,a^,...arethecoefficients intheMaclaurin seriesfor/(2^). [Note.Jacobi inoneofhislaterpapers,Astr. Nach. xxviii. (1849),col.257—270[&'es. Math. Werke, vii. (1891), pp.175—188]has criticised Carlini forstatingthat certain expansions arevalidonlywhen\z\<0'663 ButCarlini hadsome excuse forhisstate- ment because theexpansions areobtained byrearrangementsofrepeatedseries which are permissible onlyinthisdomain, althoughtheexpansions areactuallyvalid throughout the domainK.'] 17'6. Kapteynseries ofthesecond kind. Series ofthetype 2/3../^.{(^"'+»).}J-.^IC^^ +«).} havebeen studied insome detail byNielsenf.But theonlyseries ofthis typewhich have, asyet,provedtobeofpractical importance |,aresome specialseries withix=v,andwithsimplecoefficients. The resultsrequired in theapplications justspecifiedareobtainablebyintegratingMeissel'sexpansion ofI17"31 (1)afterreplacingzby2sin ^.Itisthusfound that,throughout thedomain K, \lM =„?(p.p)(2'4...(..'+r )-^/.^^"'"'' sothat ^^Zi n'+^- 2P-f^""^2.4'(l- +^2)(22+H andhence wededuce that'S,Jn-{nz)ln-'"'isapolynomialinz-ofdegree m; while thesum ofseries ofthetype 'Zn^^Jn^(nz) maybefound inasimilar manner from thecorresponding expansion '2n^^"'J.2n{2nz). *Cf.Nielsen, Ann. sci.deVEcole norm. sup. (3)xviii.(1901), p.60. tAnn. sci.deVEcole norm.sup. (3)xviii. (1901), pp.39—75. XCf.Schott, Electromagnetic Radiation (Cambridge, 1912), Chapterviii. 17-6, 17-7] KAPTEYN SERIES 573 Thus Schott* hasshewn that (2) 2J^'{nz) = 2v/(l-22)2'[Tfyn=i"^ '2^/(1-22) (3)27i'JJ{nz)=^zH4+z^) Ageneral theory resemblingthat of§16'14 isdeducible from theex- pansion which iseasilyderived from§17"5(1)and isvalidthroughout K\but it seemsunnecessarytogointodetails which thereader should havenodifficulty inconstructing,intheunlikelyevent ofhisrequiringthem. 17'7. Kcipteynseries wJiichconvergeoutside thedomain K. If Im|v/«n|=l, wehave seen that theKapteynseries%anJn{^^z) represents ananalytic functionthroughoutthedomain K.But since,when xisreal,\Jn{nx)\< 1, theseriesmayconverge alongthewhole ofthereal axis,although when\z\>1, theseries doesnotconvergeatpointswhich arenotonthereal axis. Thebehaviour ofsuch aKapteynseriesmaybesummedup^f*bysaying that itresembles apower-series throughoutthedomainKandthat itre- sembles aFourier series ontherealaxisoutside K. Asanexample,letusconsider theseries ^Jnjnx) Itisevident that, if(^= -v/^—^sin-v/r,then -^Jo «=in- since theFourier series isuniformly convergent. Now,when x>1, cf)decreases asyjrincreases from toarccos(l/a-) and then increases toirasyjrincreases from arccos(1/^')tott.Ifvibetheinteger such thattheminimum value of (f>liesbetween —2m7rand—2{m+1)tt,let thevalues of-\^correspondingtothevalues 0,-27r,-47r, ...,-liHTr, -2??i7r, ...,-27r, of </)be7o,7i,...jm, S/«, ^,/(-i,...§1,So,andthen S= TT(,.=J7r •'7mr=()JS,.+l -Soin-lM" *Electromagnetic Radiation (Cambridge, 1912), p.120. tThesuggestionofthese analogies wasmade byProfessor Hardy. 574 THEORY OFBESSEL FUNCTIONS[CHAP.XVII Nowwhenyjrliesintheintervals(jr,7r+i)and(S,.,8r+i)thesum ofthe series under theintegral signis i<f>'-- Ittc^+Itt^+r(r+1)tt^+(r+1)ircf), and, since l(-^—Xsint/t) fZ-v|r=^-\/r-+a;cosyjr, {(yfr—Xsini/r)^ c^-v^= ^^i/r*+2a;(i^rCOS •\//'—sin>^)+ ^x^ {-^jr—sinyfrcos-^/r), itmaybeshewn without muchdifficultythat m m S=:^x'+\x+S{I(8,--7,^)+X(cos 8^-cos7,)}+'2'ir^r{h,- 7,). ,.=0 j-=i Thereader willseethatalargeclass ofKapteynseriesmaybesummed by thismethod*. 17*8. Theconvergence ofKapteynseries ontheboundary ofthedomain K. With theexceptionofthepoints +1,theboundaryofKpresents no features ofspecial interest; because, bymeans ofDebye's asymptotic expansion theconsideration oftheconvergenceoftheKapteynseries SonJv+n [{v+n)z] isreducible tothat ofthepowerseries ^On(^exp V(l-g-) l""Vn i1+\/(l-z'))' andthat oftwosimilarseriesf with \/'^^ V'*''written forijn. Thepoints+1present more interest, because theordinary asymptotic expansionsfail.Butthelacunathereby producedisfilled, forrealvalues ofv, bythefollowingtheorem ofanAbeliantype: Theconvergence of i^ issufficienttoensure both theconvergence of%a^Jv+n (^+^^)(^f^d thecontinuity ofSanJv+n {{v+n)x]throughouttheinterval^ ^x^\. Sincelajn^ convergesand{n/{v+7i)|^ismonotonic, withalimit asn-*oc, itfollows^that'S,anl(v+n)^converges;andsince, by§8'54, (v+n)^Ju+n(v +n) ismonotonic, with alimit asn^x,itfollows that%anJv+n {v+n)converges. *Inthisconnexion theresearches byNielsen, Oversigt K.Danske Videnskabernes Selskabs, 1901, pp.127—146, should beconsulted. tIfaj^ndoes nottend tozero theseries cannot converge ;and ifitdoes tend tozero 2<aJ^n^isabsolutely convergent, and so,ifwereplace eachBessel function bythe firsttwoterms oftheasymptotic expansion with aremainder term, theseries ofremainder terms isabsolutely convergent. JDueallowance hastobemade fortheoriginifv<0. §Bromwich, Theory ofInfinite Series, §19. 17-8] KAPTEYN SERIES 575 Again,sinceJ.^^\iv+n)x] isafunction ofnwhich does notincrease asnincreases, forallvalues ofx intheinterval O^ic^l,itfollows from thetestofAbel'stypeforuniformity ofconvergence*that Sa„/^+„ [{v+n)x\ isuniformly convergent (and therefore continuous) throughouttheinterval ^a-$1;andthisprovesthetheorem. Byreversingthereasoning,itmaybeshewn that if^a^Jv+niv +n)con- verges,sodoes2a,i/n^,sothattheconvergenceof^dnju-^isbothnecessary and sufficient forthetheorem tobetrue; thetheorem istherefore thebest theorem ofitskindf. *Bromwich, Theory ofInfinite Series, §44. tThiswaspointed outbyProfessor Hardy. Cf.Watson, Proc. London Math. Sac.(2)xvi. (1917), pp.171—174. CHAPTER XVIII SERIES OFFOURIER-BESSEL ANDDINI 18*1.Fourier sformal expansion ofanarbitrary function. Inhisresearches ontheTheoryofConduction ofHeat, Fourier* wasled toconsider theexpansionofanarbitraryfunction/(ic)ofarealvariable of ,r intheform 00 (1) /(«)= 2a,nJo(jm'^), whereji,j^yjs,...denote thepositivezeros ofJo(2')arrangedinascending order ofmagnitude. Thenecessityofexpandinganarbitraryfunction inthismanner arises also inDaniel Bernoulli'sproblemofachainoscillatingundergi-avityandin Euler'sproblemofthevibrations ofacircular membrane withaninitial arbi- trary symmetrical displacement (§§1'3, 1"5). Inorder todetermine thecoefficients a,„intheexpansion,Fourier multi- pliedboth sides of(1)byocJq{jmx)andintegratedbetween thelimits and 1. Itfollows from§5"11 that l^^i' \Jm)> m—ti, andhence Fourier inferred that (2) a,,=j-^^ \' tf{t)Joijmt)dt. fflKJm)' Ifwenowchangethesignificanceofthesymbols J„,,sothatf j'l,jg,Ja,... denote thepositivezeros ofthefunction J^{z),arrangedinascendingorder ofmagnitude,then 00 (8) /(«;)= 2a,nJ,(jmOo),m=l where (i) am=J,^,.l\f(t)JAjmt)dt. tfv+i \Jm)J Thismoregeneralresult wasstated byLommel;]:; but, ofcourse, neither inthegeneralcasenorinthespecialcase z--=doestheprocedurewhich has been indicated establish thevalidityoftheexpansion;itmerelyindicates how thecoefficients aretobedetermined onthehypothesisthat theexpansion exists and isuniformly convergent. *LaThiorie AnalijtiquedelaChaleiir (Paris, 1822), §§316—319. fTheomission ofthesuffix v,associated withji,J2,js, ,..,should cause noconfusion, and it considerabh' improves theappearanceoftheformulae. XStudien iiber dieBesseVsclien Functionen(Leipzig, 1868), pp.69—73. 18-1] FOURIER-BESSEL SERIES 577 Infactthesimplicityoftheprocedureissomewhatdeceptive;forthe readermight anticipate that,ifthefunction/(a;)issubjectedtoappropriate restrictions, theexpansionwould bevalid forallvalues ofvforwhich the integral ItJAjmt)JAint)dtJo isconvergent,i.e.whenv^—1. But, aswaspointedoutbyDini inthecourse ofhisresearches onthe expansion,itseemsimpracticabletoestablish itexceptonthehypothesisthat i'^—^;andalthoughvarioussubsequent writers, whileprovingtheorems on thehypothesis v^—^,have stated that theextension oftheanalysisto values ofvbetween —|and—1ismerelyamatter ofdetail, their statements appeartobeopentoquestion. The fir.stattemptatarigorous proofoftheexpansions (1)and(3)is contained insome notescompiled byHankel* in]869 andpublished post- humously. Amorecomplete investigation wasgiven bySchlaflif ayear after thepublicationofHankel's work; andanimportant paper byHarnack:^ contains aninvestigationoftheexpansion (3)bymethods which differed appreciablyfrom those ofearlier writers. Afewyearsafter theappearanceoftheresearches ofHankel andSchlafli, themoregeneral expansion CO (5) f{^)= SKJA^ni^Xm=l where Xj,>2>^a,•••denote thepositivezeros (inascendingorder ofmagnitude) ofthefunction whenv^— h,andHisanygiven constant, wasinvestigated byDini§, The coefficients intheexpansionaregiven bytheformula (6) {(X„r- v-")JJ^(X„ )+X„rJ;^(X^)}6^=2\^'\tf(t)J.(X,„i)Jodt. Themode ofdetermination ofthenumbers \nsubjects /(*")towhat is known asa'mixedboundary condition,' namelythat/'{x)+Hf{x)should formallyvanish ata?=1. The^xpansion (5)wasexamined byFourier (when v=i}) intheproblem ofthepropagationofheat inacircularcylinderwhen heat isradiated from thecylinder;inthisproblemthephysical significanceofHistheratio ofthe externalconductivityofthecylindertotheinternalconductivity. *Math. Ann. viii.(1875), pp.471—491. Inthecoarse ofthispaper, Hankel obtained the integral formula of§14-4asalimiting caseof(3). tMath. Ann. x.(1876), pp.137—142. %Leipziger Bcrichte, xxxix.(1887), pp.191—214; 3Iath. Ann. xxxv.(1889), pp.41—62. §Serie diFourier(Pisa, 1880), pp.190—277. w.B.F. 37 578 /THEORY OFBESSEL FUNCTIONS [CHAP. XVni ItwaspointedoutbyDini thattheexpansion (5)must bemodified* by theinsejxion ofaninitial termwhenH+v=0:and,althoughDini'sanalysis contams anumerical error, thisdiscoveryseems tomake itadvisable to associate Dini's name rather than Fourier's with theexpansion. Theresearches which havenowbeen described depend ultimatelyonaset oflemmas which areproved byCauchy's theoryofresidues. Theuseofcom- plexvariables has,however, been abandoned, sofaraspossible, byKneserf andHobson^:, whohave constructed theexpansion byusingthetheoryof integral equationsasabasis. Onaestheticgroundsthere isagreatdeal tobesaid forthisprocedure, because itseems somewhat unnatural tousecomplexvariables inproving theorems which areessentiallytheoremsconcerningfunctions ofrealvariables. Ontheother hand, researches based onthetheoryofintegral equationsare liable togiverisetouneasy feelingsofsuspicioninthemmd oftheultra- orthodox mathematician. Thetheoryhasrecentlybeenmadedistinctlymorecomplete bythe important memoir ofW.H.Young|,whohasthrown newlightonmany partsofthesubject byusing modernknowledgeofthetheoryoffunctions of realvariables inconjunction with thecalculus ofresidues. Anearlierpaper byFilonjl which makes somepartsoftheanalysis appreciablylesssynthetic must alsobementioned here. Thequestionofthepermissibilityofterm-by-termdifferentiation ofthe expansionwhichrepresentsafunction asaseries ofBessel functions hasbeen discussed byFordii, whohasobtainedimportantresults with thehelpof quite simple analysis (cf§18"4). More reconditeinvestigationsareduetoC.N.Moore**, who, afterstudying thesummabilityoftheexpansion byCesaro's means, hasinvestigatedthe uniformityoftheconvergenceoftheexpansionintheneighbourhoodofthe origin, and alsotheuniformityofthesummabilityoftheexpansion (when notnecessarily convergent)inthisneighbourhood. Thereason whytheuniformityoftheconvergence (orsummability)of theexpansionintheneighbourhoodoftheoriginneeds ratherspecialcon- sideration isthat itisnecessarytouseasymptoticformulae for J^,{Xm^)which arevalidwhen\n^ islarge; and, asxapproaches zero, thesmallest value of m,forwhich theasymptoticformulae aresignificant,iscontinually increasing. *Details ofnecessary modifications whenH+v^0willbegivenin§18"3. Themodification wasalsonoticed byKirchboff, BerlinerSitzungsberichte,1883, pp.519—524. tArchill derMath. undPhys. (3)vn. (1908), pp.123—133; Math.Ann.i.xiu. (1907), pp.477—524- %Proc. London Math. Soc.(2)vii.(1909), pp.359—388. §Ibid.(2)xvin.(1920), pp.163—200. ilIbid.(2)IV.(1906), pp.396—430. Cf.§§19-21— 19-24. ITTrans. American Math. Soc. iv.(1903), pp.178—184. **Ibid. X.(1909), pp.391—435; xii.(1911), pp.181—206; xxi.(1920), pp.107—156. t 18-11] FOURIER-BESSEL SERIES 579 Intheexpositionwhich willbegiveninthischapter,themethods ofthe calculus ofresidues willbeused toafargreaterextent than hasbeen usual inrecent researches ;this isareversion tothepracticeofHankel and Schlafli and(inthespecialcase ofFourierseries) ofCauchy. Theadvantage ofthisprocedureisthat itresults inagreat simplificationinthegeneral appearanceoftheanalysis throughoutthewholetheory. And, althoughit seemsimpracticabletoprovecertain theorems(notablythose*relatingto fractional orders ofsummability)with thehelpofcomplex variables, thegain insimplicityissomarked that ithasbeenpossibletoinclude inthischapter verymany more theorems than would have beenpossibleifthemethods ofthetheoryoffunctions ofrealvariables hadbeen usedmoreexclusively. Asanexampleofthesimplicity produced byusing complex variables, it maybementioned thatcomparativelycrudeinequalities,such as C:exp1/(^)1 wherec^isaconstant, independentoiz,when visgivenandexceeds —^,are sufficient toprovealltherequisitetheoremsconcerning convergenceatapoint (orsummabilityatapoint)andtheyarealso sufficient toprovetheorems con- cerning uniformityofsummability throughoutaninterval ofwhich theorigin maybeanendpoint.Directproofsoftheoremsconcerning uniformityof convergence throughoutsuchanintervalrequire more elaborateinequalities, butinthiswork theuseofsuchinequalitiesisevaded bydeducing uniformity ofconvergencefromuniformityofsummability byanapplicationofHardy's convergencetheoremf. Itmaybestated herethatthetheorems ofthischapter correspond exactly tothetheoremsconcerningFourier series which aregiveninModernAnalysis. Inaddition tothememoirs which havealready been cited, thefollowing maybemen- tioned :Beltrami, R.1st.Lombardo Rendiconti., (2)xin. (1880), pp.327—337;Gegenbauer, WienerSitzungsberichte,Lxxxviii.(2)(1884), pp.975—1003;Alexander, Trans.Edinburgh RoyalSoc. xxxili.(1888), pp.313—320; Sheppard, Quarterly Journal, xxiii.(1889), pp.223—260; Volterra, Ann. diMat.(2)xxv.(1897), p.145; Stephenson,Phil.Mag. (6) XIV. (1907;, pp.547—549;Messenger.,xxxiil. (1904), pp.70—77,178—182;Rutgers, NieuwArchief, (2)vni.(1909), pp.375—380; Orr, Proc. R.Irish Acad, xxvii. A,(1910), pp.233—243;andDinnik, Kief Polyt.Inst.{Engineering Section), 1911, no.1,pp.83—85. [JahrbuchUber dieFortschritte derMath. 1911, p.492.] Theinvestigations byAlexander aremainlybased onoperational methods, while Orr dealt withexpansionsinwhich functions ofthesecond kind areinvolved. 18^"11. Thevarioustypes ofseries. Inthespecialcase ofseries ofcircular functions, itisnecessary,asthe reader willremember, tomake adistinctionJbetweenanytrigononirical ser'ies Itty+^(a,nCOS7nx+b,nsinimv), 7/1=1 *Such theorems have been investigated byMoore andYoung. tModern Analysis, §8-5. %Cf.ModernAnalysis, §9-1. 37—2 580 THEORY OFBESSEL FTr>''CTIO>'S [CHAP. XVIII andaFourier series inwhich thecoefficients areexpressedasintegrals, 0^=— If(t)cosmtdt, b^=—Ir\t)sinmtdt. TT.'-s- ITJ^^-^ Itisnecessaiytomake asimilar distinction* between thetypesofseries which willbedealt with inthischapter: anyseries ofthet\-pe X m=l inwhich thecoefficients a^merelyform agiven sequenceofconstants, will becalled aseries ofBesselfunctions. If,however, thecoefficients inthisseries areexpressible bytheformula" Oi-m—j\f{t)J,{j^t)dt, theseries willbecalled theFourier-Bessel series associated withf{x). Andif,farther, theseriesconvergestothesumf(x)foranypoint xof theinterval(0,1),theseries willbedescribed astheFourier-Besselexpansion offix). Inlikemanner, theseries «=1 whereXj,X»,Xj,...arethepositivezeros of willbecalled Dints series ofBesselfunctions. Ifthecoefficients 6,„aredetermined bytheformula^ [(x,„;--^)J,2(x^)+x.,^2j/^(xj] b„,=2x„^rtf{t) J,(X«dt JO theseries willbecalled theDini series associated withf(x). Andif,further, theseriesconvergestothesum/' (jr)foranypointxofthe interval(0,1),theseries willbedescribed astheJXniexpansion off{x). Some writers havebeen inclined toregardFouiier-Besselexpansionsas merelyaspecialcase ofDiniexpansions,obtainablebymakingH-*x :but there arecertain distinctions between thetwoexpansionswhich make this viewsomewhatmisleading (cf^1826, 1834, 18-35 ). 1812.SpecialcasesofFourier-Bessel andDiniexpansions. There areveryfewexpansionsofsimplefunctions inwhich thecoefficients assume asimpleform. One function whoseexpansionhassimplecoefficients hasalready been *Thegreater part oftheterminologyisdaetoYoung, Proe. London, Math. Soc.(2)svm. 11920), pp.167—168. tItissupposed thattheintegralisconvergent forallpositive integral values ofm. XItissupposedthattheseries ismodified, asin§18'34, when flt-f^0. 18-12, 18-2] FOURIER-BESSEL SERIES 581 investigatedin§15"42. Another isx",whichgivesrisetotheformalexpansions (I)oc„_ >r2J,{jmCC) m (2) ^"=2=\JmJ v+\\Jm) m- Itwillbeseensubsequentlythat(1)isvalidwhen ^^<1,and(2)when ^.rs:1,ifiT+z^>0.Cf.§§18-22, 18-35. Thereduction formula .'"» J•' iseasily established, sothattheDiniexpansionof.r"*-"maybedetermined when visany positive integer. TheDiniexpansionof.r^+^'i+ixnay similarly bedetermined; inthis casethegeneral coefficient isexpressibleinterms ofknown functions and Inorder tocalculate thiswhen visaninteger, McMahon* hasproposedtotabulate the fuiiction/: 1/*' ,{t)dt=,/i(.r)+J-, (.^•)+Jr,{.v)4-..., which isaspecial form ofoneofLommel's functions oftwovariables(§§16-5, 16-56). 18-2. Themethods ofHankel andSchldfli. The earlierinvestigationswhich were described in§18-1arebased onthe analysisusedbyDirichletfinhisresearches ontrigonometricalseries of Fourier'stype;thismethod ofproceedingisobviously suggested bythe factthat thetrigonometricalseries arespecialcases oftheFourier-Bessel expansion,obtainedbygivingvthevalues ±\. InthecaseofFourier's theorem, toprovethat 00 /(*)=1*^0+^(a,rtcos 7*?,^+6,„sin»i.r),m=l where «,„=-f(t)cosmtdt, 6,,,,=-f(t)sinmtdt, itissufficient toprovethat ^-^1f""f(if)=lim—/{^+cos{x—t)-hcos2{oc—t)+...+cosn{cc—t)\J (t)dt, .....that/W=, .'!!",2^./,. sW^'Iqf(t)dt *Pioc. American Assoc. 1900, pp.42—43.Thetabulation ismost simply effected byusing §10-74(3)inconjuuetion withTable I.(pp.666—697) ;seeTable VIII. tJournal farMath. iv.(1829), pp.157—169. 582 THEORY OFBESSEL FUNCTIONS [CHAP. XVLQ Inthecase ofthegeneralFourier-Besselexpansion,thecorresponding limit tobeevaluated is limi^^^%^!\f(t)JAj,J)dt, andsoitisnecessarytoinvestigatethebehaviour ofthesum I2J^{j,„a;)J^(jmt) m=l "'v+i\.lm) when nislarge;and itisinthisinvestigationthattheuseofthecalculus of residues ismore than desirable. Inthecase ofDini'sexpansion,thecorresponding sumwhich needs examination is Anapplicationofthecalculus ofresidues which willbedescribed in§§18'3— 18"33 shews that thedifference ofthetwosums isreadily amenable todis- cussion, and sowearesparedthenecessityofrepeatingthewhole ofthe analysisoftheFourier-Besselexpansionwith themodificationsappropriate tothemoregeneralcaseoftheDiniexpansion. 18*21. TheHankel-Schldflicontourintegral. Weshallnowbegintheattack ontheproblemofFourier-Besselexpansions bydiscussing propertiesofthefunction Tn(t,x),definedbytheequation /I\ rn /J.^\_V-•"v\.1m^) "v\Jmt) \^) J-nV'yX)— ^ j:^Tl~\'m=\ "v+\\Jm) where 0<a;^l, O^^^l, andtheorder visrealand issubjecttothecondition v+l^O. Themethod which willbeused isdue toHankel* andSchlaflif, though manyofthedetails oftheanalysisaresuggested byYoung's;):recent memoir. Thefunction Tn(t,a;)isobviouslyasfundamental inthetheoryofFourier- Besselexpansionsasisthefunction sin(n4-^){a:— t) sin^(iJO—t)' inthespecial theoryofFourier series. Inorder toobtain theformulae connected withTn(t,x)which aresub- sequently requireditisnecessarytoexpressthemth terra ofthesum for Tn(t,x)astheresidue atjmofafunction, ofthecomplexvariable w, which hasipolesatji,jo,jo,...jn.When thishasbeen done, weexpress *Math. Ann. win.(1875), pp.471—494. tIbid. X.(1876), pp.137—142. XProc.London Math. Sac.(2)xviii.(1920), pp.163—200. 18-21] FOURIER-BESSEL SERIES 583 Tn(t,x)astheintegralofthisfunction round arectangleofwhich oneofthe sides liesalongtheimaginaryaxiswhile theoppositesidepasses between jnandjn+\-The sidesparalleltotherealaxis arethenmoved offtoinfinity inopposite directions, sothat, inorder tosecure theconvergenceoftheintegral, itisnecessarytoprescribethebehaviour oftheintegrandas |/{w) |^oo . There arethreeintegrandswhich weshallstudy, namely (2)2[tJ^{xw) J,+,(tw)-xJ^(tw)J,+, {xw)}f{{t''-x')J^^ (w)}, (3) irw[J^(w)Y„(xw)—J^(xw)Y^(w)}J^(tw)jJ^ (w), (4)TTiu{J^(w)Y^(tw)-Jy(ttv)Y^(w)}J^(xw)/J^ (w). The first ofthese wastheintegrandstudied bySchlafli; theother twoare suggested bythework ofKneser andCarslaw which wasdescribed in§15'42. Astudyoftheasymptoticvalues oftheseintegrandsindicates that(2)is suitable fordiscussions inwhich x=^tand<x+t<2;(3)when*-^^<^< 1; and(4)when ^x<t<l. Weproceedtoverifythattheintegrandsallhave thesame residue, namely ^Ju(jmX) JAjmt)/J\+i(jm), atw=jm-Inthecase of(2),wedefine thefunctionf g(w)bytheformula 2w (5) g(w)=[tJ^(xtu) J^+i (tiv)-xJ^(tw)J^+i(xw)}, andthen, iflu= ;"„,+6,where 6issmall, wehave J.(tv)=dJ,( jm)+iO'J," (j,„)+..., sothat WJ,' (W)=e"-j^JJ^ (j„,)+d'JJ (j,n)[jmJ u"(jm)+JJ(jm)]+•••• Itiseasytoverify, byusingBessel's differentialequation,thatthecoeffi- cient of 6-'ontheright vanishes; andhence theresidue oig(w)l[wJj^(iu)] atj,nis f)'{Jni)l\jmJJ'-(jm)\, andthis iseasilyreduced to 2/^(jmX) Jv(jmt)IJ\+i (jm) byusingrecurrence formulae. Inthecaseof(3),theresidue atj^is T^i-n [J>'(jm)Y^(j,nX)-J,(jmX) Y„(j^)]J^(jmt)IJJ (jm) =--jrjmJv. (jmX) Jv(jmt) Yv(j>n)IJJ (jm)^=2J^(jniX) Jv(j,nt)IJJ' (jm), by13"63,and this istheexpression required;theintegrand (4)isdealt with inthesameway. *This ismost easily seenbywriting theintegrandintheform l-rriw {HyC") (ic)Hv^) [xw]-Hut})(xi(7)if^(2)(it;) J.,T^(tw)IJv (iv). \Theresults obtainable byusing theintegrand (2)arediscussed ingreat detail byGrafand Gubler, Einleitung indieTheorie derBesseV schen Funktionen,i.(Bern, 1898), pp.131—139. 584 THEORY OFBESSEL FUNCTIONS [CHAP. XVIII Wenext take thecontour ofintegrationtobearectanglewith vertices at±Bi,An±Bi,whereBwillbemade totend toqo,andAnischosen so thatjn<An<jn+i- When itisdesired toassignadefinite value to^„,we shall take ittobeequal to(n+^v+ ^)7r,which liesbetweenj„andjn+iwhen nissufficiently large (§15'53). Now itiseasytoverifythatthethreeintegrandsareoddfunctions ofw, andsothethreeintegrals alongtheleftsides oftherectanglesvanish *. Again,ifio=u-\-iv,itmaybeverified thatwhen vislarge, andeither positiveornegative,while u^0,then thethreeintegrandsarerespectively {e~i^-^-t'\% (e-*^-*'!"!), (e-<«-^)l«i), and so,foranyassignedvalue of^,i,theintegrals alongtheupper andlower sides oftherectangletend tozero asB^oowhen xand thave therelative values which havealreadybeenspecified. Wethusobtain thethree formulae (6) r„(«,.)-1i"-"-'?(''i*" 1rAn+'^i ri20wJ^(xOw) J^(tdw)dddw ~27ri.'^„_aoi Jo J„-=(w) {<d<x-\-t< 2;xi^t) (7) Tn(t,x)=^.w[J.{w)n{xw)-J,{xiu)F,{w)]''^)-7 ^"^, (0<t<x<l) (8) Tn(t,X)= ^.f^"^"u-{J.MY.itw)-J.(tw)F.(w)]'^-'^^. {0<X<t<\) Fromequation (6)itiseasytoobtain anupper bound for jTn{t, x) !;for itisevident from theasymptotic expansionof§7"21 that,when ;^+-|-isposi- tive(orzero)andbounded, there existpositiveconstants dandCgsuch that (^>i'J(tw) I<g^expli 7(^^)1} c,_exp[J /(w)|| (y) ^JAtw)\^ ^^l^^^i,u.wi^^,^j whenwisonthelinejoining An—ocitoAn+cciandf^O, provided thatn exceeds avalue whichdependsonv.Hence \it'-^')Tn{t,x)\^^J'j^^^^ j\xip{-{2-x-t)\vl}dv, sothat 4r- Thisinequality givestheupper bound inquestion. *Itisnecessary tomake anindentation attheorigin, buttheintegral round theindentation tends tozerowith theradius oftheindentation. 18-22] FOURIER-BESSEL SERIES 585 Itisalsoeasytoseethat fr+iT,(t,x)(t-- .^•-)dt Jo {tJ,(xw) J,+,(tw)-.rJ,+i {tiu)J^+i{xw)] J, andhence [Note. Theorems obtained byaconsideration ofintegrals involving Bessel functions ofthe firstkindonlycanusually bemade tocover theorigin,inview ofthefactthatthe constantCjinequation (9)isindependentoftintheinterval 0^<<1.Thus(11)maybe written \JITTCoJin[Z—X—I) validwhenO^.r^l, O^t^l. This extension isnotsoeasilyeffected when integrals involving functions ofthesecond kind have been used because thesimplest inequality correspondingto(9)is !Y^(tw)I<c'l'{ 1tw!""log Iiw I+ 1tic\-i}exp {[7(tw)\}, and itisasomewhat tedious matter toobtain asimple upper bound totheintegrandin (8)from thisinequality.] Equation (<j)wasusedbySchlafli toprove that,when nislarge,then ^"^''^^'^2;^)sin-4„(^—^)sin-rl,j(^+^) sin^TT{t—x)sinitt(i+x)j' but,since theorder ofmagnitudeoftheerror inthisapproximationisnot evident, weshallnext evaluate someintegrals involving Tn{t,x)bymeans of which difficulties causedbytheunknown errormaybeevaded. 18"22.Integrals involving Tn{t,x).' '' Thetwofundamental formulae which weshallnowobtain areasfollows : (1) lim[t^+'T,,{t,x)dt=a-",' (0<x<1) (2) lim I't''+'Ta{t,x)dt=^x''.' "(0<.r<l) From these itisobvious that (3)lim[f^'Tn(t,x)dt=^x".-^(0<x<l) Inthecourse ofproving (1)itwallbeapparentthat xi!t''+'T„(t,x)dt-^x''+i•'•' • Jo uniformlyasn-^z)whenxliesintheinterval• , ^^'^1-A, ^., w^hereAisanypositivenumber. 586 THEORY OFBESSEL FUNCTIONS[CHAP. XVIII Weshall alsoinvestigatetheboundedness of n IJo intheinterval inwhich <t^1. Ofthese results, (1)wasgiven byYoung, Proc. London Math. Soc.(2)xviir.(1920), pp.173—^174,andtheproofofit,which willnowbegiven,ishis.Formula(2)seems tobe new,thoughitiscontainedimplicitlyinHobson's memoir. Itisevident that f^'+1T^(t,x)dt= I^lA^cc)_ When wetransform thesumontherightintoacontourintegralafter the manner of118'21, wefindthat itisequalto 1r=^^2J^(a;w)dw1 1'-^'^+'^'•2J^{a;w) dw Intheformer ofthese twointegrals,theoriginhas tobeavoidedbyan indentation ontherightoftheimaginaryaxis. Since theintegrandisanoddfunction ofw,thevalue ofthe firstintegral reduces toiritimes theresidue oftheintegrandattheorigin,sothat /, Now11C^n-^^^i 2J(xw)diu 2711 JAn-^iWJ^{W) ^n+^i2J„(xw)dw I 2ci 4ci2c r<* —^^-r exp{—{\—x)\v\} dv CiAn (1—x)\Jx' and,from this result, (1)isevident;itisalsoevident that x^lt--^^Tn{t,x)dt-x^-^Jo tendsuniformlytozero asn-^xsolongas ^^.r^1—A. Itwillbeobserved thattheimportant expansion (4x^-^i=x\i^\'^y\, (0^^<1)m=lJm** v+i\Jm) which wasformallyobtained in§18'12, isanimmediateconsequenceof(1). Formula (2)canbeprovedinasomewhat similar manner(thoughthe details oftheproofarerather more elaborate) byusing anintegrand involving functions ofthesecond kind. Itiseasytoseethat rt''+'Tn(t,x)dt= S^''''^''^''Uf'>f':+^(J^-00)Jo w=l Jm" f+iiJm) B^x ^^JA„-Bi Jv{W) 18-22] FOURIER-BESSEL SERIES 587 Now take< ^it;^1inthelastintegral andsubstitute fortheBessel functions thedominant terms oftheirrespective asymptotic expansions,validwhen |w \ islarge (§7"21).The errorproduced therebyintheintegrand is,atmost, (l/^y-) when <a*^1;and, as«^^oo,wehave •An+'^ry, fl^\^^^^/J („_^/w^!~A,~ \n Now theresult ofsubstitutingthese dominant terms is .,fp<«+fi' sinw;(l—a:)sin(a?w—ii/TT—4:7r) , -hm -.^ ^~ '- ^-—-—*^dw li^^ TTIJAn-Bi WCOS{w—^VTT-ATt) =hm^r—.^ ^ ^^— ^ -^— ^—^^-^diu rAn+B' COS(2xw—w—^vTr-iTr) , Ji^y- JJn-Bi WCOS{w-^VTT-i Tt) Weshallhave todiscuss, almostimmediately,severalintegralsofthisgeneral type;soitisconvenient atthisstagetoprovealemmaconcerningtheir boundedness asn-^ cc . Lemma. Theintegral cos,Q\.w—\vtT-^n), [An+Bihm I ii-*-'x JAn-Bii;IVcos(w—hvir—jtt) is(l/n),as«-»-x,if-1<X<1; and theintegralisboundedif^\^1. Ifweputiv=A.n±iv, where J„,asusual, stands for(vi+iv+j)tt,theexpressionunder consideration maybewritten intheform ,.r,,,,, ,PcoshAy.^y •/^ i\ <PysinhX?'.c?y "l 2iJ,,cos(X-1)J,,. I--,-, :,T-.sm(X—1)/1„.I7,-;,— -^,-,— , L ju(J,-'+H)coshy^jo (.'l„2+?;2)coshi;J When—1<X<1,themoduhis ofthisdoesnotexceed 2 /"*coshXv.dv 2/""«'! sinhXv \dv AnJcosh V J„2Jcosh V' andthe firstpartoftheLemma isobvious. Again,if^X^1and«(1—X)=^,wehave* ^y(1—X)sinhX«=Isinh(y—^)^coshv, sotheintegraltobeconsidered doesnote.xceed(inabsolutevalue) •2A I i-2A I =277 andthesecondpartoftheLemma isproved. Itfollowsimmediatelyfrom theLemma that [%''+'Tn{t,a;)dt=la)''+ 0{l/n),Jo when <:v<1;andthis isequivalentto(2). *Thefunction ^sinh(i>-^) hasonemaximum, at|osay,and itsvalue there isequal to sinh^(v-^o)/cosh {v-fo)which islessthan sinh(v-to)- 588 THEORY OFBESSEL FUNCTIONS [CHAP. XVIII Moreover, ifweclose therangeofvalues ofxontherightsothat <a;^1, weinfer fi-om theLemma thattheintegrals ["v"^^Tn{t,w)dt, [V^'Tn{t,a:)dt Jo Jo arebounded as ??.-^oowhen <;»$1. Lastly weshall consider t''+'Tn(t,x)dt, andweshallprove that,when <^^1and<^'^1,thisintegralisa.bounded function ofn,xandt,asn-^co . Itiseasytoshewbythemethods which havejustbeen used that,when 1—x+t'^l,i.e.when t^x, then f t''+'Tn{t,x)dt i m=l Jin"v+iKjm) , 14-]- lim '-^. I>------ V-^,^.^,.w ,..,^^^=lim-^ {J„{w)Y^(xtu)—J„(xw)J\(w)] J5-*oo ^*JA„-Bi Jv(W) /1\ ,. f^'^ f''^«'+^' sinw(l—x).sin(tw-^vir- Iit) \AJ b^^TTi*'^ JA„-Bi wCOS{iu-lv'Tr-\ ir) „/l\ ,. V""^ f-^n+Bi COS(xw+tw—w—lvrr—^tt) ,= -r--hm „ '.,^ :-,^— ;r-^—-dw\AJ B^^^TTIX- JAn-Bi WCOSi^lV-\vTr-{tt) fv+h rAn+Bi(jQslyj^tW—XW—il/TT—Itt),+hm^r—r-,^ -. ,^ ,.""-'dw. s^ao lirix'! JA„-Biwcos{w—hi>'rr—\7r) Theseintegralsareofthetypeexamined intheLemmagivenearlier inthis section;and sotheoriginal integralisbounded when—1<j"+^—1^1 and—1<1+^—ic^l,i.e.when0<t^x^\. Toprovethattheintegralisbounded when <r»^^^1,wefirstshew that rt It''+'Tn{t,x)dt =lim*"'"i"'"^^'' '^-^ ^^^-^ ^^-^^^-^^J.{xw)dw B-*"x>-^p- {J^{w)F^+i(tw)-J^+i(tw)Y^{w)] "y,' ^*:An-Bi Jv{W) andthenapplythearguments justused inorder toapproximatetothe integralontheright ;thedetails oftheanalysisarelefttothereader. Ithasthereforebeenproved that, ifAbeanarbitrary positive number, then ft J^"+1Tn{t,x)dt<u, whereUisindependent ofn,xand tluhenA^o;^!, A^t^l. 18-23] FOURIER-BESSEL SERIES 589 These results constitute thenecessary preliminarytheoremsconcerning Tnit,x),andwearenow inapositiontodiscussintegrals, involving Tn(t,x), which occur intheinvestigationoftheFourier-Besselexpansionassociated withanarbitraryfunction/'(a?). 18*23. Theanalogue oftheRiemann-Lehesgue Lemma*. Weshallnowprove that, if{a,b)isanypartoftheclosed interval(0,1), such thatXisnotaninternalpointoranendpoint of(a,h),then theexistence andtheabsolute convergence ofb tif(t)dt aresufficienttoensure that, asn^cc, \'tf{t)Tn{t,x)dt=0{l),Ja 'where-f 0<.r ^\. Thereader willobserve that thistheorem asserts that theonlypartofthe pathofintegrationin ftf{t)Tn(t,x)dt Jo which isofanysignificance,asn-^oo,istheparti7itheimmediatevicinity of thepointx. Itisconvenient toprovethetheorem inthreestages.Itisfirstsupposed thatt^f{t)isbounded andthattheoriginisnotanendpointof(a,b).In thesecond stageweremove therestriction ofboundedness, and inthethird stageweremove therestrictionconcerningtheorigin. (I)Lett~''f(t)=F(t){f'-x% and lettheupperbound of\F{t)\in(a,b)beK.Divide(o,b)intopequal parts bythepoints t^,t2,...^^-i, (^o=c*,tp=b) ;and, afterchoosinganarbitrary positivenumbere,takeptobesolargethat pS(U>„-L„,) {t,n-t,n^^)<e, m=l whereUmandL,nJ^retheupperandlower bounds ofF(t)in(Y„,,_i, 1^)- Let F{t)=F{t,,^,) +co,,{t), soth?lt 10),„(t)\^U,n—Lm,in(^,„_i, tni)- It,is'then evident that (•!' P [tin tf(t)Tn{t,X)dt^XF(^,,_, )f+'Tn{t,X)(f'-of)dt Ja tn=\ Jt-m-i P I'f-m +i:t''+'Tn{t,x)(f'-x^)ay,n{t)dt,m=lJtm-i *Cf.Modern Anuli/sin, §9-41. tIfx=1,itis,ofcourse, supposed thatb<l. 590 THEORY OFBESSEL FUNCTIONS[CHAP. XVIII andhence, bytheinequalities (10)and(11)of§18'21, \\f{t)Tn{t,x)dtJaSci'Kp 7rC2^^„ (2—X—h)six + TTCo that istosaydt. tf(t)Tn(t,x)dt\^4ci- 2Kp+€ 7rc.^(2-x-b)^xlA, Now thechoice ofefixesp;when e(and thereforep)hasbeen chosen, we areatlibertytochoose AnsolargethatAn> ^Kpje. That istosay,bya suitable choice of^,1,wemaymake theintegralontheleftlessthan TTci(2—X~b)\lx' which isarbitrarilysmall.Consequentlytheintegralis(1)asA^-^00,and this isthetheorem tobeproved. (II)When F(t)isnotboundedthroughout (a,b),letitbepossibleto choose rintervals/x,such thatF(t)isbounded outside these intervals and such that "^fF(t)\dt<€. flJIX When tliesinoneoftheintervalsyu,weusetheinequality If^^{t^-X^)Tn(t,X)I<^''-- , TTCa^{2—X—b)^/x andhence, ifKistheupperbound of ii^(^) ]inthepartsof{a,b)outside the intervals/x,byapplying (I)toeach oftheseparts,wehave tf{t) Tn(t,x)dtSci" '(r+\)Kp+e irci(2—X—b)\/x Ifwetake esufficientlysmall(thus fixingK)andthen takeA^tobe sufficiently large,wecanmake theexpressionontheright (and therefore also theexpressionontheleft) arbitrarily small, andthis istheresult which had tobeproved. •b (III)Ift^-f{t)dtexists and isabsolutely convergent, wecanchoose77Jo sosmall that andthen, sincewehave rtf{t)Tn{t,x)dtJoV <dt<e, 4ci^ TTC^{2—X—b)i^xft--x^dt 18-24] FOURIER-BESSEL SERIES 591 itfollows from(II)that (r+l)Kp 3e" A,,+ 2' whereKistheupperbound of\F(t)\in(77,6)when theintervals/j,are omitted. Hence itfollows thattheexpressiononthe leftcanbemadearbitrarily small bytakingnsufficiently large,and sotheanalogueoftheRiemann- Lebesgue Lemma iscompletely proved. 18"24. TIteFourier-Besselexpansion. Weshallnowprovethefollowing theorem*, bymeans ofwhich thesum oftheFourier-Bessel expansionassociated withagiven function isdetermined : Letf{t)heafunction defined arbitrarilyin.theinterval (0,1);and let t~f{t)dtexistand(ifitisanimproper integral)letitheabsolutely convergent. 2 [^Let am=Yo7^-^ tf(t)J^ (j,nt)dt, 'J'v+\ \Jm)J where v+^^0. LetXheanyinternalpoint ofaninterval{a,b)such that0<a<h<1and suchthatf{t)haslimited totalfluctuationin(a,6). Then theseries 2a„iJw{jm^^m=l isconvergent and itssum ish,[/(^+0)+f{x—0)}. We firstobserve that,by§§18-21, 18-22, n n 2a,nJAjm^)=tf(t)Tn(t,x)dt,m=l Jo i{f(x- 0)+f(x+0)}=limx-\f(x-0)[%''+^T,,(t,x)dt +limx-'fix +0)(f+'Tn (t,x)dt. Hence, if 8n{X)= 1%"+^ [t-^f{t)-X~^f{x-0))r„{t,X)dt Jo +ft"^' [t-^fit)-X-^fix +O)}Tn{t,x)dt, itissufficient toprovethatSn{x)-*asw-*00inorder toestablish the convergenceof <x> ^(^m"f\Jm,X)m=l tothesum|{f(x+0)+f(x- 0)}. *Hobson, Proc. London Math. Sac.(2)vii.(1909), pp.387—388. 592 THEORY OFBESSEL FUNCTIONS [CHAP. XVIII Wenowdiscuss I\^^'U-'fit)-X--f{x +0)}Tn(t,X)dt indetail, andthereader cantheninvestigatetheotherintegralinvolved in Sn(x)inpreciselythesame manner. Thefunction t'"f{t)—x~''f(x +0)haslimited total fluctuation in(x,h), andsowemaywrite* r"/(t)-x-^f{x +0)=xi(0-X.{t), whereX\(0^^^1')(^{t)areboundedpositive increasingfunctions oftin{x,b), such that %i(^+0)=x.(^+0)=0. Hence, when anarbitrary positive number eischosen, there exists a positivenumber Snotexceedingb—x,such that whenever x^t^x+8. Wethenhave !\^+'[t-\f(t)-X-\f{x +0)]Tn(t,X)dt JX =fr+>[t-'fit) -X-^f{x+())}Tn{t,X)dt Jx+S +r' i"-^'%i(0Tn(t,x)dt-r^ t"^'x^(t)T„(t,X)dt. Wenowobtaininequalitiessatisfied bythethreeintegralsontheright. Itfollows from theanalogueoftheRiemann-Lebesgue lemma thatthe modulus ofthe firstcanbemade lessthan ebytaking nsufficiently large. Next, from thesecond mean-value theorem itfollows thatthere isanumber ^between and8such that ["'"^V^'X^(0Tn{t,X)dt=X.0^+8)f''"' «"+'Tn(t,X)dt, JX Jx+$ and,by§18'22, themodulus ofthisdoesnotexceed 2Ue;andsimilarlythe modulus ofthethirdintegraldoesnotexceed 2U6.Bytreatingtheintegral between thelimits andxinasimilar manner, wededuce that,bytakingn sufficient!}^ large,wecanmake thedifference between n SaraJAjmSc) and|{/(a;+0)+/(a;-0)} numericallylessthan (8t/+2)e;andthis isarbitrarilysmall. Hence, bythedefinition ofaninfinite series, wehaveproved that, inthe 00 circumstancespostulated, 2CbmJvijm^)isconvergentand itssum is M/(^+0)+/(^-0)}; andthis isthetheorem tobeproved. *Cf,ModernAnalysis, §3'64. 18-25] FOURIER-BESSEL SERIES 593 18*25. Theuniformity oftheconvergence oftheFourier-Besselexpansion. Let/(^) satisfytheconditions enunciated in§18' 24,and alsolet/(^)be continuous {maddition tohavinglimited totalfluctuation)intheinterval(a,h). Then theFourier-Bessel expansionassociatedwithf(t) converges uniformlyto thesumf(x) throughouttheinterval {a+A,i—A)whereAisanypositivenumber. Thistheorem isanalogoustotheusual theoremconcerning uniformityof convergenceofFourier series*;thediscussion oftheuniformityofthecon- vergenceoftheFourier-Besselexpansionnear^=1andnearx=requires, rather more careful consideration, inthefirstplacebecause formula118'22(1)is untrue whenx= 1,and inthesecondplacebecause itisnotpracticable toexamine thebounds of •f t''+'Tn(t,x)dt, ' whenXand taresmall, withoutusing approximationsforBessel functions of thesecond kind. The difficulties inthecase oftheneighbourhoodof^=1areeasytoover- come(cf.§18*26); butthe difficulties inthecase oftheneighbourhoodof theoriginareofagravercharacter;andthediscussion ofthem isdeferred to§18-55.' Weshallprovethetheoremconcerning uniformityofconvergence through- out(a+A,6—A)byarecapitulationoftheargumentsoftheprecedingsection. Inthe firstplace,sincecontinuityinvolvesuniformityofcontinuityf,the choice of8which wasmade in§18'25 isindependentofxwhen xliesin (a+A,6-A). Nextwediscuss suchanintegralas- . ., , r t"^^[t-\f{t)-c^-\f\x)] Tn{t,x)dt ' Jx+S- Since Bisindependentofx,itfollows from theproofoftheRiemann- Lebesgue lemma(§18-23) that thisintegraltends tozerouniformlyasn-*oc, providedthat rt''^'^{t-''f{t)-x-''f{x)]dt isabounded function ofx.^ Now ft^-^^t'^f{t)-x--f{x)]dt %C\t'^f{t)\dt+\x--f(x)\Ct'+'^dt,J:xr^'Jo' Jo and this isbounded in(a-f-A,6—A)since/(*•)iscontinuous andtherefore bounded inthis interval. *Cf.ModernAnalysis, §9-44. tCf.ModernAnalysis, §3'61. Itisnowconvenient toplace anadditional(trivial)restriction on5,namely that itshould belessthanA,inorder that theinterval {x- 5,x+d)maylieinside theinterval{a,b). w.B.F. 38 594 THEORY OFBESSEL FUNCTIONS [CHAP.XVIII Similarlytheotherintegralsintroduced in§18'24 tend tozerouniformly, andso n tends tozerouniformlyas ?i^x,andthisprovesthetheorem stated. 18'26. Theuniformity oftheconvergence oftheFourier-Bessel expansion near a;=1. Itisevident that alltheterms oftheFourier-Besselexpansionvanish at thepointa;=1,sothat, atthatpoint,thesum oftheterms oftheexpansion iszero. Sinceuniformityofconvergenceofaseries ofcontinuous functions involves thecontinuityofthesum, itisevident thatthecondition /(l-0)= isnecessaryinorder that theconvergenceoftheFourier-Besselexpansion associated withf{t)maybeuniform nearx=l. We shallnowprovethat theconditions thatf(a;)istobecontinuous in (a,1)andthat/(I)iszero,combined with theconditions stated* in§18"24, aresuffi,cientfortheconvergencetobeuniformthroughout (a-I-A,1). Theanalysisisalmost identical with thatoftheprecedingsection;wetake \''+'\t-''f(t)-w-''f{x)] Tn{t,cc)dt,IJ justasbefore, andwethen divide theinterval(0,1)either intothreeparts (0,X—B),(x—B,x+B),{x+ B,1),ifa;^1—S,orintotwoparts (0,x—B), (x— B,1),ifx"^!—B.Andwethenprovethatthethreeintegrals (orthetwo integrals,asthecasemaybe)tenduniformlytozero. Again, when /(I)=0,wecanchoose Bisothat when 1—Bi^x ^1. Then theexpression f{x)-x-''f{x)(\''+^Tn{t,x)Jodt tendsuniformlytozerof asw-*-oo when xliesin(a+A,1—8i),andthe expressiondoes notexceed (?7-|-l)eforanyvalue ofwwhen xliesin (1-Si,l). *The interval(a,b)is,ofcourse, tobereplaced bytheinterval(a,1). tBecause theintegral involved tends tox"uniformly throughout (A,1-5i),by§18-21. 18-26, 18-27] FOURIER-BESSEL SERIES 595 Hence wecanmake /(a;)-x-"f(x)!1"+'Tn(t,x)dt Jo arbitrarilysmall for allvalues ofxin{a+A,1)byachoice ofnwhich is independentofx;and thisestablishes theuniformityoftheconvergenceof tf(t)Tn(t,x)dt tothesum/(a;)in(a+A,1)inthepostulatedcircumstances. 18•27.The order-ofmagnitude oftheterms intheFourier-Bessel series. Itiseasytoprove that, ift'-f{t)haslimited totalfluctuationin(a,b), luhere(a,b)isanypart {orthewhole) oftheinterval(0,1),then ^\f{t)J^{\t)dt= oQ-^ asX,-*00 . From thistheorem weatonce obtainSheppard'sresult* that 2J,(jmX) J' \'tf{t)J.{jmt)dt=0ll-)-'0 \Jm/ v-ir\\Jm) .'o" '" Vjr when <ic^1;thisequation,ofcourse, hasawell-knownparallelinthe theoryofFourier series. Wefirstobserve that, asaconsequenceoftheasymptotic expansionof§7'21, t'^J, (t)dt<c, .Ja where cisaconstant, independentoftwhen tliesintheinterval(0,qo). Now writet^f(t)=\jr^{t)— yjr„(t),v!here-\|ri(<)and-v/^o(0^^^monotonic in{a,b);andthen anumber ^exists such that iri{t)tKT^{Xt)dt -(/ti{a)\t^J^(Xt)dt+yfr,(b) jt^J^(Xt)dt <2c{\^|r,{a)\ +\^|r,{b)\}X-^ =OiX-^). . Asimilar result holds forx/rg(t),andhence thetheorem stated isevident. Ifitisknownmerelythat- ''' • ^\if(t)dt-^ exists and isabsolutely convergent,then allthat canbeprovedisthe theorem that ..:...,'.. tf{t)J,(Xt)dt=o(ll^/X).•• Ja Quarterly Journal, xxiii.(1889), p.247. 38—2 596 THEORY OFBESSEL FUNCTIONS[CHAP. XVIII Thistheorem isduetoW.H.Young*, and itmaybeprovedinprecisely thesame manner asthetheorem of§18'23.Weshall write outtheproof when jt^f(t)iisbounded, withupper bound K,andleave thereader tocon- struct theproof,when thefunction isunbounded, onthelines of|1823. Divide(a,b)intopequal parts bythepoints ^i ,t.^,...,tp^^ {to=a,tp=b), and letthepartsbesonumerous that m=l where UmandZ,„aretheupperandlower bounds oft^f{t)in(^,„_i, t^n)- Next lett-f{t)=F{t), F{t)=F{t,,_,)¥ co,n{t), andthen \'tf{t)J,{\t)dt <irl|r" t^-J,{\t)dtW i['"'\tU,{\t)<Om{t)\dt where c'istheupper bound of\t^J^(t)\intheinterval(0,oo).Hence, by reasoning resemblingthatused in§18*23, theintegralontheleft iso(X~-^), andthis isthetheorem tobeproved. Thetheorems ofthis section canbemade tocover theclosed interval (O^a?^ 1)intheforms ;f/;./(07,(>o.^«={„%^">'•^v+i\Jm) This isevident when itisremembered that Hence thegeneral term intheFourier-Bessel series associatedwith'f{x) tends tozero{after midtiplication hy\/x)throughouttheinter'val (0$a;$1) if^'/i'"^)f^o,^ o.nintegral which isabsolutely convergent; and,ifthisfunction haslimited totalfluctuation,thegeneral term tends tozeroasrapidlyasl/jm- 18*3. Theapplication oftheHankel-Schlafii methods toDini'sexpansion. Weshallnowconsider aclass ofcontourintegrals bymeans ofwhich we canobtain theoremsconcerningDini'sexpansion, analogoustothose which havebeenprovedforFourier-Besselexpansions,either inadirect manner or bymeans ofthecorrespondingtheorems forFourier-Besselexpansions. TheDiniexpansionassociated with/(a;)is 00 Z(O^JpyKmOC), 7n=\ where Xi,X2, ^3,...are thepositivezeros(arrangedinascendingorder of magnitude)ofthefunction zJ,'{z)+HJ,{z), *Proc.London Math. Soc.(2)xviii.(1920), pp.169—171. 18-3] DINISERIES 597 whereUandvarerealconstants, and The coefficients h^aretobedeterminedbytheformula 'in1. n tJ;-(X,J)(it=t/{t)J^(\J)(it Jo 2\J!tf{t)J,(\nt)dtJosothat Beforeproceeding further, weshallexplainaphenomenon, peculiartocertain Diniexpansions,which hasnoanalogueinthetheoryofFourier-Bessel expansions. TheinvestigationofDiniexpansionsisbased onpropertiesofafunction which haspolesatthezeros of' and,whenH+v=0,this lastfunctionhasazero attheorigin. Further, \iH+v isnegative,thefunction hastwopurely imaginaryzeros. Itisonlytobeexpectedthatthese zeros should contribute totheterms oftheseries, andsuchacontribution infact ismade. liH-^v=0,aninitial term" (1)" 2{v^l)x^\\''+'f{t)dtJo :....•;, hastobeinserted onaccount ofthezeroattheorigin. IfH+v isnegative andthepurely imaginaryzeros are±i\n, thenan initial term''' ' " '- • {\r+ir)1^-(A,o)-V^.-" (^o) J must beinserted onaccount ofthezeros 4-i\0- These initial terms intherespectivecases willbedenotedbythecommon symbol ^^{x),sothattheseries which willactuallybeconsidered is mi=1 wh^re J^Q^x)iszerowhenH+vispositiveand isdefined astheexpression (1)or(2)intherespectivecasesH+v—0,H+p<0. [Note.The factthataninitial termmust beinserted whenH+v=wasnoticedby Dini, Serie diFourier(Pisa, 1880), p.268,butDinigaveitsvakieincorrectly,thefactor x"being omitted. Dini's formula wasmisquoted byNielsen, Handhuch derTheorie der Cylinderfwnktionen (Leipzig, 1904), p.354.Forcorrections ofthese errors, seeBridgenian, Phil.Mag. (6)XVI.(1908), pp.947—948; Chree, PJul.Mag. (6)xvir.(1909), pp.329—331; andC.N.Moore, Trans. American Math. ^oc. x.(1909), pp.419—420.] 598 THEORY OFBESSEL FUNCTIONS[CHAP.XVIII Wenowconsider thefunction '2wJ„ (xw)J^(tw) J^iw) [wJJ (w)+HJ^ {w)]' This function haspoles Sitji,J2,J3, ...,Xj,X^,X3,...,(0or±iXq). Theresidue ofthefunction atJ,^is Theresidue atX,„is 2X„jJ^(X^a;)(/^(X^ t) Jv(^m)V^mJv (X„i)+Jy'(X,„)4-HJy (X^)} (X^'^-V^)J„^(X„i)+X„i-Jy- (X,rt)* Theresidue attheorigin when IT+2/= is| Theresidues at+iX^when if+yisnegativearebothequalto 2Xo^7,(Xoar)/,(XoO (Xo"-Vv-) /„-(Xo)- X(,-/^'- (Xo)' Now letDnbeanumber, which liesbetween X^andX„+i, sochosen that itisnotequaltoanyofthenumbers j^ ;and let^*jvbethegreatestofthe numbersj„iwhich doesnotexceed D^^. LetSn{t,x-H)=l ^MI^f}Z^mil_^^(^^^)m=l«^v+i\Jm) whereS4o{oc,t)isdefined tobe0,2(1/+1)^"^" or 2\^I,{\x)h{\,t) (Xo-*4-V-)IJ"(Xo)- Xo'*//^(Xo)' accordingasif+yispositive,zeroornegative. Then, evidently, iV n n 1 2at„,Jy(jm^)-^oi^)- 2bmJA'^mX)= tf(t)Sn{t,x;H)dt.m=l m=l •'0 Weshallnowproveanumber oftheoremsleading uptotheresult that,when <^< 1,theexistence andabsoluteconvergenceof 'tV(t)dt I aresufficient toensure that, asn^^cc, Itf{t)Sn(t,x; H)dt=o(l). 18-31, 18-32]DINTSERIES 599 Thisequationenables ustodeduce thepropertiesofDini's series inrespect ofconvergence*from thecorresponding propertiesoftheFourier-Bessel series. 18-31. Thecontour integral forSn{t,x\ H). Itisevident fromCauchy's theoryofresidues that 1CDn+^i 2wJ„{xw)J„{tw)dw 1p*2iuJ„(xiu)/„(tw)diu '2'iri J-QoiJ„(iv)[wJJ {w)+HJ^ {w)\' where thesymbolPdenotes Cauchy's' principalvalue.' Theintegrand being anoddfunction ofw,thesecondintegral vanishes, andsowehave n\ ^(t1cBn+^i2wJ,{xw)J,{tw)dw Animmediate consequenceofthisformula(cf.§18-21)isthat c (2) |S.((,,;J)|^____i___, where Cjisindependentofn,xand t. Also l\^^^8 (tx-mdt-^^r^^+-'- 2Mxw)J^^,itio)dw andhence IJt''+^Sn{t,x; H)dt I where c^isindependentofn,xand t.{2-x-t)Bn^/x' 18*32. Theanalogue for Sn,{t, x;H)oftheRiemann-Lebesguelemma. Weshallnowprovethetheorem that, if(a,b)isanypart {ortheivhole) oftheinterval(0,1),then theexistence andabsoluteconvergence of '\if(t)dt.:.:-.; Ja aresufficienttoensu7'e that, asn-^<x, tf{t)Sn{t,x; H)dt=o{l), providedthat<x< 1.And, ifb<1,thetheorem isvalid when <^'$1. Theproofhastobedivided intothreestages justasinthecorresponding- theorem(§18-23) forTn{t,x).Weshallnowgivetheproofofthe firststage, when itissupposedthatt-f(t)isbounded anda>0. Theproofsofthere- maining stagesshould beconstructed bythereader withoutdifficulty. *Except atthepoint a;=l.ll 600 THEORY OFBESSEL FUNCTIONS[CHAP. XVIII Lett-''f(t)=F{t), and lettheupper bound of jF(t) \in(a,h)heK. Divide (a,b)intopequal partsbythepoints ^i ,^2,•••,^j>-i (^0=a,tp=h), and, afterchoosinganarbitrar}^ positive numbere,takeptobesolargethat ^{Urn—Lm){t^n—tm-i)< e,,m=l where 17^andLmaretheupperandlower bounds ofF{t)in(^,„_i, ^^)., LetF{t)=F{t,^,) +<o„,{t), sothat Ia),n(t) \^U^i-Z„,in(tm-i, tm). Then j\f{t)Sn(t,x:H)dtJa =^F(t,n^,)t"-*-'Sn(t,x;H)dt+:i r+1CO,,(t)Sn(t,x;H)dtm=l Jtm-i m=lJt.m~\ Hence, by§18-31, tf{t)Sn(t,x;H)dt (2—x—h)\/x l)rec. and ifwenowtakensolargethatDnec^>2Kp c^,wehave [\f{t) S,,{t,X ;E)dt !<^''), Ja\{2—x—b)-Jx andtheexpressionontherightisarbitrarilysmall. Hence theintegralon theleft.tends tozeroas71-*-00 . When thereader hasremoved therestrictionsconcerning boundedness andthemagnitudeofabythemethod of§18*23, thetheorem iscompletely proved. Asacorollary,itshould beobserved that rh x^tf{t)Sn(t,x;H)dt Ja tendsuniformlytozeroasw-*-00when O^aj'^l if6<l, andwhen ^a;^1—A if6^1,whereAisanarbitrary positivenumber. 18*33. Dini'sexpansion ofanarbitrary function. Animmediateconsequenceoftheresult oftheprecedingsection isthat theexistence andabsoluteconvergenceoftheintegral \^f{t)dt aresufficient toensure thattheDiniexpansionassociated with/(.r)behaves inthesame manner, asregards convergence (orsummability),astheFourier- Besselexpansion throughouttheinterval {0<x< 1). 18-33] DINISERIES 601 For itisevident that m=1 m=l tends tozeroasw^xwhen <x<1 ;andthissum(multiplied by\/x)tends uniformlytozerowhen^^a'^1—A. Now, since thenumbers X„,,andj,^which exceed |v \areinterlaced (§15"23),itfollows thatDnmaybechosen sothatn-Nhasthesame value forallvalues ofnafter acertainstage. Therefore, since n m-N+l uniformly throughout (0,1),wehaveprovedthat n XMo(x)+ Sx^{b,nJi>(\na;)-a^nJAjm^)}m=l tends tozero, asn-^oc,uniformly throughout (0,1—A).' '' ' That istosay,theseries CO m=l isuniformly convergent throughout (0,1—A)and itssum iszero. Itfollows from the'consistencytheorems'concerning convergentseries* that,when theseries is'summed' byCesaro's means, oranysimilar method, it is(uniformly) summable and its'sum' iszero. Hence,if,foranyparticularvalueofxintheinterval(0,'1—A),theseries .iwx-a-Dio I,\jviX),-. , . m=1. ' associated withf{x),isconvergent (orissummable bysome method),then the series-^ . ... . ' "'• .- -. :''' xiMo{x)+ SxibmJy(\mX)m=l isconvergent (orissummablebythesame method) andthetwoseries have the same'sum! Andif,further,theFourier-Bessel series (mtdtiplied by\/x)isuniformly convergent (oruniformly summable) throughoutaninterval (a,b),ivhere 0^a<b<l,: .,.' ; then also theDini series(mtdtiplied by\Jx)isuniformly convergent (oruni- formly summable) throughout (a,b). Inparticular, y^f(x)haslimited total fluctuation in(a,b)where -' ..'O^a<6< 1,/ *Cf.liiomwich, Theory ofInfinite Series, §100. 602 THEORY OFBESSEL FUNCTIONS[CHAP. XVIU then theseries w=l convergestothesum if/(^+o)+/(^-o)} atall'pointsxsuch thata+A^x^b —A,whereAisarbitrarilysmall;and theconvergenceisuniform iff{x)iscontinuous in{a,b). 18*34. Thevalue ofDini's series atx=1. Weshallnowcompletetheinvestigationofthevalue ofthesum ofDini's seriesbyconsideringthepointx=l; andweshallprovethetheorem, dueto Hobson*, that, if/(a;)haslimited total fluctuation intheinterval(a,1),thesum oftheDiniexpansionatir=1is/(I—0). We firstwrite Tr,{t,x; H)=T,,{t,x)-Sn(t,x; H) — <iyt'Q[^tC,tJ-f— .2 ox^2/^ \ I-v""«T'2i-\\'m=\y'^m~V-)Jt,\A^)+f^m''JVK'^n) andthenwehave ,I j-Ai+"'•^(^^sc)J^(tw)dw I«: T„{t,x-H).27ri JBn-ooiwJJ(w)+HJ^(w)' 'Dn+^ilU^ (W,t)J,.ixw)dw ,iwJJ (w)+HJy{w)'1rJJn+ where (}>(lu, a;)sTT[{wJJ (w)+HJ^(w)}Y^(xw)-[wF/(w)+HY, (w)}J,(xw)]. Theformerrepresentationof2'n(t,x; H)isvalidwhenQ<t<x^\, the latterwhen0<x<t^\. [Note. TheserepresentationsofT„{t,x\H)arestrictly analogoustotherepresenta- tions ofTn{t,x)givenby§18-21(7)and^51821(8) ;thefactthatthere isnoformula for Tn (t,X;R)analogousto§18*21(6)isthereason whyDini series were discussed in§18'33 with thehelpofthetheoryofFourier-Besselseries.] Now consider thevalue of 't t-^'Tn{t,l\ H)dt when <^<1.WehaveJo ^.+1cDn+Bi ^(^^fj^ l)J,+,{tyj)dwt"^^Tn{t,1;H)dt=liraf—.^-^^ im(iv)+HJ^{w) ^^'+lrD„+BiJ^^^(tw)dw jB^x7riJD^-BiwJJ {w)+HJ^{w) "tProc.London Math. Soc.(2)vii,(1909), p.388. 18-34] DINISERIES 603 # Foranygiven positivevalue of8,itfollows from§18'21 that this isa bounded function oftintheinterval (8,1).When 8%t^1—8,it is(1/Z)„). Andwhen ^=1,ithasthelimit 1whenn^oo . Itfollows that .'^„(1)+ ib,„J^(X,„)-f{l-0)=[\''+^{t-'f(t)-f{l-0)] Tn{t,\; H)dt.m=\ Jo Since t'"f{t)—f{1—0)haslimited total fluctuation in(a,1)wemay write itintheform^^(t)—^2(0>wherex^(0^^^clx^(0'^^^boundedpositive decreasingfunctions oftsuch that Hence, givenanarbitrary positive numbere,wecanchoose apositive number 8,notexceeding1—a,such that whenever 1-8^1-^1. Wethenhave [V+> {t-\f(f)-f(l-0)] Tn{t,\-H)dt. .'0 ., =|''V-^irv'(0-/(l-0)1Tn{t,\; H)dt Jo +rt^+'x^{t)Tn{t,l;H)dt-C t''^^x2ii)Tn{t,l;H)dt. J1-6 JIS Byargumentssimilar tothose used in§18-24, the firstintegralonthe rightiso(l)asn^X);andneither thesecond northethird exceeds 2elira 1*1"+' Tnit,l; H)dt \J inabsolute value(cf§18'24), andthisexpressionisarbitrarilysmall. Itfollows that -•'..• - . Urnrt''+'{t-''f{t)-f{l-0)]T,,(t,l;H)dt=0, n-^x J andsowehaveproved that, inthecircumstancespostulatedatthebeginning ofthissection,.,.."- m=l convergestothesum/(I—0). Thisdiscrepancybetween thebehaviours ofDini series and ofFourier- Bessel series(§18*26)issomewhat remarkable. 604 THEORY OFBESSEL FUNCTIONS[CHAP. XVIH 18'35.Theuniformity oftheconvergence ofDini'sexpansioninaninterval extendingtox=l. Because Dini series donotvanishidenticallyata;=1,itseems notunlikely that thecondition thatf{x)iscontinuous^ in(a, 1),combined with the existence andabsoluteconvergenceof \^f(t)dt, andthecondition thatf(x)haslimited total fluctuation in(a,1),maybe sufficient toensure theuniformityoftheconvergenceoftheDiniexpansion in(a+A,1). Weshallprovethat this is,infact,thecase. Thereason forthefailure intheuniformityoftheconvergenceofthe Fourier-Besselexpansion (§18"26) nearx=1wasthefactthat ft''+-'Tn(t,x)dtJO doesnotconverge uniformlytox"in(A, 1),aswasseen in§18'22.Weshall prove that,onthecontrary, ft''+'Tn(t,x:H)dtJo doesconverge uniforml}'tox"in(A, 1),andthecause ofthe failure is removed. Aconsideration of§18'26 should then enable thereader toseewithout difficultythattheDiniexpansion converges uniformlyin(a+A,1). Itiseasytosee,from§18'34, that [t''+'T„(t,x:H)dtJo isthesumoftheresidues of 77^"+![{wj;(w)+HJ, {w)] F,+,(tw)-[wY:(w)+HY, (tu)}J,+,(tw)] XJ^{xio)l{wJJ (w)+HJ^ (w)], atXj,Xo,...,X„ ,plushalftheresidues at or+iX^ifH+v^O. Hence[f^'T^(t,x;H)dt Jo isthesumoftheresidues of -{21w)(H+v)J,{xio)/{ivj; (w)+HJ, {tu)], andhence, when <j-^\, H+VC^^n+^i J^{xw) div t''+'T^{t,x; H)dt=x''-"^^ ,^; {l0)+HJ,{lv)y *Without restriction outhevalue of/(1-0). 18-35, 18-4] DINISERIES 605 andtheintegrandontheright*isoftheorder ofmagnitudeof exp{—{l—x)\ I{w) \] 10-sjx andsotheintegralontheright converges uniformlytozero likel/(Dn\/a;) whenA^iz;?;1.That istosay \''+'Tn{t,a;;H)dt converges uniformlytox"in(A,1);andwehavejustseen that this isasuf- ficient condition fortheuniformityoftheconvergenceoftheDini series associated withf{t)tothesum/(^)in{a+A,1)under theconditionspostu- latedconcerning /(O. 18'4. Thedifferentiabilitij ofFoitrier-Besselexpansions. Intheearlierpartofthischapter weobtained anexpansion which, when written infull,assumes theform (1) /(^)= SarrJ„{jm,uX)-" ' . . We shallnowstudythecircumstances inwhich, giventhisexpansion,it ispermissibletodeduce that (2) f(x)=ta.mjm, uJv (im,v«)•m=\ Thisproblemwasexamined byFordf, and hisinvestigationisanalogous toStokes' researches onthedifferentiabilityofFourier series :|:. Ford alsoinvestigatedthedifferentiabilityofDini'sexpansion whenR—-v,buthis method isnotapplicabletoother values ofH. Itisevident thatwecanprovethetruth of(2)ifwecansucceed in provingthat (3) /(^)--/(^')=-Sa,J„,,.J.+i(jm,.^y,X m=l andthenumbersJ,„,^arethepositivezeros of Nowweknowthat/'(.r) —(v/x)f{x)admits oftheDiniexpansion 00 iO^n'Jv+1\Jm,v ^)m=l inside anyinterval inwhich thefunction haslimited fluctuation, providedthat jy[f'{t)-jf(t)\dt exists and isabsolutely convergent. *Theterm inwJ„'(lo)ismore important than theterm inJ„(iv) except inthelimitwhenH isinfinite;thisshewsclearly thereason forthedifference inthebehaviour oftheDiniexpansion from that oftheFourier-Bessel expansion (of.§18-2()). tTraiis. American Math. Soc. iv.(1903), pp.178—184.+Cf.ModernAnalysis, §9-31. 606 THEORY OFBESSEL FUNCTIONS The coefficients bmaregiven bytheformula 2j^«,.I\tf (t)-vfit)} J-.+i (>,,,dt J[chap, xvni h,„.= 2fd 'J''v+i{jm,v)Jdt 2[t-^f{t)\t^^'J,^,{j,,,,t)dt <J^v+l (jm, v)[_ ^^Jm,V(^m >if{t)Ju+l(jm.ut) Jm,Itf(t)JAjm,.t)dt providedthat =0. Sufficient conditions that thismaybethecaseare (i)^"^-yXO^Oast^O, (ii)/(l-0)=0, (iii)f{t)iscontinuous intheopeninterval inwhich <^<1. These conditions combined with theexistence andabsoluteconvergenceof 'y'^j^[t-^f{t)]dt aresufficient toensure thetruth of(2)inanyinterval inwhich /'{x)-(vlx)f(x) haslimited total fluctuation. 18*5. Thesummahility ofFourier-Bessel series. Aconsideration ofthevalues ofthe coefficients intheFourier-Bessel series associated withf{x),combined withtheexpressionofT^^(t,x)asacontour integral, suggeststhat itisnoeasymatter todiscussbydirect methods the questionofthesummability,byCesaro's means, oftheFourier-Besselexpansion. Itis,however, veryeasytoinvestigatethesummability when themethod ofRiesz* isused to'sura' theseries, andthen thesummability (CI) canbe inferred with thehelpofquite elementary analysis. Theexpressionwhich willbetaken asthe'sum' oftheseriesbythe method ofRiesz is lim2 (1-T^)O'mJvijmOo); andwhen this limit exists, theFourier-Bessel series willbesaid tobesum- viable(R). Itisevident that (1) i(l-^)ar>,JAj,n^)=f\f(t)Tn(t,x\R)dt,' m=l \ -^n' JO Cf.Hardy, Proc.London Math. Soc.(2)vui.(1910), p.309. 18-5, 18-51] FOURIER-BESSEL SERIES 607 where (2) T^.{t,x\R)= i(i^fy-±^^p^^}^ andsoitwillbeconvenient todiscuss thepropertiesofT„(t,a; \R)after the manner of§18"22 before wemake furtherprogresswith themainproblem. 18'51. Theorems concerning Tn(t,x \R). When Tn{t,x\R)isdefined byequation (2)of§18'5, itisasymmetric function oftand x,andsoweshallproceedtoestablish thepropertiesofthe function onthehypothesisthat^t^x^l, andwecanthen writedown the corresponding properties when ^.r^^^1byinterchangingtandxinthe resultsalreadyobtained. We firstobserve thatTn(t,x \R)isthesum oftheresidues of J^{tw)TTW (1-£){/.{w)F,{xw)-J-,{xw)F,(w)]y^ ^"Ji)J2}Jsf'•'>Jn- Forbrevity wewrite w[J^(w)Yy(xiv)—J^,{xw)Fy{lu)]=^(w,x), andthen itisobvious that,when* t<x, ,1f^'^/ .Ju{tw)dw since 4>(w, a;)J'^(^w;)/J'p(t{;)isanoddfunction ofw,r Weshall now^obtain someupper bounds for j4>{w,x)Jy(tw)/Jy (w)\ bothwhenwisonthelinejoiningAn— ooitoAn+c»i,andwhenwisonthe imaginaryaxis;theformulae which willbediscussed arevalidwhen ^ic$1 and ^^^1,thesignofa:—^beingimmaterial. Toobtain theseinequalities, weshall useseries ofascending powersoflu when Iw|isnotlarge, andinequalitiesderived from theformulae ofChaptervii when I«; 1isnotsmall. •..-.'':•• ; *When t5;x,theintegrals taken along thelines joining±ii?toJ„±iBdonottend tozero asB-*00 .There isnoneed tomake anindentation attheorigin, because ${w,x)isanalytic at theorigin. ... 608 THEORY orBESSEL FUNCTIONS [chap,xvm We firstdealwith thefactorJt,{tw)/J^{w). Weobserve that* (1)Jy(tw) l<^^exp{-(l-01/(t^)|l whenwisoneither contour; this follows frominequalitiesofthetype §18"21(9)when |w[isnotsmall, andfrom theascendingserieswhen\w\is notlarge (i.e.lessthanj^. Wenext consider O(w,x),which isequalto liw {H,^'^ {w) F,'--^' {ayw)-^,"'(xw)H,^^ {w)] ; itisconvenient tomake twoinvestigations concerningthis function, the former beingvalidwhen—^^v^^, thesecond whenv^^. (I)The firstinvestigationisquite simple.Itfollows from§3*6and S7-33that. - - (2) ^,<i)(xw) I< \xw 1^,\Hy'»^{xw)\<L Ie-^''"' xiu\^ forallthevalues ofwandxunder consideration when—h^v^^.Hence (3) O(w,x) I<-~~exp {(1-x)\I(w)\ ]. (II)When V^iand |w |isnotlarge,itiseasytodeduce from the ascendingseries forJy(w),Y^(lu), J^,{xiv)andY^(xw) that (4) \<t>{w,x)\<k3\w\x'". If Iw Iisnotsmall, weusetheinequalities (deduced from§7"33) (5)ii7."' (to)\<h Ie^' w\^.<2»(w) I<h\e-IW W (6)gtXW I o—ixw Itogetherwith theinequalities ir^»)(xw)\<ki{\xiv r*+ Ixwj-"} !e« ^^'2'(xw)\<k^{\xw\-i+\xw j-"} Ie- Itfullows from§3*6and§7*33 that theinequalities (6)aretruewhether \xw\islargeornot.Hence, - (7)I^(w, x) I<k^ki{x-i+a;-" IwI*-"}exp{(1-x)\I(w)\ ], when V^|and\w\islarge,whatever bethemagnitudefof jxw|. Ifwenowcombine theresults contained informulae(3),(4)and(7)we deduce that,whether —h-^v^Jory^^, (8)4>(w,x)\<k, (x-^+x-")exp {(1-x)^I (w)i }, *Itissupposed thatthenumbers Aj, A-2,A3,...arepositive andindependent oflo,xand t\ their values may, however, depend onthevalue ofv, tProvided ofcourse that<x^1. 18-51] FOURIER-BESSEL SERIES 609 whenwisanypointofeither contour and<a;^1.Hence, by(1),itfollows that (9) ^{%v,x/-j~^j'"^\<ht-Hx-^ +x-'')ex^[-{x-t)\I{iu)\\, when0^x^\ andO^^^l. Wenowreturn totheintegralformula forT„{t,x \R).Ifwereplace wby An±ivand+ivinthe firstandsecond contourintegrals respectively, we deduce that,when ^t<x^l, Ir.ft. IiJ) i<-5^(..-.+.-0/;.e— cfo= ^l^^^^^ilf Wehaveconsequently provedthetwoinequalities (10)|r„(,,|^),<2Ai^^i±iC^(o,*<.,i), (11)^TAt..m<J§^,(o«.<««i). Itistoberemembered that k^isindependentofxandt,sothatwemay make |a;—^ |tend tozero, ifwedesire todoso. Oneotherpairofinequalitiesisrequiredinorder todiscuss thebehaviour ojfTn{t,X IR)when xand tarenearly equal.Toobtain them,wewrite 2'..(«,.,iJ)=2.j(l-^J*(»,.)-^-,. when ^t^xi^l; inthisintegralthecontour istaken tobearectanglewith vertices +iAn,A^±iA^- Itiseasytoseethat (9)issatisfied whether tvbeonthehorizontal sides oronthevertical sides ofthisrectangle;andthefactor 1—{w/An) doesnot exceed \/2inabsolute value atanypointofthecontour. Consequentlythemodulus oftheintegranddoesnotexceed kst-^ix-^ +x-")^-!; andsince thelengthofthecontour isQA„,weinfer that,when0^^^^^ 1, (12)• iT.,,Mm<'-^^^^^i^. andsimilarly, when ^x-^t^1, (13) \T,,(t,x\R)\< -^^^^-^. The lastfourinequalitiesaresufficient toenable ustodiscussadequately thesummability (R)ofFourier-Bessel series. Thereader willobserve that theconsideration ofsmall values ofxhasincreased thelengthoftheanalysis toanappreciablebutnottoanundue extent. W.B.F. 39 610 THEORY OFBESSEL FUNCTIONS [CHAP. XVni 18'52. Theanalogue ofFejer'stheorem. Wecannowprovethat theexistence and theabsoluteconvergence of Ciif{t)dt Jo aresufficienttoensm'e that theFourier-Bessel series associated withf{t)is summable (R)atallpointsxoftheopeninterval (0,1)atwhich thetwolimits f{x±0)exist.And thesum(R)oftheseries is' i{/(^+0)+/(^-0)}. Thistheorem isobviouslytheanalogueofFejer's theorem*concerningFourier series. Since faseries which isconvergentissummable (R),itfollows from§18'35 that,when <a;<1, limrt''+'Tn(t,x\R)dt=lim [t"-*-'Tn{t,x\R)dt —2• Hence itfollows that,when thelimits/'(ir+0)exist, then lim rt"-^'Tn(t,x\R)x-''f{x-0)dt+limft-+'Tn(t,x\R)x-^f(x +0)dt =|{/(^+0)+/(^-0)}. Wearenow inapositiontoconsider thesum>S^„{x1R),defined as I(l-Jf) amJ.Umx)- I^^"-^^T,(t,X IR)x-''f{x-0)dt m=l\ -^n/ Jo -\t''+'Tn(t,x\R)x-''f{x +0)dt, andweshallprovethat itcanbemadearbitrarilysmallbytaking?tsufficiently large. ThesumSn{x\R)isequalto [%'+'[t--f{t)-x--f{x-0)}Tn(t,x\R)dtJo +I*"+'[t-'fit)-x-^f{x +0)}Tn(t,X IR)dt JX Now,onthehypothesisthatthelimits/(ic+0)exist, ifwechoose anarbitrary positive numbere,there exists apositive number^8such that i\t-'fit)-x-^f{x +0) I<e, {x^t^x^-l), li«~''/(0-^""/(^-0)I<e, {x-^t^x-h). Wenowchoose apositivefunction ofn,say<r(n),which islessthan Sfor sufficiently large values ofn,anddivide theinterval(0,1)into sixpartsby thepointsx±8,x ±(t(n),x. *Cf.ModernAnulysis, §9-4. tCf.ModernAnalysis, §8-43. iItisconvenient totake 6lessthanxand 1-a;. 18*52] FOURIER-BESSEL SERIES 611 Intheintervals (0,x—h),{x—h,x—a{n))and also intheintervals {x+a(n),X+B),(x+8,1)weuseinequalitiesoftheformgivenin§18'51(10) and(11);andintheintervals (x—a(n), x),{x,x+a-(w))weuseinequalities oftheformgivenin§18-51(12)and(13). Itisthusfound that |Sn{x|R)Idoesnotexceed 2ks€(x-^+x-")+ +-77^ f+idt An^/xA-n \_JX-& {x—ty \J2Jx-a(n) 8^„-p+''(")- p+« dt _\/2 ia: Jx+<r(7i) {X-ty_ 2Lri Ji-nO V*^.X+& Foranygivenvalue ofe(andtherefore ofh),the firstand lastterms inthis expressioncanbemadearbitrarilysmall bytakingnsufficiently large,on account oftheconvergenceof t^f{t)\dt.II 1 Theremainingterras donotexceed 2.h€(3a;-*+x-")[1 ^An'o- {n) An \(T{n) V2 and, ifwetakecr{n)=1/^4 „,this isindependentofn,and itcanbemade as small asweplease bytakingesufficientlysmallinitially. Wecantherefore make theintermediate terms intheexpressionfor \Sn{x\R)\assmall asweplease bytakingesufficiently small, andwhen this hasbeen done, the firstand lastterms canbemade assmall asweplease by takingnsufficiently large. That istosay, 18.^{x\R)\canbemadearbitrarilysmallbytakingn sufficiently large,sothat lim8n{x\R)=0. Herice limif1- ^-f]a,,,J,(j,„x)=x~''f{x-0)lim [''f^'T,,(t,x\R)dt +^'-"/(a; +0)lim It'+'Tn{t,x\R)dt,n-^XJX since thelimits ontherightexist. 39—2 612 THEORY OFBESSEl. FUNCTIONS [CHAP. XVIII Since each ofthelimits ontherightisequalto^x",ithasnowbeen provedthat 2tCim^ V\Jm'^)m=l issummable (B)withsum^{/{a:+0)+f{x—0)}providedthatthelimits f{x±0)exist;andthis isthetheorem tobeestablished. Asacorollary,thereader should beable toprove withoutdifficulty that,if/{t)is continuous in{a, b),thesummability (R)isuniform throughout theinterval inwhich a+A^.v'^b-A, where Aisanypositive number. Cf.§18'25. 18*53. Uniformity ofsummability oftheFourier- Bessel series near the origin. We shallnowexamine theuniformityofthesummability {R)ofthe Fourier-Besselexpansion throughoutaninterval ofwhich theoriginisan end-point.Itwillbesupposedthattheexpansionismodified bybeingmul- tiplied throughout by>^x,and itAvillthenbeproved that, ift~''f(t)iscon- tinuous intheinterval(0,h),then themodified expansionisuniformlysummable throughout (0,b—A),whereAisanypositive number. Givene,wecannowchoose S(lessthanA)sothat \{tr''f{t)-x-''f{x)}\<e whenever x—S^t^^x +Bandt^O, providedthatxliesin(0,6—A). Since continuityinvolvesuniformityofcontinuity,thischoice of8maybe taken tobeindependentofx. Wenowwrite Sn{a;\R)=\f^'[t-^fit)-x--f{x)] Tn(t,x\R)dtJo andthenexamine jx^Sn{x\R)\after themanner of§18'52. Weexpress x^Sn{x \R)asthesum ofsixintegrals (some ofwhich areto beomitted when x<8),andweseethat jx^Sn(x |R)\doesnotexceed AnP +2k,e{x^^+x)\ {''-''(»)dt 3^„'2rx X-& +^/Cc +3-4,1" IX-\-<T(n) dt+(t-x) •x+S:,+-7f|dt dt V2Jx"' }x^^{n){t-Xy_ P-S \[t-^f{t)-x-^f{x)]\dt. Inthisformula anyofthelimits ofintegrationwhich arenegativearesupposed tobereplaced byzero. 18*53, 18-54] FOURIER-BESSEL SERIES 613 Now thisupperbound for\xiSn(x\ R)\doesnotexceed and, sincex~''f{x)isbounded (becauseitiscontinuous),thiscanbemade arbitrarilysmallbyachoice ofwwhich isindependentofic. Consequently x^Sn{x\R)tends tozerouniformlyasw-* oo . Now ithasalreadybeenshewn(§18"22) that Jo isuniformly convergentin(0,1—A),and so,sinceuniformityofconvergence involvesuniformityofsummability, x^-^f{x)\ t''+'Tn(t,x\R)dtJo tendsuniformlytox^f{x)in(0,6—A). Hence, sincex^Sn{x\R)tends tozerouniformly, xi[tf{t)Tn{t,x\R)dtJo' tends uniformlytox^-"f(x)if+'^Tn (t,x \R)dt,i.e.tox^f{x)in(0,h-A)."Jo ;. Ithastherefore beenprovedthat 00 Sa„,x^J^{j,nOc) isuniformly summable {R)in(0,6—A)withsumx\f(x), providedthat •1 t^f{t)dt'' I exists and isabsolutely convergent, andthatt~''f{t)iscontinuous in(0,6). 18"54. Methodsof'summing'Fourier-Bessel series. Weshallnowinvestigatevarious methods ofsummingtheFourier-Bessel series* 00 m=0 onthehypotheses (i)thatthelimits/(a-+0)exist, (ii)that ^Ctif(t)dtJo exists and isabsolutely convergent,and(iii)thattheseries issummable(R). Itconduces tobrevitytowrite/^(if)inplaceofdm^^Jvi'^),sothat/„i(a;) tendsuniformlytozero(§1827)asm-^cc when xliesin(0,1), *Thefactor x-isinserted merely inorder thatthediscussion maycover theinvestigation of uniformityofsummability near theorigin. 614 THEORY OFBESSEL FUNCTIONS [CHAP. XVni Consider firstthelimit limS(lJf)f^{a>) whichgivesthemost natural method(ofRiesz'type)forsummingtheseries. Since (jJAn)-^!,itisevident that limi(^^^)Ma^ exists and isequalto limI(l-i^)/^(^). Again, since/n(a;)=o(l),itiseasytoseethat n 2fm{x)=o{n),m=\ SOthat andthereforelim(-^^-^)IA(^)=0, w-».oo \Jn/m=l limIfl-^fVm(^)=limI(l-^)/„(^);Jr thelimit ontherightexists inconsequenceofthehypothesesmade atthe beginningofthesection. Again,since Jnn \nj whether mbeo{n)or0{n),itfollows that limSpf-^V«.(^)=0, n-»aom=\\Jn "/ andso limS(l-^)/^(^O=limI(l- ^f)U(^). Consequentlythehypothesesthat thelimits/(^r+0)exist(0<a'<1) and thattheintegral exists and isabsolutely convergentaresufficienttoensure that 00 m=\ issutnniahle (G1)withsum^x^[f{^+0)+/ (^'~ 0)|- Bythesamereasoning,iif{x)iscontinuous in(a,h),thesummability (C1) isuniformin(a+A,6—A) ;and, ifa=andt~"f{t)hasalimit ast--*O.'the summability (Cl)isuniformin(0,6—A). 18-55, 18-56] FOURIER-BESSEL SERIES 615 18'55.Uniformity ofconvergence oftheFourier- Besselexpansionnear the origin. Wecannowprove, byusing Hardy's convergence theorem*, that, ift^f(t) haslimited totalfluctuationin(0,6),while/(^)isalsosubjecttotheconditions of§18-53,then 00 2^a^yiX''Ji,\jq^i,x) 111=1 isuniformly convergentin(0,6—A)withsum x^f{x). Leth(t)beanauxiliaryfunction defined tobeequaltof(t)in(0,6)and equaltozeroin(b,1);and lettheFourier-Bessel series associated with h{t)be 00 Hl=l CO Then, by118*54, Sam^c^Jv (jm^)isuniformly summahle(C*1)throughout (0,6—A)withsumx^f(x), and,bySheppard'stheorem(§18*27), am/\//i„,is 0(l/ni),while(jm^)^ Jvijm^)isabounded function ofxand ni.Hence, by Hardy's convergence theorem, 2a^x^J^ijrnx)m=l isuniformly convergent throughout (0,h—A),withsumx^f{x). Again 2{am-0im)0C^Jv{jmX)=X^\tf{t)Tn(t,x)dt,m=\ Jb andthistendsuniformlytozero in(0,6—A)asn^-oo byananalogueofthe Riemann-Lebesgue lemma(§18'23). Hence %(ijnX^Jv{jm^)tendsuniformlytothesumx^f(x)in(0,6—A)as m=l n -JO :andthis isthetheorem tobeestablished. 1856.Sunimahility ofDini series. Except whenx=\, thesummability (C1)oftheDini series associated with f{t)maybeinferredbycombiningtheresults of§18'33 and§§18*51—18*53. Thesummability (Cl) may, however, beestablished independently ffor allpointsxsuch that <^^1byreplacing Anandthefunctions J^(iv)and F^{'w),which occur in§18*5,byDnandthefunctions wJJ (lo)+HJ^ (in)and wYJ (w)+HY^(w)respectively;thedetails oftheanalysis maybelefttothe reader, andhewillfindthatwhen x=ltheexpression ^{f{x+0)+f{x-0)] must bereplaced by/(I—0). *Cf.Modern Analysis, §8-5. tOfcourse onthehypotheses concerning f(t)which wereassumed in§18-53, 616 THEORY OFBESSEL FUNCTIONS [CHAP, XVIIT Theuniformityofthesummabilityintheinterval (a+A,1)when/(so) iscontinuous in(a,1)maybedealt with inthesamewayastheuniformity ofconvergence wasdealt with in§§18"33, 18"35. ThesummabilityofDini series (and ofFourier-Besselseries) byamodifi- cation ofAbel's method isofsomephysical importance. Thus, inFourier's* problemoftheConduction ofHeat inaninfinite solidcylinderofradiusunity, thetemperaturevatdistance rfrom theaxis satisfies theequation dv_J(d^v 1dv \ dt~W'-^rd^V with theboundaiycondition dv=0, iftheinitial distribution ofheat issymmetrical. Normal solutions ofthedifferentialequation satisfyingtheboundary condition are ^0(^w?') exp(-A;\,,rO' andsothetemperaturevisgiven bytheseries f 00 Sb,nJo(Knr) exp(-k\Jt), where thecoefficients b,naretobedetermined from theconsideration that 00 SbmJo(Xmr)m= 1 istheDini series associated with theinitialtemperature f(r).Itisevident thattheinitialtemperatureisexpressibleas 00 limSZ>„;Jo(\nr)exp{—kX^' t) ; and thislimit exists when theDini series issummable(R). 18*6. Theuniqueness ofFourier-Bessel series andDini series. Ithasbeenshewn byYoung Jthat theexistence andtheabsolute con- vergenceof \\\f(t)dtJ aresufficient toensure thatifallthecoefficients a^oftheDini series {orthe Fourier-Besselseries) associated ivithf(t)arezero, then thefunction f{t)must beanull-function. *LaTheorie AnalytiquedelaChaleur(Paris, 1822), §§306—320. Cf.Rayleigh, Phil.Mag. (6) XII.(1906), pp.106—107 [Scientific Papers,v.(1912), pp.338—339] ;andKirchhoff, Berliner Sitzungsbericlite, 1883, pp.519—524. +Inthisphysical problem,H>0,andsothere isnoinitial term tobeinserted. tProc.London Math. Soc.(2)xvm.(1920), pp.174—175. 18-6] FOURIER-BESSEL SERIES 617 Toprovethistheorem weobserve that,whenp—0,1,2, ...,wemaywrite where thecoefficients a,,,,aredetermined bytheformula 2 u'^''v+iijm). andtheseries ontheright converges uniformlyin(0,1—A)and oscillates boundedlyin(1—A,1).Itisthereforepermissibletomultiplytheexpansion byt^f(t)andintegrate term-by-term. Itfollows that m= J =0. Since alltheintegrals rtu+.p+.f(t)dt (p=1,2,2,...) Jo arezero, itfollows thatt^f{t)isanull-function, byLerch's theorem*, and thetheorem stated isprovedforFourier-Bessel series. Thetheorem forDini series canbeprovedinpreciselythesameway,and itistheoretically simpler because theDini series associated with f^'^P does not failtoconvergeuni- formlyin(1—A,1). Itispossibletoconstruct atheoryofseries ofBessel functions ofthetypes m—l w=l (wherethecoefficients a^and bm.areanyconstants) which resembles Rieraann's theoryoftrigonometricalseriesf.• • . Such atheory is,however, moredirectlyassociated with Schlomilch's series ofBessel functions, which willbediscussed inChapterXix;and it seems convenient todefer theexamination oftheseries ••, 00 00 ,\ -' 711=1 Wl"1 byRiemann's methods to§19"7,when thediscussion oftheseries forms asimple corollaiT tothediscussion ofSchlomilch series,' *Lerch, Acta Mathematica, xxvii.(1903), pp.345—347; Young, Messenger, xh,(1910), pp.37—43. Cf.§12-22. tCf.Modern Analysis, §§9-6— 9-632. .'- . ,' CHAPTER XIX SCHLOMILCH SERIES 19'1.SchlomilcJisexpansion ofafunction ofareal variable. InChapterXViiiwedealt with theexpansionofafunctionf(x)ofthe realvariable xintheform 00 f(x)= 2a^J^(jmx),m=l where/,„isthemthpositivezeroof/„{z),sothat, forlargevalues ofm, jm=(m+lv-i)7r+0 (l/»0. That istosay,theargumentoftheBessel function inaterm ofhighrank in theseries isapproximately proportionaltotherank oftheterm. Inthischapter weshall discuss theseries inwhich theargumentofthe Bessel function ineachterm isexactly proportionaltotherank oftheterm. Bychoosingasuitable variable, such aseriesmaybetaken tobe Sa,ftJy{nix). Itwillappear subsequentlythat itisconvenient toaddaninitial term (§19"11;cf§18'33); andtheanalysisissimplified bymakingaslightmodi- fication intheform ofthecoefficients intheseries(§19'2). Series ofthistypewere firstinvestigated bySchlomilch *.Theyarenot ofsuchgreat importancetothePhysicistasFourier-Bessel series, though Rayleighfhaspointedoutthat(whenv=0)they presentthemselves naturallyintheinvestigationofaperiodictransverse vibration ofatwo- dimensional membrane, ifthevibration iscomposedofanunlimited number ofequal one-dimensional transverse vibrationsuniformlydistributed indirection throughthetwodimensions ofthemembrane. Apartfromapplicationsthe seriespresentvarious features ofpurely mathematical interest; and, inparticular,itisremarkable thatanull-function canberepresented bysuch aseries inwhich thecoefficients arenot allzero (§19-41). Insomerespectstheseries aremoreamenable toanalysisthan Fourier- Bessel series, butthetwotypesofseries havemany propertiesincommon; andthereader willberightwhen heinfers from acomparisonofthe arguments j^^xandmxthat therelevantrangeofvalues ofxis(0,ir)for Schlomilch series, correspondingtotherange (0,1)forFourier-Bessel series. *Zeitschrift filrMath, imdPhys.ii.(1857), pp.155—158; Schlomilch considered only the special cases j'=and v=l. tPhil.May. (6)xxi.(1911), pp.567—571 [Scientific Papers,vi.(1920), pp.22—25]. 19-1, 19-11] SCHLOMILCH SERIES 619 19*11. Schlomilch's expansioninaseriesofBesselfunctions oforder zero. Wenowstateandprovetheexpansiontheorem discovered bySchlomilch. Thetheorem isconcerned with theexpansionofanarbitraryfunction j{x) oftherealvariable x,and,withmodernterminology,itistothefollowing effect : Letf{x)heanarhitrai'y functioyi,withaderivatef{x)which iscontinuous intheclosed interval(0, tt)andwhich haslimited totalfluctuationinthis intei'val. Thenf{x)admitsoftheexpansion (1) f{x)=^a^+SarnJoimw), where a,=2/(0)+-rf\if{usin</>)d(f)du, (2)' ''''''' 2f^ (' TTI Iuf(usin^)cosmud^du; (m>0) JoJ{) and thisexpansionisvalid, and theseries isconvei^gent, throughouttheclosed interval(0,tt). Schlomilch'sinvestigationisbased onadiscussion oftheintegral equation (3) f(x)=^["(/(xsin6)dO, ofwhich heprovedthatacontinuous solution is (4) g{x)=/(0)+Xr/ (xsin<^) fZ</). Weproceedtoverifythat thefunction g(x)defined by(4)actuallyisa solution 01(3);wesubstitute thevaluegiven by(4)intheexpressiononthe rightof(3),andthenweseethat tW fj(xsin6)d6="j f{0)+xsmd f' (xsin sin(f))d(f)TTJ'TT./ .'dO 2r ri'^ fi'^=/"(O)+"fixsine sin<^)sin6d4>dd.TTJ{,Jo Nowreplace^byanew variable xdefined bytheequation sinX=sin^sin^ 620 THEORY OFBESSEL FUNCTIONS andchangetheorder oftheintegrations. Wededuce that -g{wsme)dd-f(0)=~' f'{xsine sin<b)sinOdcbde TT. TTJoJo[chap. XIX _2wri'r« sin^cosXdxdd ~7rj, Jo-^^^''"^\/(sin^^-sin^;^) 2a; TT.'o)JYsinx) —arcsm =^I/(^sinx) =a;f'{xsinx)QosxdxJosin^cosXdddx \/(cos-X—cos'^^) cos 6'"'^^ /cos c'X^"^, cosvav Vcosx/Jx-^'^ =/(^)-/(0), and so,wheng(x)isdefined by(4),g(x)isasolution of(3). Now itiseasytoverifyfrom(4)that,when/'(x)isacontinuous function with limited total fluctuation intheinterval(0, tt),soalso isg(x); and therefore, byFourier's theorem, g(x)isexpansibleintheform g(x)=^tto+^«mcosmx, where»»=i 2f^^m—^19 ('^)COSmudu 2 /•'^ ttJo/^O)+i^ I' /'(usin<^)(i(^COSmudu, andthis series forg(x)converges uniformly throughouttheinterval(0, tt). Henceterm-by-term integrationsarepermissible, andsowehave f(cc)=-g(xsin6)dO ="- li«,o+S«,«cos(wa;sin6)\dd m=\ — 2'-*'om=l and this istheexpansiontobeestablished. Itiseasytoverifythat the values obtained forthe coefficients amarethesame asthosegiven by equation (2). When therestrictionconcerningthelimited total fluctuation off(x) isremoved, theFourier series associated withg(x)isnolonger necessarily convergent, though thecontinuityoif(x)ensures that theFourier series 19-2]'SCHLOMILCH SERIES 621 isuniformly summable (C1)throughout (0,tt);andhence, byterm-by- term integration, theseries 00 isuniformly summable (Cl) throughout (0,tt),withsumy(^);anapplication ofHardy's convergence theorem* thenshews thattheadditional condition ttrn=0{\l\/m) issufficient toensure theconvergenceoftheSchlcimilch series tothesum f{x)when xliesinthehalf-openinterval inwhich <x^tr. Forfurther theoremsconcerningthesummabilityofSchlomilch series, thereader should consult amemoirbyChapman f. [Note.Tlieintegral equation connecting /(.r)andg(x)isonewhich wassolved in1823 byAbel, JotcrnalfiirMath. l.(1826), p.153. Ithassubsequently beeninvestigated Jby Beltrami, 1st.Lombardo Reiidiconti, (2)xiii. (1880), pp.327,402;Volterra, Ann. diMat. (2)XXV.(1897), p.104;C.fi.Smith, Trans. American Math. >Soc. viii.(1907), pp.92—106. Theequation —" I'f'{xsindsiiKJ))sin6<^0dd=f(x)-/(O)""./ ./ ti . .,. . , ismostsimply establishedbythemethod ofchanging axes ofpolar coordinates, explained in§3'33;thismethod wasusedbyGwyther, Messenger,xxxiii.(1904), pj).97—107,but inview ofthearbitrary character off{x)theanalytical proof giveninthetextseemspre- ferable. Inconnexion with thechangesintheorder oftheintegrations, cfModernAnalysis, §4'51.. : .V. 19*2. Thedefinition ofSclduniilch series. WehavenowinvestigatedSchlomilch'sproblemofexpandinganarbitrary function intoaseries ofBessel functions oforder zero, theargumentofthe function inthe{m4-l)thtermbeing proportionaltom;andtheexpansionis valid fortherangeofvalues (0,tt)ofthevariable. Such seriesmaybegeneralised byreplacingthefunctions oforder zeroby functions ofarbitraryorderv;andafurthergeneralisation maybeeffected by takingthegeneralterm tocontain notonlythefunction J^{nix)butalsoa function which bears totheBessel function thesame kind ofrelation asthe sinedoes tothecosine. The lattergeneralisation is,ofcourse, suggested by thetheoryofFourier series, andwearethus ledtoexpecttheexistence of expansionsvalid fortherangeofvalues (—tt,tt)ofthevariable. Thefunctions whichnaturallycome under consideration forinsertion are *Gi.Modern Analysis, %%-b. tQuarterly Journal, xliii.(1011), p.34. JSome interesting apphcationsofFourier's integral theorem totheintegral equation have beenmade bySteam, Quarterly Journal, xvii.(1880), pp.90—104. 622 THEORY OFBESSEL FUNCTIONS [CHAP. XIX Bessel functions ofthesecond kindandStruve's functions; andthetypesof series tobeconsidered maybewritten intheforms*: * +Z T(v+1) r>t=i (ima;)" 2^n+^a^nJv(tnx)+hm"H-v(mx) Series oftheformertype(withv=0) havebeen considered byCoatesf; buthisproofofthepossibilityofexpandinganarbitraryfunctionf(x)into such aseries seems tobeinvalidexceptinthetrivial case inwhichf(^)is defined tobeperiodic (with period 27r)andtotend tozero asa;-*oo . Series ofthelattertypeareofmuchgreater interest, andtheyform a directgeneralisationoftrigonometricalseries. Theywillbecalledgeneralised ScJilomilch series. Twotypesofinvestigation suggestthemselves inconnexion withgeneral- isedSchlomilch series. The first istheproblemofexpandinganarbitrary function intosuch aseries; andthesecond istheproblemofdeterminingthe propertiesofsuch aseries withgivencoefficients and, inparticular,the construction ofanalysis (resemblingRiemann'sanalysisoftrigonometrical series) with theobjectofdeterminingwhether ageneralisedSchlomilch series, inwhich thecoefficients arenot allzero,canrepresentanull-function. Generalised Schlomilch series have l)een discussed inaseries ofmemoirs byNielsen, Math. Ann. Lii.(1899), pp.582—587;Nyt Tidsskrift,x.B(1899), pp.73—81;Oversigt K.Danske Videnskahernes Selskahs, 1899, pp.661—665;1900, pp.55—60;1901, pp. 127—146 :Ann. diMat.(3)vi.(1901), pp.301—329. Nielsen^hasgiventheforms forthe coefficients inthegeneralised Schlomilchexpansionofanarbitraryfunction andhehasinvestigatedwith greatdetail theactual construction ofSchlomilch series whichrepresent null-functions, but hisresearches areofadistinctlydifferent character from those which willbegiveninthischapter. Theinvestigationwhich weshallnowgiveofthepossibilityofexpanding anarbitraryfunction intoageneralisedSchlomilch series isbased onthe investigation given byFilon§forthecase y=inhismemoir onapplications ofthecalculus ofresidues totheexpansionsofarbitrarj'^functions inseries of functions ofgivenform. Itseems tobeofsomeimportancetogivesuchan investigation!!because there isnoobvious method ofmodifyingthesetof *Thereason forinserting thefactor .r"inthedenominators istomake theterms ofthesecond series one-valued(cf.§19*21). tQuarterly Journal, xxi.(1886), pp.189—190. JSee e.g.hisHandbuch derTheorie derCylinderfunktionen (Leipzig, 1904), p.348. §Proc.London Math. Soc.(2)iv.(1906), pp.396—430. IIIthastobeassumed that-i<f<J.The results which willbeprovedin§§19'41—19'62 suggestthat itisonlytobeexpected that difficulties should arise forother values of i'. 19-21] SCHLOMILCH SERIES 623 functions Jy{mx), ll^(mx)soastoobtain asetwhich isanormalorthogonal setfortheinterval (—tt,tt);andconsequentlythere isnomethod ofobtaining thecoefficients inaSchlomilchexpansioninsosimpleamanner asthat inwhich thecoefficients inaFourier-Besselexpansionareobtained(§18"1). Theinvestigation,which forms thelatterpartofthechapter, concerning therepresentationofnull-functions bygeneralisedSchlomilch series, isof exactlythesame character astheexpositionofRiemann's researches on trigonometricalseriesgiveninModernAnalysis, §§9'0—9'6"32. 19*21. Theapplication ofthecalculusofresidues tothegeneralised SchU'miilchexpansion. Weshallnowexplainthemethod*bywhich itispossibletodiscover the values ofthe coefficients inthegeneralised Schlomilchexpansion which representsanarbitraryfunction/(a;),when theorder voftheBessel functions liesbetween —|and|.When thishasbeen done,weshall notconsider the validityoftheprocesses bywhich thediscoveryhasbeenmade, butweshall prove directlythat theSchlomilch series inwhich thecoefficients have the specifiedvaluesactuallydoesconvergetothesumf{x). This isanaJogoustotheprocedurewhich isadoptedinDirichlet'sproofofFourier's theorem :intheexpansion /(.r)=5ao+2(a„iC0S7)i.r-|-/3„(SinmA')m—1 thevalues ofthecoefficients arediscovered bymultiplyingtheexpansion bycosmsand l^v sinmx,andintegrating,sothat thevalues ofa„jand/3„;aretaken tobegiven bythe equations 1/""• If-^ «»»=-! f(t)COH 7)1tdi, (3m=- If(t)sin7ntdt. 'rJ-TT* '^J-TV Wethen taketheseries inwhich thecoefficients have these values, namely 1 /"tt 1"^T'^-— /f{t)dt+-2 fit)conm{x-t)dt, andprovethat itactually convergestothesum/(.r). Itconduces tobrevitytodealwith thepairoffunctions J^,(mx)±iH^(mx) (Imx)"' instead ofwith thepairoffunctions '^Jy(mx)l(^mxy, 'H.^(mx)j(\mxy. Weshall write (A\ J^(z)+i-a,(z) _ *Apart from details ofnotation, thefollowing analysisisduetoFilon; itwasgiven byhim, inthememoirjust cited, forthespecial case ^=0,buttheextension tovalues ofvbetween ±h presents nodifficulty. 624 THEORY OFBESSEL FUNCTIONS[CHAP. XIX sothat*<j>v{^)isanalyticanduniform forallfinite values ofthecomplex variable2;andevidently J^(7nx)±m^(mx) ,, Wenowobserve that(—)"*(pv(nix)istheresidue at ^^=mofthefunction 7r<j)v{xz) suxirz wherem=0,+1,±2,...;andsoweshall consider theintegral 1f J,;.^,(^ ziTi Jc sinTTZ inwhich thecontour Cisacircle, ofradius M-\-\, with itscentre atthe origin,andMisanintegerwhich willbemade totend toinfinity. Thefunction F{z)isassumed tobeone-valuedthroughoutthe2^-plane, andtobeanalyticatinfinity (cf§19"24);itsonly singularityinthe finite partoftheplaneisanessentialsingularityattheorigin. ByJordan's lemma, theintegraltends tozero asMtends toinfinity, providedthat v>— |. Itisevident, bycalculating residues, that S{-y[F(m) <}),(mx)+F{- m) (f),(-7nx)] OT=1 isequaltotheresidue attheoriginof -F(z)'^i^^\smirz that istosay (2) i(-)"^{F(m) (f>,(mx)+F(-m) <j>,(-mx)] »i=i 1/<"+'r-...-^*.('t^) I^'cm TT'z27ri'sinTTZ Theproblemofexpandinganarbitraryfunction f{x)intoageneralised Schloinilch series isconsequentlyreduced tothedetermination oftheformof F{z)insuch awayastomake 2771 difi"erbyaconstant fi'omf(x).''F{z)'^^^dz(0+) ^iz)-SmTTZ *The insertion ofthefactor {hzf inthedenominator makes<p^,(z)amenable toCauehy's theorem when thecontour ofintegration completely surrounds theorigin. 19-22] SCHLOMILCH SERIES 625 19*22. Theconstructionofthefunction F{z). Wenowtakethecontourintegral 1i^'^^j.^.^'^^^^i^^) ]^(^)27rzJ sinTT^'dz. and, inorder tocalculate itinasimple manner, weshallsupposethatF{z)is expansibleinaseries ofFilon'stype* (1) ^w=i?'»-*#,«=1z where-v|r„{z)denotes thesum ofthose terras intheexpansionoftt"'sinttz whosedegreedoesnotexceed n,andthecoefficientsp^willbedefined later. Thereader willobserve that With thisdefinition ofi^(2^),itisevident that, forsmall values of|^;, smTT^ „=i (£r"+ismTT^) 2.j1 7r'^+^ cos1nvr+{z)]^., «=i [2"+i {n+l)\ smirzj^"^^^ Itfollowsimmediatelythat ZiriJ smTT^ ^,.^xVJt>7i7i-^COs|?i7rI PrJMxf andconsequently weproceedtoidentify 00_s Pn{\ixy ,^^V{^n+l)T{^n +v+\)- with/(a;)~/(0).Forthispurpose wehave toassumetemporarilythat/(a;) hasdifferential coefficients ofallorders attheorigin, andthenwedefine the coefficients'p^bytheequation.. (3)/'-'(Q>_ PnihiTr._i2 3 ^ Wenext transform thisequation defining pninsuch awaythatthesum ofthe series, bywhichF{z)isdefined, isexpressibleinacompact symbolic form;thetransformation oftheseries forF{z)canbeeffectedbyexpressing *This type ofseries isfundamental inFilon's theory, and isnotpeculiar toSchlomilch expansions; thus, inhisworkonFourier-Bessel series, sin ttzisreplaced byz~''J^,{^z) and'/'„(;) denotes thesum oftheterms whose degree doesnotexceed nintheexpansion ofthat function. w.B.F. . 40 626 THEORY OFBESSEL FUNCTIONS [chap. XIX thecoefficients pninaformwhich involves nonlyasanexponent. Forthis purpose wemake useofEulerianintegralsofthe first kind, and, inorder thattheymaybeconvergent, weshall findthat itisnecessarytosupposethat —\<V<^.Wethenhave ^_r(i)r(i. +.+i) ^-w{r^V"^' (0)£(i-^r*-"^'^-"- dt andsoweobtain thesymbolicformulad dti'iv du^dt. M= W P»= -,M^(t^) 1.(1-'')-'-'I.^2.D''f{tu) M=dt, whereDstands fordjdu. Now, ifwearrangetheseries indescending powersofz,itiseasytoverifythat ^ylrn(z)D''_sinhTrD „=1i"2"+» TT(t>-Z))' andtherefore (5)^<^)=r5^/i(i-«r'-',^[*"sinhTri) ,,,' 7r(D—izydt. M=0 Again,aconsideration of(2)shews thatweneed tosumtheseries ^Pntt"cos\nir andweareable toeffect ourpurpose bymakinguseofformula(4),whence wefindthat (6)2Pn'Jr^COS^ntr - T{\-v)]^^^^dtY \(sinh irD 77Dl\f{tu)dt M= Wehavenowobtainedsymbolic expressionsforallthecoefficients inthe generalisedSchlomilchexpansionoff{x),but itisnecessarytotransform theseexpressionsintomore useful forms, byfindingthesignificancetobe attached tothesymbolic operator -—j,—:-t ,both forgeneralvalues ofzand forthevalue zeroofz. 19-23] SCHLOMILCH SERIES 627 19'23.Thetransformation ofthesymbolic operatorsinthegeneralised Schlomilchexpansion. Weproceedtoobtain aninterpretation*ofthesymbolic expression sinhttD . , , J{tu)7r(D—iz)• M= Theusualinterpretationof j.—-fit^Ois gizu[ e-^'''f(tv)dv, Ja where aisaconstant ofintegration;andtherefore ^izur* e-'^^fitv)dvsinhTrZ),,.sinh Tri) IT{D—iz)"TT ^,<^^'^^^(D +i^) [\-,..f(tv)dv.IT Now,bythesymbolicform ofTaylor's theorem, wehave where;^{u)isanarbitraryfunction ofu;andhence itfollows that 628 THEORY OFBESSEL FUNCTIONS[chap. XIX at0,±1,±2,...isconcerned, wemayomit thesecond terrti ontherightin(1), andcalculate theresidues of TTff)^{xz)F{z)sinTTZ whereF{z)isdefined bytheformula dtpe-i^^f{tv)dvdt. Again,from^19'22 (6)andequation (1)ofthissection wehave » pnir^^cos^nir _r(l)/(0) pdt^^ ("^^Z. (r^+i)!-Tjf:r^]^^-'^'dt""' - 2ra-.,ra) /;^^-^'-It^/:/(^^>H'^- The firstterm ontherightin(3)isequaltoF(v+l)/(0), exceptwhen• v=0;when v=0,thevakie oftheterm inquestioniszero. Wethusobtain theexpansion (4)f{x)= <f>,(0)F(0)-\- i(-)'«[F(m) 4>,{mx)+F(-m) cf>,(-mx)].m=\ Inthespecialcase inwhich y=0,themodified form of(3)shews thatan additionalterm/(0) must beinserted ontherightin(4), When wechangethenotation tothenotationnormallyused forBessel functions andStruve's functions, theexpansion becomes (5) where f/(^)=2^0 r(i;+l)'^Jlo^mJv{mx)+6,„H^ {mx) {\nfixy (6) \\'-= r(i-l)r(i) /o<i-^^>"-'^iWjj^''^'''""^^ This isthegeneralisedform ofSchlomilch'sexpansion.dt, dt. 19•24.TheboundednessofF{z), as\z\-»- cc. Weshallnowprove that,when thefunctionf{x)isrestricted inasuitable manner, thefunction F{z)isbounded when 12 |-*-oc ,whatever bethevalue ofargz.Theleader willremember that theassumptionthatF{z)isbounded wasmade in§19'21 tosecure theconvergenceofthecontourintegral. Wetaketheseries of§19-22(1),bywhich F{z)wasoriginally defined, namely "Pn^niz) n=l-n+l' *When Visnegativeitisnecessary touseamodifiedexpressionfortheintegrals ;cf.§19'3. tWhen v=0,theexpression foroqhastobemodified bytheinsertion oftheterm2/(0),in consequence ofthediscontinuityinvalue oftheexpression ontherightof(3). 19-24, 19-3] SCHLOMILCH SERIES 629 anddivide itintotwoparts, namelythe firstiVterms andtheremainder oftheterms, whereNistheinteger such that iV^7r|2l<i.V+l. When n^JV,theterms of(//„(z)donotexceed tt**"^ 12|"/n !,andtherefore, when n^JV, ,71+1n7r"-l|3h/(»!)<:!'-LS' : \z\.{n-i)\ When n^JV,wehave |-^n(s) |<tt~'sinh tt 12 1,andtherefore ' ^^' i0i„=i (n-l)! Trl^jA+i „=o |2|" c. sinh Tr\z\bince 7liV-+l tends tozeroas 12 ]^-o),itisevident thatasufficient condition forF{z)tobebounded as 12 1-*-Qcisthattheseries 00 2\Pn\n=l should beconvergent ;andthis istheeasei{f{x)issuch that ISconvergent.27.''+ i|/(»)(0)| 19*3. Theexpansion ofanai^hitr^ary functionintoageneralised Schlomilch series. Now that theforms ofthecoefficients inthegeneralisedSchlomilch expansionhavebeen ascertained byFilon's method, itisaneasymatter to specifysufficient conditions forthevalidityoftheexpansionandthen to establish it. Thetheorem which weshallprove*isasfollows: LetVbeanumber such that—^<v<\; and letf{x)bedefined arbitrarily intheinterval (—tt,tt),subject^tothefollowingconditions: (I)Thefunction h{oc), defined bytheequation h{x)=^vf{x) +xf{x), exists and iscontinuous intheclosed interval (—tt,tt). (II)Thefunction h(x)haslimited totalfluctuationintheinterval (—tt,tt). (III) IfVisnegative Itheintegral isabsolutely convergenttuhenAisa(small) number eitherj)ositiveornegative./: *Theexpansionisstated byNielsen, Haiidbuch derTheorie derCi/linderfunktionen (Leipzig, 1904), p.348; buttheformulae which hegivestorthecoefficients intheexpansion seem tobe quite inconsistent withthose given byequation (2). tThe effect ofconditions(I)and(II)ismerelytoensure theuniformityoftheconvergence ofacertain Fourier series connected with }i(x). tIfvispositive, thisLipschitz condition issatisfied byreason of(II). 630 THEORY OFBESSEL FUNCTIONS[CHAP. XIX Thenf{x) admitsoftheexpansion , . .,^_ lap ^argJ^(mx)+6mH^ (mx) (!)f{x)-^-^^^^y^^ ^^—y^, where I rnMirsec^""*"^ d)d w/zew wi>0;thevalueofaoisobtainedbyinserting anadditional term 2r(^+i)/(0) ontherightinthefirst equation ofthesystem (2). Weshall base theinvestigationonadiscussion oftheintegral equation (3) /(^)=r7-ZW7r^ Tcos-%(a.sin^)^6^; itwillbeprovedthatacontinuous solution isgiven bytheformula (4)g{x)=^T{v +\)f{()) +rTfl^) Ifsec-+i(^^[sin- 4>{/(^sin <^)-/(O)]] dc^. [Note. The(absolute) convergenceoftheintegral contained inthisformula issecuretl bycondition(III).Itshould beobserved thattheaggregateofterms containing /(O)in equation (4)maybeomitted when vispositiveinview oftheformula /,'^''P—d^'^'P- rW)- which isvalidonlywhen vispositive.] Weproceedtoverifythat thefunctiong(x)defined by(4)actuallyisa solution of(3),bytaking g(x)tobedefinedby(4),substitutingintheexpres- sionontherightof(3),andreducingtheresulttof(x). Theresult ofsubstitution is 2cosvirf^ C^'^ d, cos-esec-+^ d>~[sin-6 [f{xsin6sin<h)-/(O)}]d^dd +/(0). Hence wehave toprovethat 2cosvTT fi"'r^ /7^ I cos-(9sec-+i(^^[sin-(/>{/(a;sin6'sin</))-/(O)}]rf0d<9 =/(^)-/(0)- Replace <^onthe leftbyanewvariable -y^defined bytheequation sin;^=sin^sin</>, changetheorder oftheintegrationsintheresulting absolutely convergent integral,andthenreplace^byanewvariablei/rdefined bytheequation cos6=cos;^sin>/r. 19-3] SCHLOMILCH SERIES 631 Wethusdeduce that JoJnd cos-"dsec-"-^!4>,^[sin^" (^{f{xsin6sin</))-/(O)}] cZc^ci^, sin^cos-"^rf-r., f/./•X/-/^MT ,/i 7 (75?^iSrs» «>T.3^[«"''Xl/(^«m X)-/(0)1]Mdx Jx •i- d Itan-'"\|/-rtY.I ^^tan-^"^/.rf^/..^[sin- ;,<:[/(^^sinx)-/(O)}] (^x =ir(^+i)r(i- 1.){/(..) -/(O)},• andhence theformula tobeestablished isevident;andso,when[/(,c)is definedby(4),thenequation (3)issatisfied. Now, byFourier's theorem, g(a;)=l^o+2(o.,ncosmx+b^sinmx), ''m=l where 1f" g(u)cosmudtii (5)TT'_ 1f" ^m=1 5^(")Sinmudu; and itiseasytoverifythatwhenf(x)isacontinuous function with limited total fluctuation intheinterval (—tt,tt)soalso isg(x),andtherefore the expansionforg(x)isuniformly convergentwhen—ir+S^x^tt—S,where S isanarbitrarily smallpositive number. Replacexbyxsin intheexpansionofg(x),multiply bycos-*' 0,which hasanabsolutely convergent integral,andintegrate term-by-term:wededuce atonce that /•/A= t^o ,Vctm^.>(mx)+bmilu {mx) andthisexpansion converges uniformly when—tt+S^x^tt—S. The values ofa,^andb^given byformula(5)areeasilyreconciled with thosegiven byformula(2). Itshould benoticed that,bytheRiemann-Lebesgue lemma, «,„and6,„, arebotn (1/?n)whenmislarge.Thisseems tobeconnected with thefact thatwhen wecome todealwith anySchlomilch series(§19"6"2) weare unable tomakeanyprogresswithoutassumingthatS6,„/?yiisconvergent (or someequivalent hypothesis);thisassumptionwillappearin§19'62 tobe necessarybecause the diff"erentialequationwhich Struve's function satisfies isnothomogeneous,sothat Struve's function isnotofatypewhich occurs in solutions ofLaplace's equationorthewaveequation;there would conse- 632 THEORY OFBESSEL FUNCTIONS[CHAP. XIX quently seem tobereasons ofaphysicalcharacter forthelimitations which havebeenplacedonf{x)inorder toensure theexistence oftheSchlomilch expansion. [Note.Just asin§19-11,ifcondition(II)concerningthelimited total fluctuation of ivf{x)+xf'{x)isnot satisfied, then allstatements made inthissection uptothispoint about convergenceofseries have tobereplaced bystatements aboutsummability (Cl).] There isoneimportant consequencewhich follows from thefactthat a„i and6^areboth (1/m)when2vf(x) +xf' (x)haslimited total fluctuation in(—TT,tt),namely,that intheneighbourhoodsof—ttandtt,thegeneral term oftheSchlomilch expansionis(1/m''"*'^), andsotheexpansion repre- sents acontinuous function;hence theexpansion converges (uniformly)to thesum/(^) throughouttheinterval (—tt,tt). 19*4. Special functions i^epresented bySchlomilch series. There areafewproblemsofMathematicalPhysics (other than theproblem mentioned in§19"1)inwhich Schlomilch series occur inanatural manner, andweshallnowgiveanaccount ofvarious researches inwhich Schlomilch series aretobefound. Averysimpleseries is 1+ie-'"'Jo(mp); this series isconvergentwhenpand zarepositive, and, ifpandzdenote cylindrical-polar coordinates, itisasolution ofLaplace's equationatallpoints ofspaceabove theplanez=0. Various transformations oftheseries have beengiven byWhittaker*; thus,bychangingtoCartesian coordinates(x,y,z)andusing §2'21,wehave (1)1-f^e-^^J,{mp)=^ :,.,^ ^r—-. „,=i'^27rj_„1—exp}—(2:-f-ia;cos M+t?/smw)} When X- -\-y^\-z"-<1,theintegrand maybeexpandedinascending powers Q){z-Vixcos 11-hiysinu. Ifthis isdone,wegetf (2)1-fSe-'-/„ {mp)=^ ^-^ '^'^-.-^-+'- ,„=! llTJ^T,Z+IXCOS,u+iys\T\u2 +s—2~--^—~I(z+ixcosu+iysinu)^"~^ du r2,„=i {2m)\ where(/•,6)arethepolarcoordinatescorrespondingtothecylindrical -polar coordinates{p,z),andB^,Bo,B^, ...areBernoulli's numbers. *Math. Anil. lvii.(1903), pp.341—342. tCf.§4-8andModernAiiahjsis, §§7-2,18-31. 19-4] SCHLOMILCH SERIES 633 Another transformation oftheseries, alsogiven byWhittaker,isobtained from theexpansionfor1/(1—e~^)inpartial fractions; thisexpansionis 1 1 ) l-e whence wededuce that1112,^=7+9+^.=11^—2m7ri t+^rmri) (3)1+ie-^-Vc, (mp)= ^.+^ 100 +:^ ,tiLVi(2w7ri +2)-+X-+if] ^/[{2m'I^i-zf+x'+y-^_ Itfollows that theseriesrepresentstheelectrostaticpotentialdue toa setofunitcharges (some positive andsomenegative)attheoriginandata setofimaginary points. Thereader mayfind itinterestingtodiscuss theLijjschitz-Hankel integral ofI13*2 asalimitingform ofaseries ofWhittaker'stype. Some other series havebeenexaminedbyNagaoka*inconnexion with a problemofDiffraction. Onesuch series isderived from theFourier series for thefunction which isequaltol/\/(l—^')intheinterval (—1,1). TheFourier series inquestionis 1 (4) ,,^ ^;r=iTT+TTSt/,,(?H7r) COSWtTT.r, \/(l-X-) ,„=i and itconverges uniformly throughouttheinterval (—1+A,1—A),whereA isanypositive number. Multiply bye"**andintegrate,andwethen obtain theformula(alsodue toNagaoka) (5)a:£,axidx le.axi TTJV(l-^")21 a+2SJo(m-rr)acosTUTTX—min sinmirx m=l q2_^y^2^2 The series ontherightin(.5)converges uniformly throughouttheinterval (—1, 1)andsowemaytake—1and 1aslimits ofintegration. Hence, forallvalues(realandcomplex)ofa, (6) Jo(«)=sma a1+let?t(-)'"Jo(m7r) 1=1 a"—m-TT- Amoregeneral result, validwhen /^(y+1)>0,is sina 0) J„(a)= ar(i.+1)W) „,=im"(a--m'TT') *Journal oftheColl.ofSet.,Imp. Univ. ofJapan,iv.(1891), pp.301—322.Some ofNagaoka's formulae arequoted byCinelli, Nuovo Chnento, (4)i.(1895), p.1;">2. 634 THEORY OFBESSEL FUNCTIONS[CHAP. XIX Thisexpansion isalsoobtainable byexpressing ——^asasum ofpartial fractions*. Variousrepresentationsoftheintegral onthe leftof(5)were obtained byNagaoka; theformula quoted seems tobethemostinterestingofthem. Finally weshallgivetheformula f <«)J,^^|^^="-^-(— > This isdeducible from theFourier series ^cos(2m—l)a;tt, ^. , . byreplacingwbyxsindandintegratingwithrespectto6from to^tt. Asanexampleofthecalculation ofthesum ofaSchlomilch serieswhen thevariable liesoutside theinterval (—tt,tt),weshall take7r<x< lir,and then, \if{(t)denotes thesum oftheFourier series, weseethat 9f/•arc sin(7r/x) rin- 1=^ + /(.rsin^)rf^T(Jo.'arcsin(Wj-) J =-^(7r-2^sin^)rf^ +-f"" \(2xsine-Sir)dO, ""•'arcsin(n/x)^ SOthat,when tt<x<'lir,wehave /Q\ ^Jo[(2m- 1)a;}.,, .,^- /7r\ tt^ ^^^ix~(2^;r:riy-=V(^--)-Ix-^arccos(-j+^. 19'41. Null-functions expressedasSchlomilch series. Weshallnowprovetheremarkable theorem that (1)^+i(-r Jo(wA-)=0, providedthat <^c<tt;theseries oscillates when x=Qanddivergesto +00when x=7r. Thistheorem hasnoanalogueinthetheoryofFourier series, and, infact, itisdefinitely known]: thataFourier cosine-series cannotrepresentanull- functionthroughouttheinterval(0,tt). *Cf.ModernAnalysis, §7-4. tThiswas setasaproblemintheMathematical Tripos, 1895. JCf.ModernAnalysis, §§9-6— 9-632. 19-41] SCHLOMILCH SERIES 635 Itiseasytoprove (1)byusingParseval'sintegral; whenMisalarge integer, wehave 1-^^2r^'^ii'^' )=-+2 (-)™ J,(mx)=- ]o+S(-ycos(mxsm«)[dt (_).w |-JTcog|(J,/^1 ),^.gint] TTJo cos(|*sin^) (_)3/ /•xcos(ilf+i)u duIrx JoTT^0 cos|w' V(^"''^—'^^) =0(1), asiHf^00 ,bytheRiemann-Lebesgue lemma*, which isapplicablebecause theintegral f''du JocosIII. exists and isabsolutely convergent when <^<tt. Hence wehaveprovedthat lim 3/-H.-001-^' +S(-y»J,(mx) 9, —m=l= when <a;<TT;andthis isthetheorem stated. Itiseasytoproveinasimilar manner that (i)when— |< i^$|and<a;<tt,(ii)when v>hand<a;^tt. ByusingPoisson'sintegralwehave(since v>—^) \^1{-rJ.{mx) 2j'i'^(1'^^1 ^(I;+|)^(l)^^^/o cos|w'^^ =0(1), asM-*oo,providedthattheintegral '"=(x--u^y-i cosilldu J ^"»f' exists>ndisabsolutely convergent;andthis isthecasewhen xandvsatisfy theconditions stated. Thetruth of(2)isnowevident. Ifnisapositive integer,and ifvissolargethat v—2n>— |,theoperator d {d xdx\dx *Cf.Modern Analysis, §9-41. 636 THEORY OFBESSEL FUNCTIONS [CHAP. XIX maybeappliedntimes toequation (2).The effect ofapplyingtheoperator once tothefunction J^{mx)l{\mxYistomultiplythefunctionby—m-;and therefore, when <a;<tt, that istosay,=0, providedthat either (i)—\<v—1ni^^ and^a;<tt,or(ii)v-2n>|and ^.«$tt. Theformulae giveninthis section aredue toNielsen* Math. Ann. lii.(1899), pp.582—587;twoother papers byNielsen onthissubject were publishedatabout the same time, A^t/t Tidssh-ift,x.P>(1899), pp.73—81;Ooersigt K.Danske Videnskabernes Selskahs, 1899, pp.661—665. Inthe firsttwoofthese three papers integralvalues ofv onlywere considered, theextension togeneralvalues ofvbeingmade inthethirdpaper. Shortlyafterwards tNielsen gaveaformula forthesumoftheseries in(2)whenx>n; thisformula iseasily obtained from theintegralofDirichlet's type (-)M/-•^cos(J/+i).. _ , byconsideringthebehaviour oftheintegrandat z(=7r,Stt, Stt, Itisthusfound that,when xispositive andqistheinteger such that (25'-l)7r<.r<(2^+l)7r, then _J__ I{-r,h{r,ix) _%T{\)'1f{^n-lYn^- y-h ^^>r(v+l)"^„,=, {hmxY xT{v+i)n=x\X'j Theimpoi'tanceofNielsen's formulae liesinthefactthattheymake it evident that,when afunction f{x)isdefined fortheinterval (—tt,tt),ifthe function canberepresented byaSchlomilch seriesthroughouttheinterval (except possiblyatafinitenumber ofpoints)therepresentationisnotunique andthere areanunlimited number ofSchlomilch series which areequalto thefunction y(a;)throughouttheinterval, exceptatafinitenumber ofpoints, namelythepoints already specified togetherwith theorigin and(when —^<y^I)theend-points+tt. Theconverse theorem, that theonlySchldniilch series withnon-vanishing coefficientswhichrepresent null-functionsatallpoiiits ofthe interval —'7r<x<7r, {whe7il—^<V^^)excepttheorigin areconstantmultiples of h V{-)'^^h{mx) *Formula(1)wasrediscovered byGwyther, Messenger, xxxin.(1904), p.101. tOvcrsigt K.Danske Videnskabernes Selskahs, 1900, pp.55—60;seealsoalater paper by Nielsen, Ann. diMat.(3)vi.(1901), pp.301—329 formore compHcatedresults. Cf.§19-4(9). JThetheorem isuntrue whenlof ;cf.formula(3).Itwould beinteresting toknow whether anySchlomilch series other than theonegiven canrepresent anull-function when^--:>'<f. 19-5] SCHLOMILCH SERIES 637 is,ofcourse, ofamuchdeeper character, and itseems thatnoproofofithas yetbeenpublished. Weshallnowdiscuss aseries ofpropositionswhich lead uptothistheorem;theanalysiswhich willbeused resembles, initsmain features, theanalysis*, duetoRiemann, which isapplicabletotrigonometrical series, * 19"5. Theorems concerningtheconvergence ofSchlomilch series. Weshallnowdiscuss thespecial typeofSchlomilch series inwhich v=0, andinwhich Struve's functions donotappear;theobjectoftakingthispar- ticular case istoavoid thelossofclearness due tothegreater complication intheappearanceoftheformulae inthemoregeneralcase. With afew exceptions,thecomplicationsinthegeneralcase arecomplicationsindetail Only;those which arenotmatters ofdetail willbedealt withfullyin §§19-6— 19-62. The seriesnow tobeconsidered is 00 (1) i«o+ ScimJoimx),m=l inwhich thecoefficients a^arearbitrarily givenfunctions ofni. We shall firstprovetheanalogueofCantor's lemma t,namelythat the condition thata^Jo('''*')^asni-*oo,atallpoints ofanyintervalofvalues ofuc,is sufficienttoensure that a,n={\fm). [Note.Iftheoriginisapointoftheinterval inquestion, then thetheorem that «m=0(l) isobviously true.] Takeanyportion Joftheinterval which does notcontain theorigin,and letthisportionbecalled /j.Letthelengthof/jbeL^. Throughout /jwehave(c£§7'3) a„i,Jo(mx)=a,„("^^J•[-^i^mx, 0)cos{mx—\ir)—Q{nix, 0)sin{mx— \'tt)\; and, asni-^ oo, P(mx,0)^l, Q(mx,0)^0. Hence, forallsufficiently largevalues ofm,(sayallvaluesexceeding mo) atallpointsof/j. Nowsupposethata^isnoto(\/w);wehave toshew that thishypothesis leads toacontradiction. *Cf.ModernAnalysis, §§9-6— 9-632. tIbid. §9-61. XSinceJg(>nx)isaneven function of.r,theportion maybesupposed tobeontheright of theorigin without loss ofgenerality. 638 THEORY OFBESSEL FUNCTIONS [CHAP. XTX Ifamisnoto(\/m),apositive number emust exist such that Km I>e\/m whenever misgiven anyvaluebelongingtoacertainunending sequence* mi,Wg, tris, Letthesmallest member ofthissequence which ^ceeds both moand27r/Zibecalled ?h/. Thencos(mi' x—^tt)goesthroughallitsphasesinI^,andsotheremust beaportion fof/j,say/g.such that Icos(m^'x- l-rr) \^|^S,\sin(niiX-^tt)|<^ atallpointsof/g.IfXgisthelengthofI^,then L2=^Tr/m^'. I^Text letthesmallest member ofthesequence virwhich exceeds both mj' and27r/Zy2 becalled wio'. Then cos(mzx—^tt)goesthroughallitsphasesinlo,andsothere must beaportionof/o,sayI^,such that Icos{7110X—Iv) I^^\/S, Isin(m^'x—{tt) \^^ atallpointsofI3.IfL3isthelengthofI3,then Z3=jTr/mg'. Bycontinuingthisprocess, weobtain asequenceofintervals I^, I.-,,I^,... such thateach iscontained initspredecessor ;there istherefore apointX which liesinside allthese intervals, andatthispointwehave Icos{mX—itt)I^|-\/3, Isin{inX—\ir) \^\, whenmhasanyofthevalues m^,m^,mj, Forsuch values ofmweconsequentlyhave 2 ittm/„i^^) I^ I«mIa/rmrXj X[P(mX, 0). Icos(mX-^tt) ]- |Q(mX, 0) ,. ,sin(mX-^tt) |] V3- 4VUxj' andthis isinconsistent with thehypothesisthatamJoii^^) tends tozero at allpointsof/j. Thecontradiction which hasnowbeen obtained shews that a„,must be (s/m). Thenexttheorem which weshallproveisthat, iftheSclildmilch series converges throughout any interval, then thenecessary andsuffi.cientcondition *Itissupposed thatm\<.m^<m^< tThere are,infact, atleasttwosuch portionsofIj ;inorder that Zvmaybeuniquelydeter- mined, wetake I^tobethatportion which liesonthe leftoftheothers. 19-51] SCHLOMILCH SERIES 639 that theseries should converge foranypositivevalue ofx(whether apoint of theinterval ornot)isthat theseries TO=iV\nnrxjcos{mx—Itt)+7-—sin(mx—^tt) 11tt/y should beconvergent forthatvalueofx.• This theorem isevident from the fact that thegeneralterm ofthe trigonometricalseries differs from a^nJo{mx) byafunction ofmwhich is (amni~^)—(m~-);andSo(m~-)isaconvergentseries. 19"51. Theassociatedfunction. Letthesum oftheseries m=1 atanypointatwhich theseries isconvergent,becalled/(a-). Let (1) F(x)=iaoX^- i«^^-^). ThenF(x)willbecalled thefunction associated with theSchlomilch series whose sumisf{x). Itiseasytoseethat, iftheseriesdefiling f{x) convergesatallpoints ofany intet^al, then theseriesdefining F(x)converges forallreal values ofX. ForamJo(nix)-^asm^- ooatallpointsoftheinterval, andtherefore (§19-5) am=o('sJm). Again, by§2*5(5),forallrealvalues ofx IJo(^^)k15 andconsequently amJo{in^) Sincem? \m^ ioi—\ =1Wv00 2 isconvergent,itisobvious that the series ontherightin(1)must be convergent. Itisevident, moreover, notonlythattheconvergenceisabsolute, butalso that itisuniformthroughout anydomain ofvalues oftherealvariable x. 640 THEORY OFBESSEL FUNCTIONS [CHAP. XIX 19"52.Lemma I. We shallnowprove that, if"F(x)isthefunctionassociated ivith the Schlomilch series whosesum isf(x),andif ,,.^,.(cc+a)F (or.+2cc)+(x-a)F(x- 2a)-2xF(x) (1) G(x,a)=—^,, then (2) limG(^,a)=^/(a;) atanypointxativhich theseriesdefining f{x)isconvergent, providedthat* dm=(\/m). Itiseasytodeduce from(1)that G(x,a)=^a,x-1^^^ X[{x+a)Jo(mx+2ma)+{x—a)J^(mx—2ma)—2xJo{mx)]; and,fromI'Hospital's theorem, itfollows that lim\(x+a)Jq{mx+2ma)+{x—a)Jq{mx-2ma)—2xJf){mx)] =lim-—— [Ju{mx+2ma)—Jo{mx—2ma)„^oom-a. +2m{x+a)Jo'(mx+2?«a)—2m{x—a)Jo{mx—2/»a)] =XJo"{mx)+Jo'{mx)/m =—xJo{mx). Consequentlythelimits oftheindividual terms oftheseriesdefining G(ie,a) aretheindividual terms oftheseriesdefining xf{x). Itistherefore sufficient toprovethat theseries forG{x,a)converges uniformlywithrespecttoainanintervalincludingthepointa=when x hasanyvalue such thattheseriesfor/(ir)isconvergent. Itmaybeassumed, without lossofgenerality,thatxispositive f,andwe shall then take |ct |sosmall that itdoes notexceed Ix;weshallnowprove thattheseries forG{x,a)converges uniformlywhen— |a;^a$|a?. Byobservingthat a^X+a—\/\x{x+2a.)]=^t- ;^.v ^^^- ^^x±a+\/{x{x±2a)} <itt'/x, andthattheseries am Jq(mx±2ma)X im^[x ±a+\/{x{x ±2a)]] *Sincewearenotassuming more than theconvergence off(x)atasingle point,itisnot permissibletoinferfrom§19'5thata,„must beo{^/m). tThefunctions under consideration areeven functions ofx;andsinceG(0,a)=0,thespecial case inwhich x=needs nofurther consideration. 19-52] SCHLOMILCH SERIES 641 isuniformly convergent (upperorlowersigns throughout being taken), we seethatG{x,a)differs from W{x+2a).Jo(7nw+Ima)+\lix—2a).Jo{mx—2ma)—2\Jx.J,,(ma;)] bythesum oftwo series, each ofwhich isuniformly convergent. Itistherefore sufficient toestablish theuniformityoftheconvergenceof thelastseries which hasbeen written down. Now takethegeneralterm ofthis series, namely _^im^'r^(^. ^2a).J„(mx+2mo)+V(^-2a).Jo(mx-2ma)-2V^•-h("ix)], andwrite itintheform ttrV\miT) |_^ ^^^mxJVma / y(2x\ cosimx—l-7r) sin2ma — )'6m{x?-4o-)m-a. (2x\ sin(ma;—jtt)cos2ma XtmrJ' i*>mx{a;-—4a-)* ?h- +"""VimTT.24>{mx)—<I>{mx+2/?ia)-^{mx—2mo) 4m^a^ where$(y)isdefined bytheformula <!>^;i/)=[P(y,0)- 1]cos{1/_i,r)- |1+Q(^, 0)|sin{y-Itt). Thegeneralterm isthusexpressedasthesum offour terms, andwe proceedtoprovethateach ofthefour series, ofwhich these terms arethe general terms,isuniformly convergent. The firsttwoseries areprovedtobeuniformly convergent,inconnexion with thetheoryoftrigonometrical series*; and the third isobviously uniformly convergentfrom thetestofWeierstrass. Todealwith thefourth series, weobserve that,bythe firstmean-value theorem, numbers f6and 6iexist such that^-1<^<1, -l<6'i<l, *Cf.Modern Analysis, §§9'62, 9'621. Ithasbeen thegeneral (butnotinvariable) custom to obtain various propertiesoftheseries without establishing theuniformity oftheirconvergence. Theconvergenceoftheseries forf(x)isrequiredtodealwith the first series;thesecond series canbedealt with inconsequence ofthelessstringent hypothesis that«„j=o(,^'w). tThenumber 6isafunction ofavariable twhich willbeintroduced immediately. W.B.F. 41 --J642 THEORY OFBESSEL FUNCTIONS[CHAP. XIX forwhich 2<I>(mx)-<I>(mx+2ma)—O(mx—2?Ha) =2ma I{4>'{mx-2mat)-<!>'(mx+2moit)}dt Jo =-2ma I4<mat <!>"(mx-2ma9t) dt Jo =—4tm.^a^ <t>"(7nx—2madi). Since <1>"(_?/)=(l/^/")when yislarge,itisevident that 2<i>"(mx-2mae,) isuniformly convergentwithrespecttoa. HenceG(x,a)isexpressedasthesumofsixseries each ofwhichconverges uniformlywithrespecttoawhen—lx<a< ^x;andtherefore limG(x,a) isequaltothesumofthelimits oftheterms oftheseries forCr{x,a),i.e. it ISequaltoxf(x), providedthattheseries for/(a;)isconvergent;andthis is thelemma tobeproved. 19"53.Lemma II. Weshallnextprove that, with thenotationo/§§19"51, 19"52, thecondition that arn=o{\/m)issufficienttoensure that ,.(x+a)F{x+2a)-\-(x-a)F(x-2a)-2a;P(x)_ a-*o a forallvaluesofx. Asin§19"52, weneed considerpositivevalues ofxonly;andweexpress theseries foraG{x,a)asthesum ofsixseries each ofwhich iseasilyseen to beuniformly convergent when—\x<a< \x,byapplyingthetheorems con- cerning trigonometricalseries which wereused in§19'52. Hence lim[aG(x,a)] »a—lim(looo)—1lim"I[{x+a)Jq(mx+2ma)a^o m"ia^o4/>ra +(x—a)Jo(mx—2ma)—2xJo(mx)]=0, andthis isthelemma tobeproved. 19-54. Theanalogue ofRiemann's theorem* ontrigonometricalseries. Wecannowprove that, iftwoSchlomilch series ofthetypenowunder consideration(i.e.with v=0,andwith Struve's function absent) converge *Cf.ModernAnalysis, §9-63. 19-53, 19-54] SCHLOMILCH SERIES 643 andhave thesame sum-functionthroughouttheinterval(0,it),then corre- spondingcoefficients inthetwo series areequal. Theformal statement of thetheorem isasfollows: TivoSchlo milch series, ofthespecial type,whichconverge andareequalat allpoints oftheclosed interval(0,tt),with thepossible exception ofafinite numberofpoints,must havecorresponding coefiicients equal,unless theend- points and TTarebothexceptional points. Ifthesepointsareexceptional points,thetwoseriesmay differ byaconstant multiple oftheseries i+I{-y'J.imx). ?n=l Letthedifference ofthetwoseries be 00 |«o+2amJo{mx), in=1 and letthesumofthis series bef(x),sothatf{x) convergestozero forall values of^between and tt,excepttheexceptionalvalues. Let^1,^2beanypoints (excepttheorigin)oftheinterval(0, tt),such thatthere arenoexceptional pointsinside* theinterval(^j, ^.,). Weproceedtoprove that,ifF{.r)isthefunction associated with the Schlomilch series forf(x), thenF(*•)isalinear function oflogicinthe interval(fj, ^^).This istheanalogueofSchwarz' lemma-f-. If^-1, orif^=-1,and if <^{x)=eF(-)-F(I:)- tlWi)'^^^-^-^^^'^^ +A^ _logOr/|,) _' ^' log(6/1.)^^"^ ^'\ then(p(x)iscontinuous when^^^x$|.,,and Ifthefirstterm of^{x)isnotzero:j: throughouttheinterval(|^i, I2).there willbesomepointcatwhich itisnotzero. Choose thesignof6sothatthe firstterm of(^(c)ispositiveatc,andthen choose hsosmall that<^(c)is stillpositive. Since(^{x)iscontinuous in(^1, fa),itattains itsupper bound which is positivesince(c)ispositive.Let itattain itsupper bound atCi,sothat ^1<Cl<^ NowbyLemma I(§19-52) lim(^•i+Q)<^(Ci +^"H( Ci-a)(/)(ci-2a)-2c i0(ci)^^^^ *Thepoints fj,^.,themselves maybeexceptional points, tCf.Modern Analysis, §y-(J31. XIfitiszerothroughout (^j, ^.,),thenr(.c)isobviously alinear function oflog.r. 41—2 644 THEORY OFBESSEL FUNCTIONS [CHAP. XIX But (f)(C]+2a)^(f)(Ci), (^(Ci—2a)^(f>{Ci),sothelimit onthe leftmust be negativeorzero. This contradiction shews that the firstterm of^(x)must bezerothroughout (|i, fa),that istosaythatF(x)must bealinear function oflogx;andthis isthetheorem tobeproved. Hence thecurve whoseequationisy=F{x)consists ofasetofsegments oflogarithmiccurves withequationsofthetype y=Alogx+B. Now, by§19"51,P{x)iscontinuous in(0,tt),andsotheselogarithmiccurves areconnected attheexceptional points;andthecurve y=F{x)cannot have anabrupt changeofdirection atanexceptional point, because, byLemma II, lim a-*.0F(f+2«)-F(|) _ T(il-T(i-M^ ^2avb /2a=0, evenwhenfisanexceptional point;that istosay fF'(f+0)=fF'(f-0). Hence theconstants AandBcannot bediscontinuous attheexceptional points, and sothey have thesame values for allvalues ofxinthe interval(0, tt). Consequently, when <a;<77, %af)X^—z ;;^=Alogx+B. Makex-^0; theseries onthe lefthasalimit, namely because itisuniformly convergent.Therefore A\ogx+Bhasalimitwhen x-*0, andsoAiszero. Consequently, when ^x^tt, —=laox--B ; andtheseries ontheleftconverges uniformly throughout (0,tt),sointegrations term-by-termarepermissible. Replace xbyxsin0,multiply bysin6,andintegratefrom to^tt.Then, by§1211, ^aox'-B= i""'„f'Vo(m^sin6')sin^(^^ *ttmSinmx m=lrri^x Hence, when ^x^tt, " cimsmmx_ ^.3/3 m=l m^ 19-6] SCHLOMILCH SERIES 645 Multiply bysinmxandintegi'atefrom tott;itisthen evident that TTfto TT-B—To7r*ar, 2m" m 2ni^ Since «,„isgiventobeo(\/m),thisequationshews that B=j\TT'tto ,a,n=(-)'"«t). Hence wemust have f(w)=a.1+S(-rJo(mw) 2m=l From theresults contained in§19'41concerningthebehaviour ofthe series ontherightatw=andata;= tt,itisevident thatf{a:)cannot bea convergentSchlomilch series ateitherpointunless Uqiszero;and thisproves thetheorem stated atthebeginningofthis section. 19*6. Theoremsconcerningtheconvergence ofgeneralisedSchlomilch series. Weshallnowstudy brieflytheseries Weshall firstprove that, ivhenv<\,thecondition that the(m+\)thtermof theseries tends tozeroasm-^ ccatallpoints ofanyinterval ofvaluesofx issufficient toensure that a,n={m^+^X by,={m^+i). [Note.Iftheoriginisapointoftheinterval inquestion, then thetheorem that isobviously true.] Since theseries under consideration isunaffectedbyachangeinthesign ofXifthesignsofallthecoefficients 6,„arealsochanged,nogeneralityis lostbyconsideringaninterval ontherightoftheorigin. We callthisinterval /j ;and, atallpointsof/j,wehave,by|10'41(4), a-,nJv{mx)+h,„H„{mx) c^^m[P(n),r, v)cos(mx— |/'7r—jtt—tjyt) {^m.x)" (Imxy+'^x/ir —Q{mx, v)sin{mx— |;/7r—jtt— 7/„()]+h,n•{ni~^), where a,„=c„,cos77„,,6,„=c,„sinr}„^. We-^owsupposethata^and 6„,arenotboth o{m"'^^); wehave toshew that thishypothesisleads toacontradiction. Ifa,„and h,„arenotboth0(7/^"+*),apositive number emust exist such that c,„>€m"-^^ whenever misgiven anyvaluebelongingtoacertainunending sequence m,, m.,,m.j,.... 646 THEORY OFBESSEL FUNCTIONS [CHAP. XIX Wenowprove, exactlyasin§19'5, that, atsomepointXof/j,the inequalities Icos(inX—IvTT-Itt— J7,n) i^2\/3,Isin(mX—^vir—^tt— 7;,„) |^\ aresatisfied whenever mhasanyvaluebelongingtoasequence (w/)which isasub-sequenceofthesequence (m^). Forvalues ofmwhichbelongtothissub-sequence wehave ;a,„J,(mZ) +6>»H,(mZ)l ^ig(x/3-1) and, since j^—^isnegative,theexpressionontheright cannot bearbitraril}- small. This isthecontradiction which issufficient toprovethata„,,and6.„, must bothbeo(m''+*)ifthe(m-t-l)thterm oftheSchlomilch series tends to zeroatallpointsofJj. Thereader maynowprove (asin§19"5) that, ivhen v<\,ifthegeneralised SchldmilcJi sei^iesconverges throughout anyinterval, thenecessary andsufficient condition that itmayconverge foranypositive valueofa;{luhether apoint of theinterval ornot)isthat theseries S in=lm"^\/(«D 1'°'^^^^^-^''^-^'^-^-^ sm(ma;-hvTT—^TT-7}„M+^^ ' should heconvergent forthatvalueofx. 19'61. Theassociated function. liBtustake—^<v<-^,and letthesum oftheseries i«„ ^a,„Jy(mx)+b^H^(7nw) atanypointatwhich theseries isconvergentbecalledf^{x). Let OqX^__^a,nJ„(mx)+hmHp{mx) 11) '•'^*'>-81^1^+2) Zi i^^.(^mxy Then Y„(x)willbecalled thefunction associated with theSchlomilch series whose sum'\^fv(x). Itiseasytoprove that, iftheseriesdefining f^(x) convergesatallpoints ofanyinterval, then theseriesdefining F^(x)converges forallrealvaluesofx. Theonly respectinwhich theproofdiffers from theanalysisused in §19'51 isthattheadditional theorem thatIIy(x)/x''isabounded function of therealvariable xhastobeused. 19-61, 19-62] SCHLOMILCH SERIES 647 Again,let (2)G, (cc,a)=i[(x-^2va +a)F,(a;+2a) +(x- 2va.- cl)F^{x-2a)-2,xF^(a;)]/a-. Then, justasin§19'52, wemayprovethat* 2'^h limG^{x,a)=xf^{x)--^^— ,.xt^.ix ^— a^o"^ r(i/+|)r(^)„,=i m atanypoint xatwhich theseriesdefining f^{x)isconvergent, iprovidedthat a„,andh^(ii'&both o{nV^^) andthat theseriesShmlmisconvergent.m=1 Further wemayprovethat lim[aG;,(.'r, a)]=0, '''^ ''a-*0 j)rdvided onlythat a.,nand 6„,areboth o(m'"^^), whether theseries —bmlmis m=1 convergentornot. 19-62. Theanalogue ofRiemann's theorem. Wecannowprove that, iftwogeneralisedSchlOmilcJi seriesofthesame order v(ivhere—o<''<i)converge andhave thesamesum-functionatall points oftheclosed interval (—tt,tt)luith thepossible exception ofafinite numberofpoints (itissupposedthat theoriginand. thepoints ±irarenotall exceptional points), andifthecoefiicients ofthetermscontainingStruve'sfunc- tions inthetwo series aresufficiently nearly equal feach toeach, then all corresponding coefficientsinthetwoseries areequal. Letthedifference ofthetwoseries be ; . ^tto V'^»t-^fi'»i-^)+bmH^(mx) and letthesum ofthis series bef„{x),sothattheseries forf„(x) converges tozeroatallpointsoftheinterval (-tt,tt)withafinitenumber ofexceptions. Theconvergenceoftheseries fovfy(x) nearly everywhereintheinterval (-TT,tt)necessitates theequations «.„,=(m''+^), &„,-(m-'+i). Thestatement thatthecoefficients ofthetermscontainingStruve's func- tions inthetwoseries aretobesufficiently nearly equalistobeinterpretedto mean iksci b,a.^asw-*ooinsuch awaythat1~isconvergent. Wenow discuss thefunction Fp(a;)associated with theSchlomilch series iovfv{x).Itcanbeprovedjthat iftheinterval(|^i,^2)issuch thattheorigin *Thepresenceoftheseries ontherightisduetothelack ofhomogeneityintliedifferential equationsatisfied byStruve's function. }Thisstatement willbemade definite immediately.'' XItseems unnecessarytorepeat thearguments already used in§19"54. 648 THEORY OFBESSEL FUNCTIONS[CHAP. XIX andtheexceptional points (ifany)arenotinternalpointsoftheinterval, then F^(x)isalinear function* ofa;~^'' intheinterval. Itmaythen beshewn thattheexceptional pointsdonotcause anydiscontinuityintheform ofF„(x), andhence wededuce that (F^{w)=Ax—'''-{-B, (0<x^tt) If, (a;)=A'\x [-2-+B', (0>x^-7r) where A,B,A',B'areconstants. Now take theequation aoX^ ya„iJ„(mx)+6,„H,(mx)F.{X)-gp^~^- ^^^^^ mKi^vixy' replacea;by«sin6,multiply bysin-''^^^/cos-" 0,(which hasanabsolutely convergent integi^al)andintegratefrom tohrr.The series forF„(a;sin6^) converges uniformlyinthisinterval ofvalues of0,soterm-by-term integra- tions arepermissible. Itisthusfound that sin-''+^^ ,^ anx^ fi''sin^-''+^0 ,F^(^^^"^)-^^^^^^= 8r(.+2).L cos-"6 '^-"UrnJv(tnxsin6)+6,„H,(mxsin6)sin''+i 6dd dd -2 „i=1J wi^(^mxycos-" 6^ _aQX-V{\—v)T{l—v) ^->«„jSin7H.x--F 6,„(1—coswia;) 12r(i) r(i) ,Zx m'x When wesubstitute forF^{xsin6)wededuce that /Tx V«msinr«^-+ 6^(1-cosw«) a^x"" Ax^-""" V(%)^„, ..«i, m'=12-r(|-.)"-'ft''r(. +l). whenO^x^ir \andasimilarequation maybeobtained when 0^,i'^ —vr. Since «,„and 6„,areboth o(7?i"+^),itispermissibletodifferentiate(1)twice term-by-termwhen >f>—^;but itmay onlybedifferentiated once if Ifwedifferentiate, twice oronce asthecasemay be,theresultingseries onthelefttends toalimit asx-^0, buttheresulting expression ontheright fails todosounlessAiszero. Weinfer that^=0,andinlikemanner A'must bezero;thecontinuit}'^ ofF„(ii';)attheoriginthenshews thatBand B'must beequal. Itnow follows from (1)that /o\ V"">^^^^"-^+br„(l-C0.Smx) a^af „j^. ,,, (2)S^-, =^-Bxr{.+l) when—TT^x^'ir. *When Viszero x''^"hastobereplaced bylogx. 19-7] SCHLOMILCH SERIES 649 Multiply (2)bycosmx andintegratefrom—tttott;andthen (3) b,,=0. Again, multiply bysinma; andintegrate; andthen (-)'«a,„=«„+vf-{2Br(v +l)-^ao7r% Thisequationisinconsistent with thefactthat a„,iso(^"+2)unless 25r(;- +l)=la„7r=, andthen a^=(— )'"ay. Hence theseries forf^(x)must reduce to rh ,^(-r-/.(m^- y Now atleastoneofthepoints 0,tt,—ttisnotanexceptional point;and theseriesfor/i,(a;)cannotconvergeatthatpointunless ««iszero, sothat a,^ isalso zero. Wehave thereforeproved that, iftheseries%b„Jmisconvergentallthe coefficients «,„and!),„must vanish;that istosay,thetwoSchlomilch series withwhich westarted must havecorrespondingcoefficientsequal. And this isthetheorem tobeproved. Wehave therefore established forSchlomilch series inwhich—\<v<\ theoremsanalogoustotheusual theoremsconcerningtherepresentationof null-functions bytrigonometricalseries. 19"7. TheoremsofRiemannstypeconcerningseriesofBesselfunctions and Dims seriesofBesselfunctions. Weshall now^giveaverybrief sketch ofthemethodbywhich theseries discussed inChapter xviii, namely 00 X m=l m=\ (inwhich v>—^)maybeinvestigatedafter themanner ofRiemann's investi- gationoftrigonometricalseries. Themethod isidentical with themethod ofinvestigationofSchlomilch seriesjustgivenin§§196—19"62, thoughthere arevariouspointsofdetail*, whiclv^o notarise inthecase ofSchlomilch series, duetothefactthatJ,„ andX,,,arenotlinear functions ofm. *These pointsofdetail areverynumerous andthere isnospecial difficulty indiscussing any ofthem;but itisatedious andlengthy process tosetthem outinfull,andtheydonotseem to beofsufficient importancetojustify theuseofthespace which theywould require. Thereader who desires toappreciate thedetails necessaryinsuch investigations mayconsult thepapers by C.N.Moore, Trans. American Math. Hoc. x.(1909), pp.391—435; xii.(1911), pp.181—206; XXI.(1920). pp.107—156. 650 THEORY OFBESSEL FUNCTIONS [CHAP. XIX Inthe jfirstplace,itiseasytoprove bythemethod used in§19'5 that iftheseries 00 00 m=\ m=l converge throughout anyinterval ofvalues ofx,then am=o{s/m), hm=o{s/in). Nextweconsider theassociated function;wewrite andthen thefunction associatedwith/(.t')isdefined bytheequation F{x)= 2 m=1Jm"•^' Itmaybeproved that,whenxhasanypositivevalue forwhich theseries defining /(^)isconvergent, and iftheexpression ]-„[(a-+2m+a)F{x+2a)-2xF{x)+{x-%>a-a)F{x- l-x)] isarrangedasaseries inwhich themthterm has a„,forafactor, then the latter series isuniformly convergentwithrespecttoainanintervalcontaining thepointo=0,andthat itslimitwhena^-0 is—x^~''f{x). Itmayalsobeproved that,whether theseries forf{x) convergesornot, thecondition that «,„=o{\Jm)issufficient toensure that ~[{x+2m+a)F{x+2oi)-2xF{x)+{x-2m~tx)F {x-2a)]4a tends tozero w'ith a. Theproofsofthese theoremsdejjendonanumber oflemmas such asthelemma* tliat 2siiiVm+i« sinVm« isabounded function ofa;proofsoftheleiumas canbeconstructed onthelines ofthe proofsinthespecial (trigonometrical)case inwhich v=^. Itnow follows intheusualmanner(cf.§19'54) that,when/(a;)isanull- functionthroughouttheinterval(0,1),thenF{x)satisfies thedifferential equation andsot F{x)=A+Bx--", whereAandBareconstants. Thisequationisvalidwhen <x^1. *Cf.Modern Analysis, §9"62. tWhen v=0,F(x)^A+B'logx. 19-7J SCHLOMILCH SERIES 651 Now sincev>-^, JAjm-f')KJwi'yisbounded whenO^ic^l whatever bethevalue of«i;and so,when* v<^,theseries m-lJm' KJvv^) converges uniformly when ^,r^],bythetestofWeierstrass. HenceF(x)isacontinuous function ofxintheclosed interval and soB iszerowhen vispositive:andBiszero inthecase v=0. Foranyassignedvalue ofnmultiplyingtheseries fori''(A')byx""^^Jv{jn^') doesnotdestroytheuniformityofitsconvergence; and,whenweintegrate, wefindthat Cln.JJ'(jn)=jn{Ax"+Bx'") XJ^ (j^X)dx J Now,when nislarge,By^^"^'-(A+B)J:(j.,) I-A'(i..)l-y(^.y. and sotheformulajustobtained for6„isinconsistent with theequation f'rt=o{\/n) when v>—^unless bothA+BandBarezerof ;andthen Unis zero. Hence aseries ofBessel functions(inwhichv>—\)cannotconvergeto thesum zero atallpointsoftheinterval (0,1),withthepossible exceptionof afinitenumber ofpoints (the originnotbemganexceptional point| when V>^),unless allthecoefficients intheseries arezero. * We infer thattwo series ofBessel functions, inAvhich i>>—^,cannot convergeandbeequalatallpointsoftheinterval(0,1),with thepossible exceptionofafinitenumber ofpoints,unlesscorrespondingcoefficients inthe twoseries areequal. Dini's series§ f(x)=%b,,,J^{\„ix)m=l maybedealt with inthesame manner. The associated function isdefined bytheequation F(x)=l^^_- ^^,m=1i^in•'"' *When v'^h,theconvergence oftheseries iovf(x)jx^ at.(=0 issufficient toensure the uniformityoftheconvergence. tAnexception might occurwhen r-1=-\;butthis isthetrigonometricalcase. XThe series divided bya,"tlienhastoconvergeattheorigin. §Itissupposedforthepresent thatif+0O, sothatnoinitial term isinserted. 652 THEORY OFBESSEL FUNCTIONS [CHAP. XIX and itisinferred thatwhenf{x)isanull-function throughout (0,1),then constants AandBexist such that F{x)=A+5a;-^ n K —A,,,=\f((Ax"+Bx-") xJ^(\nx)dx J andB=when v'^0. Now, Avhen nislarge sothat, if5^0,-/.+i(^n)=(\«-*), J.-1(X,)=(X„-3), ?>n~7ri?X,iiz+l 2''r(i;)' andthis isinconsistent with theequation bn=(\/n), since v>—^. HenceBiszero,andtherefore hn(i-^,^JJ'{K)+JJHK)—AXn^v+i \\i)' Thisequationisinconsistent with theequation K=(\/'') unlessAiszero, since J^+i(\n)isnotzero;andthen bniszero. Wenext consider whathappenswhen^+1'iszeroornegative;inthese cases Dini's series assume theforms 00 m=l 00 bolu0^o^)+ 2bmJ,.(\n^)>m=l respectively. Inthesecond ofthetwocases theprevious argumentsareunaffected by theinsertion ofaninitial term;the firstofthetwocases needs more careful consideration because the initial term tobeinserted intheassociated function is _bpX^ 4(i.+l)' andhence, whenn^l, /•If 5^v+-i\ bn-Jj" (X,/)=X,;-I\Ax"+^"+Bx-")- xJ^{Xnx)dx -'-^^ IFiTTT)-^^ ^^>|+ 207Tr)• 19-7] SCHLOMILCH SERIES 653 Since hn=o{\/n)weinfer first thatB=0,byconsideringtheterm in {•i^n}", andthen that 1^= ;andso6„=forallvalues ofn. Weinfer also that, asinthelimitingcase ofseries ofBessel functions, Dini's series ofBessel functions cannotrepresentanull-functionthroughout theinterval(0,1),andthat iftwoofDini's series (with thesame vandH) convergeandareequalatallpointsoftheinterval(0,1),with theexception ofafinitenumber ofpoints,thencorrespondingcoefficients inthetwoseries areequal. CHAPTER XX THETABULATION OFBESSEL FUNCTIONS 20*1. TablesofBessel Functions andassociatedfunctions. Itisevident fromaconsideration oftheanalysiscontained inChapters vii, VIIIandXVthatalarge partofthetheoryofBessel Functions hasbeen con- structedexpresslyforthepurposeoffacilitatingnumericalcomputations connected with thefunctions. TotheMathematician suchcomputationsare oflessinterest andimportance*than theconstruction ofthetheories which makethempossible;buttothePhysicist numerical results haveasignificancef which formulae mayfailtoconvey. AsanapplicationofvariousportionsoftheTheoryofBessel Functions, ithasbeen considered desirable toinsert thisChapter,which contains an historical account ofTables ofBessel Functions which havebeenpreviously published, togetherwith acollection ofthose tables which seem tobeofthe greatestvalue forthepresent requirementsofthePhysicist. Thereader willnotbeconcerned with themonotony andtechnical irrele- vance ofthisChapterwhen herealises that itcanbereadwithout theefforts requiredtomaster theprevious chaptersand toamplif}^ argumentssoruth- lesslycondensed. The firstTables ofJq(x)andJ^(x)werepublished byBessel himself in hismemoir onPlanetary Perturbations, i?e?'/i?ierAbhandlungen, 1824[1826], pp.46—52.These tablesgivethevalues oiJn{x) andJ^ix) totenplacesof decimals forarangeofvalues ofxfromx=tox=3"20with interval 0*01. Ashort Table ofJi,{x)andJ^ix) tofourplacesofdecimals wasconstructed byAiry,Phil.Mag. (3)xviii.(1841), p.7;itsrangeisfrom a;=toa;=10*0 with interval 0"2.Airy|hadpreviously constructed aTable of2/,{x)lx,of thesamescope. Thefunction Ji{.v)lx wassubsequentlytabulated tosixplacesofdecimals byLommel, Zeitschrift furMath, undPhys.xv.(1870), pp.164—167,witharange fromx--0 to.r=20'0 with interval 0"1;thisTable, with aTable ofJ^(^)/^^ wasrepublished byLoumiel, Miinchener Abhandlungen,xv.(1886), pp.312—315. *Of.Love. Proc. London Math. Sac.(2)xiv.(1915), p.184. tCf.Lord Kelvin's statement "Ihavenosatisfaction informulas unless Ifeeltheir arithmetical magnitude—atallevents when formulas areintended fordefinite dynamical orphysical problems." Baltimore Lectures (Cambridge, 1904), p.76. +Trans. Camb. Phil. Soc. v.(1835), p291.ATable of2Ji (.r)/xand itssquare, tofourorfive placesofdecimals, inwhich therangeisfrom tothecircular measure of1125°(with interval15°), wasgiven bySchwerd, DieBeugungserscheimmgen (Mannheim, 1835), p.146. 20-1] TABULATION OFBESSEL FUNCTIONS 655 Inconsequenceoftheneed ofTables oft/„{x)withfairly largevalues of nand .?•forAstronomicalpurposes, Hansen constructed aTable ofJo(*') and Ji(x)tosixplacesofdecimals with arangefrom x=to a;=100with interval 01; thiswaspublishedinhisErinittelungderahsoluten Storungen inEllipsenvonbeliehigerExcentricitdt undNeigung (Gotha, 1843). Hansen's Table wasreprinted bySchlomilch* andalsobyLommelf whoextended it toX=20. These tables, however, aresuperseded byMeissel'sgreatTable ofJo(x) a,ndJi(x) totwelveplacesofdecimals;]:, publishedintheBerliner Abhand- lungen, 1888;itsrangeisfrom a;=tox=15-50 with interval O'Ol. Meissel's Table wasreprintedinfullbyGrayandMathews, A'Treatise anBessel Functions (London, 1^95), pp.247—266,andanabridgementofit isgiveninTable Iinfra, pp.666—697. ATable ofJq(x)and Jj(x)totwenty-one placesofdecimals, fromx= to .j;=6"0with interval 0*1,hasbeen constructed byAldis, Proc. RoyalSoc. Lxvi.(1900), p.40. ATable of./((mtt)tosixplacesofdecimals fovn=l, 2,3,...,50hasbeencomputed by Nagaoka, JuurnaloftheColl.ofSci.Imp.Univ.Japan,iv.(1891), p.313. Thevalue of./o(40)wascomputed byW.R.Hamilton from theascending series, Phi. Mag. (4)xiv.(1857), p.375. ATable ofJi(x)tosixplacesofdecimals from .r=20'lto.^=41with interval O'lor 0'2hasbeenpublished bySteiner, Math, undjyaiurwiss. Berichte ausUngarn,xi.(1894), pp.372—373. The earliest table offunctions ofthesecond kind wasconstructed by B.A.Smith, Messenger,xxvi. (1896), pp.98—101; this isaTable tofour placesofdecimals ofNeumann's functions F'"*{x)and F<'*{x).Itsrangeis from ^=to a;=l"00with interval 01andfrom^=1"0 tox=10"2with interval O'l. Amore extensive table ofthese functions isgivenintheBritish Asso- ciationReport, 1914, pp.76—82;this isaTable tosixplacesofdecimals whoserangeisfromx=toa;=15"50 with interval 0"02; ayearlater atable waspublished,ibid. 1915, p.33,inwhich thevalues ofF'"'{x)and F'^'{x) weregiventotenplacesofdecimals forarangefromx=tox—(rOwith interval 0*2andfromx=6"0tox=160with interval 05. Shortlyafter theappearanceofSmith's Table, anelaborate tablewascon- structedbyAldis, Proc. RoyalSoc.Lxvi. (1900), p.41,ofHeine's functions§ Ga{xf~am\ Gi(x)totwenty-one placesofdecimals; thereader should be * Zeitschrift furMath, undFhijs.ii.(1857), pp.1.58—165. tStudien iiber dieBessel'schen Functioneii (Leipzig, 1868), pp.127—135. XMeissel's Table contains amisprint, thecorrect value ofJg(0-62) being +0-90G18..., not +0'90518.... A.nailditioual misprint wasmade inthereprintoftheTable hvGrayandMathews. «sThese functions were alsotabulated byB.A.Smith, Fhil.Mag. (5),xlv.(1898), pp.1*22— 128;thescopeofthistable isthesame asthat ofhisTable ofI'W[x)and i'(^)(.r). 656 THEORY OFBESSEL FUNCTIONS[CHAP. XX reminded that these functions areequalto—^7rYo(x) and—^tt Fj(a;)re- spectively.TherangeofAldis' Table isfromx=boa:=6with interval O'l. Another table ofthese functions with asmaller interval waspublishedin theBritish Association Report, 1913, pp.116—130; this tablegivesthe functions tosevenplacesofdecimals forarangefrom a;=toa;=16'00 with interval 001.TheReportfor191o, p.33,contains atable totenplacesof decimals fromx=65to^=15"owith interval 0*5. Thefunctions Yq(x)andFj(x)aretabulated tosevenplacesofdecimals inTable It'iifra ;this table hasanappreciable advantageover theBritish Association Tables*, inthat theauxiliarytables makeinterpolationatrivial matter; intheBritish Association Tablesinterpolationisimpracticable. Bymeans oftherecurrence formulae combined with theuseofthetables which havenowbeen described, itisaneasymatter toconstruct tables of functions whose order isanyinteger. Such tables ofJ^ix) were constructed byHansen andreprinted bySchlomilch andLommel after their Tables of Jq(x)andJ^(x).Subsequently Lommel, Milnchener Abhandlungen,xv.(1886), pp.315—316,publishedaTable ofJn{x) tosixplacesofdecimals, inwhich n^O,l, 2,..., 20,andx=0,\, 2,..., 12;thisTable isreprintedinTableIV infra, pp.730—731.ATable ofJn{x)ofpracticallythesamescopewasalso published byMeissel, Astr. Nach. cxxviii.(1891),col.154—155. Amuch more extensive Table ofJn{x) wascomputed byMeissel, but it seems thatheneverpublishedit.Hecommunicated ittoGrayandMathews forpublicationintheir Treatise, pp.267—279. This tablegives Jn(^)to eighteen placesofdecimals whenn=0, 1,2,...,60,andx^O, 1,2,...,24. Some graphsof/„(x)were constructed, withthehelpofthelast-mentioned table, byHague,Froc.Phys.Soc.xxix. (1917), pp.211—214. ThecorrespondingTables offunctions ofthesecond kind arenotsoex- tensive. TheBritish AssociationReport, 1914, pp.83—86contains Tables ofGn{oc) tofiveplacesofdecimals forfw=0,1,2,13fortherangea;=to6*0 with interval O'landx=6"0to16"0with interval 05. Similar TablesfofF*"'(a;)tosixplacesofdecimals (with theintervals intheearlierpartequalto0*2)appearedintheReportfor1914, pp.34—36. Some values ofHankel's functionY„(a') hadbeengiven previously by Nicholson, Proc.London Math. Soc.(2)xi.(1913), pp.113—114. ATable ofYn{x) toseven(ormore) significant figuresiscontained in TableIVinfra.This hasbeencomputedfrom Aldis' Table ofGo(x)andGi(x). *Inthecourse ofcomputing TableI,asmall part oftheBritish Association Table of(?„(.r\ andGj(.r)waschecked, andthelastdigitsinitwerefound tobeunreliable inabout 5%ofthe entries checked. +Forthelarger values ofnthefunctions arenottabulated forsmall values ofx. 20-1] TABULATION OFBESSEL FUNCTIONS 657 Tables of\ogiQ['\/(^'jrx) .\H^^^i (x)\']toeight significant figuresai^egiven intheBritish AssociationReport., 1907, pp.94—97.The valuesassigned toVare0,|,1,1|, ...,6|,andtherangeofvalues ofxisfrom ;»=10to100 (interval 10)and100to1000 (interval 100). For thisrangeofvalues ofx, theasymptotic expansion (§7'51)givessorapidanapproximationthat the Table isoflessvalue than atable inwhich thevalues ofxandtheintervals areconsiderablysmaller. Functions ofthe firstkindwithimaginary argumenthavebeen tabulated intheBritish AssociationReports. TheReportfor1896, pp.99—149,con- tained aTable of/„(x)tonineplacesofdecimals, itsrange beingfromx=0 toX=5"100 with interval 0001.ATable of/j(x)ofthesamescopehadbeen published previouslyintheReportfor1893, pp.229—279;anabridgement ofthis(with interval O'Ol) wasgiven byGrayandMathews intheir Treatise, pp.282—284. Tables of/„(x)and/j(x)totwenty-one placesofdecimals havebeen con- structedbyAldis, Proc. RoyalSoc.lxiv.(1899), p.218.Therangeofthese Tables isic= tox=Q'0with interval 0*1;Aldis alsogave (ibid. p.221)the values of/„(x)and /j(x)forx=7, 8,9,10,11. Extensive tables connected withIo{x) and 1^(x)have beenpublished by And'mg, Sechsstellige TafelnderBessel' schen Funktionen imagindren Arguments (Leipzig, 1911). These tablesgive log,o /o(^-c)andlogio {^i(ic)/^}fromx=0 tox=10"00 with interval O'Ol.The}-alsogivethevalues ofthefunctions x/(2?r^).e-*/o(.«), V(27ra;).e-^/j {x), logio [V*"•U(«)}andlogio^x.I^{x)] forvalues ofxfrom a;=lO'O to ^;=50"0 (interval O'l),.t=50to a-=200 (interval 1),^=200tox=1000 (interval 10),andforvariouslargervalues oix. Table IIinfra, pp.698—713,givesthevalues ofe~^/o(;c) ande^^Ii{x)\ these havebeencomputed,forthemostpart,byinterpolationinAldis' Table. The earliest tables offunctions ofthetypeK^ (*')were constructedby Aldis, Proc. RoyalSoc.lxiv. (1899), pp.219—221. Thesegive K^ia;) and /Tj(x)totwenty-one placesofdecimals forvalues ofxfrom x= tox=6'0 with interval O'l,and also tobetween seven andthirteensignificant figures fromx=5"0tox=12*0with interval O'l. Thevalues ofe-^Ko{x) ande^K^ (x)inTable IIinfra werecomputedwith thehelpofAldis' Table, likethevalues ofe~^Io (x)and e~-^/i (x). Bymeans ofrecurrence formulae, /«(.«) hasbeen tabulated totwelvesigni- ficantfiguresforn=0,1,2,...,11over therangeofvalues ofxfromx= tox=GO with interval 0'2.These Tables ofIn(x) werepublishedinthe British AssociationReport, 1889, pp.29—32,andreprinted byGrayand Mathews intheir Treatise, pp.285—288.Anabridgement (tofivesignificant figures)ofthese Tables hasbeengiven byIsherwood, whoadded tothem w.B.F. 42 658 THEORY OFBESSEL FUNCTIONS [CHAP. XX Tables ofKn{x)tofivesignificant figuresforn=0,1,2,...,10overtherange ofvalues ofxfromx=Qtox=6*0with interval 0"2.Isherwood's Tables were publishedintheMem. andProc, Manchester Lit.andPhil. Soc,1903—1904, no.19. Tables ofe~^In(x) andKn{x)tosevenplacesofdecimals aregivenin TableIVinfra, pp.736—739. The earliest Tables ofBessel functions oflargeorder were constructed by Meissel, whohascalculatedJ2n(")totwelvesignificant figuresfor ?i=10, 11,...,2\,Astr.Nach. cxxix.(1892),col.284; Meissel alsocalculated /n(lOOO) tosevensignificant figuresforu=1000, 999, ...,981, ibid, cxxviii.(1891), col.154—155. Thevalues of/„(n), Jn-i{n), FW(w), F'"-i)(?i), Gn{n), Gn-,{n) tosixplacesofdecimals forvalues ofnfrom n=l*to 7?=50(interval 1), n=bOtow=100 (interval 5),?i=100 ton=200(interval 10),??=200 to n=400(interval 20),n=400 ton=1000(interval 50),n=1000 ton=2000 (interval 100)and forvariouslargervalues ofn,aregivenintheBritish Association Report, 1916, pp.93—96. Tables ofJ,i(w), Jn'{n), Yni''^), J^/(^0tosevenplacesofdecimals aregiven inTable VIinfra, pp.746—747. The functions ber(x),bei(x),ker(x)andkei(x)have beenextensively tabulated onaccount oftheirimportanceinthetheoryofalternatingcurrents. Abrief Table ofber(x)andbei(x),computed byMaclean, waspublished byKelvin, Math, andPhys. Papers,iir.(1890), p.493. Tables ofJ^{x \/i) andV2Ji(^VOtotwenty-one placesofdecimals have been constructed byAldis, Proc. RoyalSoc.LXVI.(1900), pp.42—43; theirrangeisfrom x=to a;=6*0with interval 0"1.These areextensions oftheTable of J^{x sji)tonineplacesofdecimals fortherangefromx=toa;=60with interval 0'2publishedintheBritish Association Report, 1898, p.228,and reprinted byGrayandMathews intheir Treatise, p.281. Tables ofber{x),bei(a;),ker{x)andkei{x)tofoursignificant figuresfor a;=1,2,3,...,30,havebeenpublished bySavidge,Phil.Mag. (6)xix.(1910), p.53. The functions hex{x),bei(a;),ber'(a;)and bei'(a;)aretabulated tonine placesofdecimals, from a;= tox^10*0with interval 01intheBritish AssociationReport, 1912, pp.57—68;andaTable ofker{x),kei(x),ker'{x) andkei'{x)ofthesamescope (exceptthatonlysixorsevensignificant figures weregiven) appearedintheReportfor1915, pp.36—88.Tables ofsquares andproductsofthefunctions tosixsignificant figuresfi:om a;=toa;=10"0 with interval 02weregivenintheReportfor1916, pp.118—121. Thefunctionst7±(M+j)(a;) havebeen tabulated tosixplacesofdecimals by Lommel, Munchener Abh. xv.(1886), pp.644—647, forn=0,1,2,...,6 with x=l, 2,...,50,and(inthecase offunctions ofpositive order) w=7,8,...,14I 20-1] TABULATION OFBESSEL FUNCTIONS 659 witha;=l,2, ...,20, andw=15,...,34withsmallerrangesofvalues ofx;see TableVinfra, pp.740—743.ATable ofthesame functions tofourplacesof decimals with n=0, 1,2andfromx=to a;=8-0with interval 0-2 isgiven byDinnik, Archiv derMath, undPhys. (3)xx.(1913), pp.238—240. Functions related tot/±(»+j) {x)haverecentlybeen tabulated intheBritish AssociationReports. Thenotation used is V(iTTX).Jn+l {x)=Sn{x), (-fs/{^TTX).J-n-l {^'^^^n(«X En(x)= ICn(x)+iSn(x) I, andthefunctions tabulated areSn(x), G,i{x), En^{x), Sn(x), Cn{x), En'(x), and theirlogarithms.IntheReportfor1914, pp.88—102,thefunctions are tabulated tosevensignificant figuresfor /;=0, 1,2,...,17and .r=1,2,3,...,10, and intheReportfor1916, pp.97—107, forn=0,1,2,...,10 anda;=l-l, 1-2 .1-9 Functions oforder +|,+f,havebeen tabulated byDinnik, Archiv der Math, undPhys. (3)xviii. (1911), pp.337—338, tofourplacesofdecimals; thefunctions tabulated arer(l±l)J^i{x), T(l±§)t/±s(a?)from a7=to j;=8'0with interval 0*2;andDinnik hasalsotabulatedI±^(x), I^^^x),ibid. (3)xxii. (1914), pp.226—227 andJ±i(x), J±^{x),ibid.(3)xxi.(1913), pp.324—326. Allthese tables have therangea;=toa:;=8 with interval 0"2.TheTables ofr(l+^)J±^{x)arelessextensive thanTable IIIinfra, pp.714—729;but,with theexceptionofDinnik's tables, there exist no tables offunctions ofordersf,jandf. Inconnexion with functions oforder +^,Airy's Table ofhisintegral I cos^TT(^U^—771W)dw must bementioned;Airycalculatedbyquadraturesandbyascendingseries thevalues ofthisintegralforvalues ofmfrom—5*6 to+56with interval 0"2;aseven-figure Table fromm=—4tom=4isgivenintheTrans. Gamb. Phil. Sac. VI.(1838), p.402,andafive-figureTable fromm=—B'Qtom=5-6, ibid. VIII. (1849), p.599. Apartfrom thework ofEuler described in§15'5theearliestcomputation ofthezeros ofJq{x) andJi{x)isduetoStokes, Trans. Gamb. Phil. Soc. ix." (1856), p.180[Math, andPhys. Papers,ii.(1883), p.355]. Stokesgavethe values ofthe firsttwelve zeros (divided bytt)ofJq(x)andJj(x)tofourplaces ofdecimals. Inthesamememoir hegavethefirstfiftyzeros ofAiry's integi-al, andth^first tenstationary pointsofthisintegral. The firstnine zeros ofJo(x), Ji(x), ...,J^{x) werecomputed byBourget, Ann. sci.deI'Ecule norm.sup.ill.(1866), pp.82—87.Bourget'sresults are giventothreeplacesofdecimals;some corrections inhisTables haverecently beenmade byAirey*. *Phil.Mag. (6)sxxii.(1916), pp.7—14. 42—2 660 THEORY OFBESSEL FUNCTIONS [CHAP. XX Bourget's Tables have been reprintedsofrequentlythat theirauthorship hasbeen overlooked bythewriters ofthearticles onBessel Functions intheEncycloplidieder Math. Wiss.andtheEncydopediedesSci.Math. The first fivezeros oiJi{x) andJo{x) weregiventosixplacesofdecimals byLommel, Zeitschrift fiirMath. und.Phys.xv.(1870), p.167andMunchener Ahhandlungen,xv.(18S6), p.315. The firsttenzeros ofJo{x) werecomputedtotenplacesofdecimalsby Meisselj Berliner Ahhandlungen,1888. The firstfiftyzeros (and theirlogarithms)ofJo{x) weregiventotenplaces ofdecimals byWillson and Peirce, Bulletin American Math. Soc. III.(1897), pp.153—155; theyalsogavethevalues ofJi{x) andlogjJi(j;)|atthese zeros toeightandsevenplacesofdecimalsrespectively. The firstfiftyzeros ofJ,{x)andthecorrespondingvalues ofJq{x)were computedtosixteenplacesofdecimals byMeissel*, KielProgramm, 1890; thisTable isreprinted byGrayandMathews intheir Treatise, p.280. Tables ofroots oftheequation Jn{X)Yn(kx)-J,,(kx)Yn(x)= havebeen constructedbyKalahne, Zeitschrift furMath,undPhys.Liv.(1907), pp.55—86;thevalues taken forkare 1'2,1'5and20,while nisgiventhe values 0,h,1,f,2,|. Dinnik inhisTables offunctions offractional order mentions thevalues ofafewofthezeros ofeach function, whileAirey,Phil.Mag. (6)XLI.(1921), pp.200—205,hascomputedthevalue ofthesmallest zeroofJ^,{x)forsmall fractional values ofvbyEuler's method. Rayleigh,Proc. London Math. Soc. x.(1878), pp.6—7[Scientific Papers, I.(1899), pp.363—364],hascalculated that (1-x^)xl,{x)lh {oo) hasamaximum when x'^=0'4858. Airey, Archiv derMath, undPhys. (3)xx.(1913), p.291,hascomputed the firsttenzeros ofSxJ^ (x)—2J^(x)andof2xJo (x)—J^(«)tofourplaces ofdecimals. Inhismemoirs onDiffraction, MiXnchenerAhhandlangen,xv.(1886), Lommel haspublishedtables connected with hisfunctions oftwo variables, butthese tables aresonumerous thatadetailed account ofthem willnotbe givenhere. HisTable ofFresnel'sintegrals (p.648) tosixplacesofdecimals from x=tox=50*0with interval 0'5(with auxiliarytables forpurposes ofinterpolation) must, however, bementioned, andwith ithisTable ofthe firstsixteen maxima andminima oftheseintegrals. *Jahrbuch ilber dieFortschritte derMath. 1890, p.521. Inconsequence oftheinaccessibility ofMeissel's table, thezeros ofJi(.r)wererecomputed (totenplacesofdecimals)forinsertion in Table VII, p.748. 20-2] TABULATION OFBESSEL FUNCTIONS 661 Lommel's form forFresnel'sintegrals was adifferent form wastabulated earlierbyLindstedt, Ann. derPh>/sikunci Chemie, (3)xvii.(1882), p.725. Definingthefunctions M{x)andN{x)bytheequations /ooCOSf^dt=M(x)cosx^—N(x)sinx-, X 00 sint-dt=M{x)sinx^+N{x)cosx-, andwritingA'={(y+ 2)'""P.Lindstedt tabulated M{x) andN{x)tosixplaces ofdecimals from3/=tot/=9with interval 0"1. Thefunction I{x)defined as ITJXt," hasbeen tabulated tofourplacesofdecimalsbyStruve, Ann. derPhysik und Chemie, (3)XVli. (1882), pp.1008—1016, from^=to4*0(interval 01),from X=4-0to7-0(interval 02)andfromx-7-0toll'O(interval 0-4). Atable ofvalues oftheintegi'al jx inwhich thelimits areconsecutive zeros(uptotheforty-ninth)ofJ^(x),has beenpublished bySteiner, Math, undNaturwiss. Berichte cutsUngarnXI. (1894), pp.366—367 ;thisintegraloccurs intheproblemofDiffraction by aCircularAperture. NoTables ofStruve's functions seem tohavebeen constructed before the Table ofH„(x)andH^{x)which isgivenonpp.666—697. 20*2.Description oftheTables contained inthisbook. Preliminary considerations onthemagnitudeandcharacter ofthetables tobeincluded inthisbook ledtothefollowingdecisions : (I)Thatspacedidnotusuallyadmit oftheinclusion ofmore thanseven places^decimals inthetables. (II)That thetables should besoconstructed astominimise thedifficulty ofmaking interpolations.Inparticular,itwasdecided thatatable with a moderately largeinterval (suchas002), togetherwithanauxiliarytable to facilitateinterpolation,would bemore useful than atable with asmaller interval (suchasO'Ol), occupyingthesamespaceasthe first table and its auxiliary,inwhichinterpolationwasimpracticable. 662'THEORY OFBESSEL FUNCTIONS[CHAP. XX (III) That incomputing tables, calculations should becarried toten placesofdecimals inorder toensure thatthenumber ofcases ofinaccuracy inthelastfigureofthepublishedresults should betrivial*. This does not applytotheauxiliarytables ofangleswhich areentered inTables Iand III. Inorder toobtainseven-figure accuracy,itisnotsufficient totabulate to tenths ofasecond ofarc,because thedifferencesperminute ofarcinaseven- figuretable ofnatural sinesmaybeaslargeas00002909;ontheother hand, anerror ofahundredth ofasecond doesnotaffect thevalue ofthesineby more than 0'00000005. Hence, forseven-figure accuracy,itwasconsidered adequatetocomputetonineplacesofdecimals thesines (orcosines) ofthe anglestabulated andthen tocomputetheanglesfrom Gifford's Natural Sines (Manchester, 1914) ;these areeight-figuretables withanintervalfof1". Theanglestabulated mayconsequently frequentlybeinerror astothe lastdigit, but, inallprobability,theerror never exceeds aunit(i.e.a hundredth ofasecond ofarc). ^Wenowproceedtodescribe thetables indetail. Table IconsistsprimarilyofTables ofJq(^). J^o(^^),J\{^)and F,{x)from x=to16"00 with interval of0"02.The values ofJo{x) andJi{x) upto 15"50 aretaken from Meissel'sTable;):,while thevalues ofY^ix) andYi{x) werecomputed partly byinterpolationinAldis' Table of(ro{x)andG^{x)and partlyfrom theasymptotic expansionsofJ^{x)+Yq-(x)and Ji^(x)+Y^-(x) givenin§7*51. Theauxiliarytables§givethevalues ofl//,i*"(^)| andarg^„<^'(a;)for n=andn=l. Inthese tables the first differences aresufficiently steady (exceptforquitesmall values ofx)toenableinterpolationstobeeffected withbut little trouble onthepartofthereader; thus,whenxisabout 10 thesecond differences of |iTo'^' (^) Idonotexceed 0'0000009. Thevalues ||of |^w"' (x)\andarg^„'^' (x)canconsequentlybecomputed bythereader foranyvalue ofxlessthan 16,with theexceptionofquite small values. Thecorrespondingvalues ofJn{^) andYn{x) canthen be calculatedimmediately bytheuseofseven-figure logarithmtables. *Thetables were differenced before removing thelastthreefigures, and itwasfound thatthe ten-figureresults were rarelyinerrorbymore than aunit inthetenth place ;soitishoped thatthe number oferrors inthelastfigure retained doesnotexceed about oneinevery thousand entries. tNotables with asmaller interval have been published ;theuseofanytables withalarger interval andagreater number ofdecimal places would have very greatly increased thelabour of constructing theauxiliary tables ofangles, andtheincreased accuracy soobtained would beofno advantagetoanyone using theauxiliarytables forpurposesofinterpolation. +Imust hereexpress mycordial thanks tothePreussische Akademie derWissenschaften zu Berlin forpermitting metomake useofthisTable. §Theidea ofconstructing theauxiliarytables grew outofaconversation withProfessor Love, inthecourse ofwhich heremarked that itwasfrequently notrealised howclosely Bessel functions ofanygiven order resemble circular functions multiplied byadamping factor inwhich therate ofdecayisslow. IITheremarks immediately following ofcourse presuppose thatnisor1. 20-2] TABULATION OFBESSEL FUNCTIONS 663 The relation between thevarious functions tabulated maybeexpressed mostbriefly byregarding|Hn^^^ {x) \andarg Hn^^'' {x)asthepolarcoordinates ofapointinaplane;then theCartesian coordinates ofthispointare./«{x) andYn(«)•Thus, from theentryforx=800, +01716508 =0-2818259 cos412° 28'40"-60, +0-2235215 =0-2818259 sin412' 28'40"-60. Table Ialsocontains thevalues ofStruve's functions Hq{x) and Hi(.x'). These functions areincluded inTable I,instead ofbeingcontained ina separate Table, tofacilitateinterpolation; by §10-41(4),the difference H„(^)—F,i(.r)isapositivemonotonic function, and itvariessufficiently steadilyforinterpolationtobeeasywhen xisnotsmall. TheTables ofStruve's functions werecomputed bycalculatingthevalues ofHq(^) andHq(x)directly from theascendingseries when x=l,2,3,...,andthen calculating Ho" (^),Ho'" (•«),...forthese values of,rfrom thedifl'erential equation >510-4(10)andthe equationsobtained bydifferentiatingit. Afewdifferential coefficients areadequatetocalculateHo(.^^) andHq' (a-)l)yTaylor's theorem forthevalues 0*5, 0'6, 0"7,...oix.Interpolationtofiftieths oftheunit isthen effected byusing Taylor'stheorem inthesamemanner. Thisprocess, thoughitseems atfirst sighttobecomplicated andlengthy, is,inreality, anextremely rapidonewhen amachine* isused. Itisverymuch more effective than theuseofasymptotic expansionsorthe process suggestedintheBritish AssociationBeport, 1913, p.116.Asanexampleofthe rapidityoftheprocess,itmaybestated thatthevalues ofe~^/o(.r) ande~^/i (x)inTable II took lessthan afortnighttocompute;ofcourse thetimetaken over thistabulation was appreciably shortened bytheuseofAldis' Table asaframework forinterpolation. Table IIconsists ofTables ofe~*/«(x),e~^/j(x),e*Kq(x),and e^K^(.r), andaTable ofe^isinserted, incasethereader shouldrequirethevalues of thefunctions'!" I(,{x), Ii{x), Kq{x) andK^{x);thefunctions aretabulated from to16-00 with interval 002. Interpolation bydifferencingiseasyinthecaseofthe firstfourfunctions throughoutthegreater partoftherange. TheTable ofe^wasconstructed with thehelpofNewman's Table ofe~^, Trews. Camb. Phil. Soc. xiii.(1883), pp.145—241. Unlike theother Tables inthisbook, theTable ofe^isgiventoeight significant figures^,andcare hasbeen taken thatthelastdigit givenisaccurate inevery entry. Interpo- lationjn thisTable is,ofcourse, effected bymultiplyingordividingentries byexponentialsofnumbers notexceeding0-01;suchexponentialscanbe calculated withoutdifficulty. *Themachine onwhich thecalculations were carried out isaMarchant Calculating Machine, 10X9recording to18figures. tThese functions werenottabulated because tables ofthem areunsuited forinterpolation, iNinefigures aregiveninpartsoftheTable toavoid spoilingitsappearance. 664 THEORY OFBESSEL FUNCTIONS [CHAP. XX Newman's Tablegives e~^toalargenumber ofplacesofdecimals, buttheactual number ofsignificant figuresinthelatter partoftheTable issmall;andlessthan halfof theTable ofe^wasconstructed bytheprocessofcalculating reciprocals ;therestwascon- structed fromNewman's Table byusing thevalues ofe^^and e^^given byGlaisher*, and thevalue ofe"!^"^given byNewman inashort table ofe~*^with interval O'l.These ex- l^onentialswei'eemployed because thetenthsignificant figuresinallthreeandtheeleventh significant figuresinthefirstandthird arezero. Table IIIconsists ofTables ofJ'i(^), Yx{x),\H^^^ {x)\and |argifj'" (a;)| ofthesamescopeasTable I,andinterpolationsareeffected inthemanner already explained. ATable ofe*K^{oc)isalsoincluded. These Tables are ofimportanceindealingwithapproximationstoBessel functions oflarge order(§8"43), andalsointhetheoryofAiry's integral. Thereader caneasily computevalues ofJ_i{x)from thistablebymeans oftheformula /_j{x)= I^j<i) {x)Icos{60°+argHf^ {x)]. TableIVgivesthevalues ofJn(^),Yn{oc),e~^In (x)andKn(x)forvarious values ofxand n.Thevalues ofJn{x)aretaken fromLommel's Table f,with some corrections, buttheremainder ofTable IV,with theexceptionofsome values ofKn{x) taken from Isherwood's Tablef,isnew; theyhavebeen con- structed inpartbymeans ofAldis' Tables offunctions oforders zeroandunity. TableVisLommel's Table-f-ofJ±(n+h)i^)^^^ Fresnel'sintegralswith some modifications andcorrections. TableVIgivesthevalues ofJn(n), Yn(n), Jn(n),Yn(n)andn^Jn{n), n^Yn(n), n^Jn(n), n^Yn (n)forn=l,2,3,...50.Interpolationinthetables ofthelastfouroftheeightfunctions iseasy. Table VIIgivesthe firstfortyzeros ofJn(^)andF„(.r)forvarious values ofn;partofthisTable istaken from theTables ofWillson and Peirce"f". Fortyzeros ofvariouscylinderfunctions oforder one-third arealsogiven. Table VIIIgivesthevalues of ^rJo(t)dt, hrYo(t)dt, -'O ^0 from a;=to50with interval1,togetherwith the firstsixteen maxima and minima oftheintegrals. Theformer table ofmaxima andminima canbe used tocomputethecoefficients (c£§18'12)incertain Fourier-Bessel series forwhich i'=0, *Trans. Camb. Phil Soc. xiii.(1883), p.245. tImust here express mycordial thanks totheBayerische Akademie derWissenschaften zu Miinchen, totheManchesterLiterary andPhilosophical Society, and totheAmerican Mathe- matical Societyforpermitting metomake useofthese Tables. Thenon-existence ofadequate trigonometricaltables ofangles inradian measure hasmade itimpracticable tocheck thelast digits intheentries inthegreater partofTable V. TABLES OF BESSEL FUNCTIONS 666 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 667 Table I.Functions oforder unity X 668 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 669 Table I.Functions oforder unity X 670 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 671 Table I.Functions oforder unity X 672 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 673 Table I.Functions oforder unity X 674 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 675 Table I.Functions oforder unity ,1' 676 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 677 Table I.Functions oforder unity X 678 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 679 Table I.Functions oforder unity X 680 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 681 Table I.Functions oforder unity ,r 682 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 683 Table I.Functions oforder unity X 684 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 685 Table I.Functions oforder unity X 686 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 687 Table I.Functions oforder unity X 688 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 689 Table I.Functions oforder unity X 690 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 691 Table I.Functions oforder unity X 692 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 693 Table I.Functions oforderunity X /iW 13 13 13 13 13 13' 13' 13 13' 13' 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13' 13' 13' 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 1402 04 06 08 10 12 14 10 18 20 ^4 26 2^ 30 3^ 34 3t) ?>^ 40 4^ 44 48 50 5^ 54 56 58 60 62 64 66 68 70 72 74 -% So •82 •84 •86 •88/ •90 92 94 96 98 00-0-0660609-o-o6i784i-0-0574892-0-0531781-0-0488525 -0-0445140-0-0401645-0-0358056-0-0314391-0-0270667 -0-0226902-0-0183113-0-0139317-0-0095532-0-0051775 -0-0008063 +0-0035587 +0-0079157 +0-0122630 +0-0165990 +0-0209219 +0-0252301 +0-0295218+0-0337954 4-0-0380493 +0-0422817+0-0464911 +0-0506758 +0-0548341 +0-0589646 +0-0630655 +0-0671353 +0-0711725+0-0751755+0-0791428 +0-0830728 -f0-0869640+o0908150+0-0946243+0-0983905 +0-1021121 +0-1057877+0-1094160 -+o-1129955 +o-1165249 +0-1200029 +0-1234282 +0-1267995 +0-I30II56 +0-1333752Y,W 2112796 2123920 2134183 2143582 2152115 2159780 2166575 2172499 2177551 2181729 2185034 2187466 2189025 2189712 2189527 2188473 2186550 2183761 2180108 2170223 2163997 2156920 2148996 2140229 2130625 2120188 2108924 2096838 2083936 2055711 2040400 2024302 2007421 1989768 1971349 1952173 1932249 1911585 1890191 1868077 1845252 1S21726 1797510 1772613 1747048 1720824 1693954 1666448i«:mi 2213665 2211959 2210257 22085592206866 2205176 2203490 2201807 2200129 2198455 2196784 2195117 2193454 2191795 2190139 2188487 2186839 2185195 2183555 2181918 2180285 2178655 2177029 2175407 2173788 2172174 2170562 2168954 2167350 2165750 2164153 2162559 2160969 2159382 2157799 2156220 2154644 2153071 2151502 2149936 2148374 2146815 2145260 2143708 2142159 2140614 2139072 2137533 2135998 2134466argH J{x) 6i2°38'i3'^5 613° 46'49-71 614° 55'25-99 616°4'2'-30 617° i2'38'-63 6i8°2i'i4'-99 6i9°29'5i'^38 620°38'27'-8o 621° 47'4-24 622°554o'^7i 624° 4'i7'-2i 625°12'53-73 626°2i'3o'^28 627° 30'6-85 628° 38'43-^45 629°47'2o'^o8 630° 55'56-73 632° 4'33-41 633 13I0''I2 634°2i'46'^85 635° 30'23'-6o 636^39' o'^38 637° 47'37-19 638°56'i4'-o2 640° 4'50-87 641° 13'27-75 642° 22'4"66 643° 30'41'-59 644°39'i8'^54 645° 47'55-52 646° 56'32-52 648° 5'9-55 649 1346'-6o 650° 22'23^67 651° 31' o'.77 652° 39'37-89 653°48'i5'.04 654°56'52'-2i 656° 5'29-41 14'6'-62 657° 658° 22'43' 659° 31'21' 660° 39'58' 661° 48'35' 662°57'13' 664° 5'50' 665° 14-27' 666°23' 5' 667° 31'42'86 12 41 72 05 18 60 668°40'2o'-'o4HiW X +o +o +o +o +o +o +o +o +o +o 4-O +o +o +o +o +o +o +o +o +o +o ^'o +o +o +o +o +o +o +o +o +o +o +o +o +o +o +o +o +o +o +o +o +o +o +o +o +o +o +o +o4290341 4279106 4268732 4259222 4250579 4242805 4235900 4229868 4224709 4220422 42I70IO 4214472 4212807 4212014 4212094 4213044 4214062 4217547 4221096 4225507 4230776 4236901 4243876 4251699 4260365 4269869 4280206 429I37I 4303358 43I6I6I 4329775 4344I9I 4359404 4375406 4392190 4409748 4428071 4447152 4466981 4487550 4508850 4530871 4553603 4577036 4601160 4625965 4651439 4677571 4704350 473176613-02 13-04 13-06 13-08 13-10 13-12 13-14 13-16 13-18 13-20 13-22 13-24 13-26 13-28 13-30 13-3^ 13-34 13-36 13-38 13-40 13-42 13-44 13-46 13-48 13-50 13-52 13-54 13-56 13-58 13-60 13-62 13-64 13-66 13-68 13-70 13-72 13-74 13-76 13-78 13-80 13-82 13-84 13-86 13-88 13-90 13-92 13-94 13-96 13-98 14-00 694 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 695 Table I.Functions oforderunity X 696 TABLES OFBESSEL FUNCTIONS Table I.Functions oforder zero X TABLES OFBESSEL FUNCTIONS 697 Table I.Functions oforder unity X 698 TABLES OFBESSEL FUNCTIONS Table II.Functions ofimaginary argnment, and e^ X TABLES OFBESSEL FUNCTIONS 699 Table II.Functions ofimaginary argument, and e*^ X 700 TABLES OFBESSEL FUNCTIONS Table II.Functions ofimaginary argument, and e^ X TABLES OFBESSEL FUNCTIONS 701 Table II.Functions ofimaginary argument, and e* X 702 TABLES OFBESSEL FUNCTIONS Table II.Functions ofimaginary argument, and e^ X TABLES OFBESSEL FUNCTIONS 703 Table II.Functions ofimaginary argument, and e^ X 704 TABLES OFBESSEL FUNCTIONS Table II.Functions ofimaginary argument, and e^ X TABLES OFBESSEL FUNCTIONS 705 Table II.Functions ofimaginary argument, and e^ X 706 TABLES OFBESSEL FUNCTIONS Table II.Functions ofimaginary argument, and e'" X TABLES OFBESSEL FUNCTIONS 707 Table II,Functions ofimaginary argument, and e* X 708 TABLES OFBESSEL FUNCTIONS Table II.Functions ofimaginary argument, and e^ X TABLES OFBESSEL FUNCTIONS 709 Table II.Functions ofimaginary argument, and e* X II-02 11-04 11-06 11-08 II-IO II-I2 IT-I4 II-I6 ii-i8 II-20 11-22 11-24 11-26 11-28 11-30 11-32 11-34 11-36 11-38 11-40 11-42 11-44 11-46 11-48 11-50 11-52 11-54 11-56 11-58 11-60 11-62 11-64 11-66 11-68 11-70 11-72 11-74 11-76 11-78 11-80 11-82 11-84 11-86^ 11-88 11-90 11-92 11-94 11-96 11-98 12-00-"/oW o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o oI2i169 I2I5039 I2I39I2 I2I2789 I211669 I2I055I 1209437 1208326 I2072I8 I206II3 1205011 1203912 I2028I7 I20I724 1200634 1199547 1198463 I197382 1196303 1195228 II94I56 1193086 1192020 1190956 1189895 II88837 II87782 II86729 II85680 II84633 II83589 II82548 II8I509 1180473 1179440 II784IO II77382 1176357 II75335 II743I5 II73298 II72284 II7I272 II70263 1169256 I168252 II6725I 1166252 1165256 II64262AW o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o1159603 II5863I I157662 1156694 II55730 II54767 II53807 II52849 1151894 II5094I 1149990 1149042 I148096 II47I52 1146211 II45272 1144335 II4340I II42468 II4I538 II406IO 1139685 1138762 II3784I II36922 113(^005 1135090 II34I78 II33268 II32360 II3I454 1130551 1129649 II28750 II27852 II26957 1126064 11^5173 1124284 11-23398 II225I3 II2I630 1120750 III987I 1118995 III8I20 III7248 II16378 III5509 1114643K, 0-3734632 0-3731319 0-3728014 0-3724717 0-3721430 0-3718151 0-37I488I 0-37II6I9 0-3708367 0-3705122 0-3701886 0-3698659 0-3695440 0-3692229 0-3689027 0-3685833 0-3682648 0-3679470 0-3676301 0-3673140 0-3669987 0-3666843 0-3663706 0-3660578 o-3'^57457 0-3654344 0-3651240 0-3648143 0-3645054 0-3641973 0-3638900 0-3635834 0-3632777 0-3629727 0-3626684 0-3623650 0-3620623 0-3617603 0-36I459I 0-3611587 0-3608590 0-3605600 0-3602618 0-3599643 0-3596676 0-3593716 0-3590763 0-3587818 0-3584880 0-3581949e""K^{x) 0-3900543 0-3896788 0-3893043 0-3889309 0-3885586 0-3881873 0-3878171 0-3874480 0-3870799 0-3867128 0-3863468 0-3859818 0-3856178 0-3852548 0-3848929 0-3S45320 0-384I72I 0-3838132 0-3834553 0-3830984 0-3827425 0-3823875 0-3820336 0-3816806 0-3813286 0-3809775 0-3806275 0-3802783 0-3799302 0-3795830 0-3792367 0-3788914 0-3785470 0-3782035 0-3778610 0-3775194 0-3771787 0-3768389 0-3765001 0-3761621 0-3758251 0-3754890 0-3751537 0-3748194 0-3744859 0-3741533 0-3738216 0-3734908 0-3731608 0-3728318X 61083-680 62317-652 63576-552 64860-883 66171-160 67507-906 68871-656 70262-956 71682-362 73130-442 74607-775 76114-952 77652-576 79221-262 80821-638 82454-343 84120-031 85819-368 87553-035 89321-723 91126-142 92967-012 94845-070 96761-068 98715-771 100709-962 102744-438 104820-013 106937-518 109097-799 III30I-72I II3550-I65 115844-030 118184-235 120571-715 123007-425 125492-340 128027-453 130613-780 133252-353 135944-229 138690-485 I4I492-2I8 144350-551 147266-625 150241-608 153276-690 156373-085 159532-031 162754-791n II 11 II II II II' II II II II II II II II II II II II II II 11 II II II II II II II II II II II II II II II II II II II II II II II II 11 II 11 1202 04 06 08 10 12 14 16 18 20 24 26 28 30 32 34 36 38 40 42 44 48 50 52 54 56 58 60 •62 •6466 68 70 74 7^78 80 82 86 88 90 92 94 96 98 00 710 TABLES OFBESSEL FUNCTIONS Table II.Functions ofimaginary argument, and e^ X TABLES OFBESSEL FUNCTIONS 711 Table II.Functions ofimaginary argument, and e^ X 712 TABLES OFBESSEL FUNCTIONS Table II.Functions ofimaginary argument, and e^ X TABLES OFBESSEL FUNCTIONS 713 Table II.Functions ofimaginary argument, and e* X 714 TABLES OFBESSEL FUNCTIONS Table III.Functions oforder one-third X TABLES OFBESSEL FUNCTIONS 715 Table III.Functions oforder one-third X 716 TABLES OFBESSEL FUNCTIONS Table III.Functions oforder one-third X TABLES OFBESSEL FUNCTIONS 717 Table III.Functions oforder one-third X 718 TABLES OFBESSEL FUNCTIONS Table III.Functions oforder one-third X TABLES OFBESSEL FUNCTIONS 719 Table III.Functions oforder one-third X 720 TABLES OFBESSEL FUNCTIONS Table III.Functions oforder one-third X TABLES OFBESSEL FUNCTIONS Table III.Functions oforder one-third721 X 722 TABLES OFBESSEL FUNCTIONS Table III.Functions oforder one-third X TABLES OFBESSEL FUNCTIONS Table III.Functions oforder one-third723 X 724 TABLES OFBESSEL FUNCTIONS Table III.Functions oforder one-third X TABLES OFBESSEL FUNCTIONS 725 Table III.Functions oforder one-third X 726 TABLES OFBESSEL FUNCTIONS Table III.Functions oforder one-third X TABLES OFBESSEL FUNCTIONS 727 Table III.Functions oforder one-third X 728 TABLES OFBESSEL FUNCTIONS Table III.Functions oforder one-third X TABLES OFBESSEL FUNCTIONS 729 Table III.Functions oforder one-third X 730 TABLES OFBESSEL FUNCTIONS Table IV.Values ofJ„(.r) X TABLES OFBESSEL FUNCTIONS Table IV.Values ofJ„(.r)731 X 3 4 6 7 8 9 lo II 12./o(.v; +0-765198 +0-223891 -~0-260052-0-397150-0-177597 +o-i50()45 +0-300079 +0-171651 -0-090334-0-245936-0-171 190 +0-047689,/,(v) +0-440051 +0-576725+0-339059-o-o()6o43-0-327579-0-276684 -0-004683+0-234636 +0-245312 +0-043473-0-176785-0-223447,/.( +o-i14903 +0-352834 +0-486091 +0-364128 +0-046565-0-242873 -0-301417-o-112992 +0-144847 +0-254630 +0-139048-0-084930/= +0-019503 +0-128943 +o-309o()3+0-430171+0-364831+o-114768 -o-i(>7556-0-291 132-0-180935 +0-058379+0-227348 +0-195137./4(-V) +0-002477+0-033996 +0-132034+O-28II29 +0-391232 +0-357642 +0-157798-0-105357-0-265471-0-219603-0-015040 +0-182499/5W +0-000250 +0-007040+0-043028 +0-132087 +0-261141 +0-362087 +0-347896+0-185775-0-055039-0-234062-0-238286-0-0734718 9 10 II 12 .r 732 TABLES OFBESSEL FUNCTIONS Table IV.Values ofJ^{x)and7„(a:) X TABLES OFBESSEL FUNCTIONS 733 Table IV.Values ofYjx) X 734 TABLES OFBESSEL FUNCTIONS Table IV.Values ofY^{x) X TABLES OFBESSEL FUNCTIONS 735 Table IV.Values ofY^{x) X 736 TABLES OFBESSEL FUNCTIONS Table TV.Values ofe-^/„(a;) X TABLES OFBESSEL FUNCTIONS 737 Table IV.Values ofK^{x) X o-i 738 TABLES OFBESSEL FUNCTIONS Table IV.Values ofK,,[x) X TABLES OFBESSEL FUNCTIONS Table IV.Values ofK,,{x)739 X 740 TABLES OFBESSEL FUNCTIONS Table V.Values ofJj^i^^i)(x) X TABLES OFBESSEL FUNCTIONS 741 Table V.Values ofJic^+i ){x) X 742 TABLES OFBESSEL FUNCTIONS Table V.Values ofJ^/,j^.^\[x) X TABLES OFBESSEL FUNCTIONS 743 Table V.Values ofJ^^^i{x) X 744 TABLES OFBESSEL FUNCTIONS Table V.Fresnel's integrals X TABLES OFBESSEL FUNCTIONS 745 Table V.Fresnel'sintegrals X 746 TABLES OFBESSEL FUNCTIONS Table VI.Functions ofequalorderandargument TABLES OFBESSEL FUNCTIONS Table VI.Functions ofequal order andargument747 3 4 5 6 7 8 9 lo II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50-y..in) 0-7812128 0-6174081 o-53854if> 0-4889368 0-4536948 0-4268259 0-4053710 0-3876699 0-3727057 0-3598142 0-3485399 0-3385583 0-3296303 0-3215755 0-3142546 0-3075580 0-3013982 0-2957040 0-2904173 0-2854894 0-2808800 0-2765546 0-2724839 0-2686456 0-2650095 0-2615652 0-2582933 0-2551791 0-2522100 0-2493744 0-2466622 0-2440643 0-2415724 0-2391794 0-2368784 0-2346635 0-2325292 0-2304705 0-2284828 0-2265620 0-2247042 0-2229059 0-2211637 0-2194748 0-2178364 0-2162458 0-2147007 0-2131988 0-2117381 0-2103166-n'iY„(n) 0-7812128 0-7778855 0-7767114 0-7761387 0-7758072 0-7755941 0-7754469 0-7753399 0-7752590 0-7751961 0-7751458 0-7751049 0-7750711 0-7750426 0-7750184 0-7749976 0-7749796 0-7749638 0-7749499 0-7749374 0-7749266 0-7749168 0-7749079 0-7748999 0-7748925 0-7748859 0-7748798 0-7748742 0-7748690 0-7748642 0-7748598 0-7748557 0-7748519 0-7748483 0-7748450 0-7748419 0-7748389 0-7748362 0-7748336 0-7748312 0-7748289 0-7748267 0-7748246 0-7748227 0-7748208 0-7748191 0-7748174 0-7748158 0-7748143 0-7748128y,/ in) 0-8694698 0-5103757 0-378I4I2 0-3069147 0-2615525 0-2297650 0-2060642 0-1876060 0-1727588 0-I605I49 0-1502159 0-I4I4I2I 0-1337852 0-1271029 0-1211915 0-II59I84 0-1111803 0-1068955 0-1029987 0-0994367 0-0961658 0-0931499 0-0903586 0-0877663 0-0853514 0-0830953 0-0809819 0-0789973 0-0771295 0-0753678 0-0737029 0-0721267 0-0706318 0-0692116 0-0678605 0-0665732 0-0653451 0-0641718 0-0630496 0-06I975I 0-0609450 0-0599565 0-0590071 0-0580942 0-0572157 0-0563695 0-0555539 0-0547671 0-0540074 0-0532735«^Y,/(w) 0-8694698 0-8101709 0-7865654 0-7733765 0-7647843 0-7586672 0-7540520 0-7504241 0-7474840 0-7450441 0-7429809 0-7412092 0-7396683 0-7383135 0-73711x2 0-7360358 0-7350670 0-7341890 0-7333887 0-7326559 0-7319817 0-7313591 0-7307820 0-7302453 0-7297446 0-7292763 0-7288371 0-7284242 0-7280352 0-7276680 0-7273206 0-7269914 0-7266790 0-7263820 0-7260991 0-7258295 0-7255720 0-7253259 0-7250904 0-7248647 0-7246483 0-7244405 0-7242407 0-7240486 0-7238636 0-7236853 0-7235134 0-7233475 0-7231873 0-72303243 4 5 / 8 9 10 II 12 13 14 15 16 ;i 19 20 21 22 23 24 26 27 28 29 30 31 32 33 34 35 37 38 39 40 ii 42 43 44 45 46 4<48 49 50 Forvalues ofnexceeding 50,thefollowing approximations maybeused ^\ath seven-figure accuracy: 0-77475 9002 1T^__i^"1^0-01016 59059F_12131 L 225W-'J ij L I.l625?t2j'X 0-71161 34100 z 713 L 3150^2JJ0-1549 518004fj947 L69300 Irtd• 748 TABLES OFBESSEL FUNCTIONS Table VII. Zeros, j^,„,y^,„,ji,n,Vi,n,ofJ^{x),J\ (.r),Ji(x),J\{x) n TABLES OFBESSEL FUNCTIONS 749 Table VII. Zeros, j.;,,„,y.^, ,,,j^,„,y^,„,ofJ^(.r);Y^(x),Jg(x),Y^{x) 3 4 5 6 7 8 9 lo II 12 13 14 15 16 17 18 19 20 21 22 ^3 ^4 25 26 27 28 29 30 32 33 34 35 36 37 38 39 40.h. 5-i35<J223 8-4172441 11-6198412 147959518 17-9598195 21-1169971 24-2701123 27-420573() 30-5092045 33-7165195 36-8628565 40-0084467 43-1534538 46-2979967 49-4421641 52-5860235 55-7296271 58-8730158 62-0162224 65-1592732 68-3021898 71-4449899 74-5876882 77-7302971 8o'8728269 84-0152867 87-1576839 90-3000252 93-4423160 96-5845614 99-7267657 102-8689327106-01 10655 109-1531673 112-2952406 ^15-4372877 118-5793107 121-7213115 124-8632917 128-00525303-3842418 6-7938074 10-0234780 13-2099868 16-3789666 19-5390400 22-6939559 25-8456137 28-9950804 32-1430023 35-2897939 38-4357335 41-5810149 44-7257771 47-8701227 51-0141287 54-1578545 57-3013461 60-4446401 63-5877658 66-7307471 69-8736034 73-0163509 76-1590031 79-3015713 82-4440651 85-5864927 88-7288612 91-8711766 95-0134441 98-1556685 101-2978536 104-4400031 107-5821201 110-7242073 113-8662672 117-0083021 120-1503138 123-2923041 126-4342746Jan 6-3801 619 9-7610231 13-0152007 16-2234640 19-4094148 22-5827295 25-7481667 28-9083508 32-0648524 35-2186707 38-3704724 41-5207197 44-6697431 47-8177857 50-9650299 54-III6I56 57-2576516 60-4032241 63-5484022 66-6932417 69-8377884 72-9820804 76-1261492 79-2700214 82-4137195 85-5572629 88-7006678 91-8439487 94-9871177 98-1301857 IOI-273I62I 104-4160552 107-5588722 IIO-70I6I97 113-8443033 116-9869284 120-1294994 123-2720205 126-4144954 129-5569276J's,« 4-5270247 8-0975538 11-3964667 14-6230726 17-8184543 20-9972845 24-1662357 27-3287998 30-4869896 33-6420494 36-7947910 39-9457672 43-0953675 46-2438744 49-3914980 52-5383976 55*6846964 58-8304911 61-9758587 65-I2o86l2 68-2655491 71-4099642 74-5541409 77-6981084 80-8418910 83-9855095 S7-1289817 90-2723230 93-4155465 96-5586637 99-7016848 102-8446186 105-9874728 109-1302542 112-2729691 115-4156229 118-5582204 121-7007659 124-8432635 127-98571679 10 II 12 13 15 16 18 19 20 21 22 24 25 26 27 28 29 30 3i 34 i5 36 37 38 39 40 750 TABLES OFBESSEL FUNCTIONS Table VII. Zeros, J4,„,y^,„,J5.,,,yr^,„,ofJ^[x),Y^{x),J,{x),Y^{x) n I TABLES OFBESSEL FUNCTIONS 751 Table VII. Zeros, ji,3,„, ?/i;3,„,ofJy.^ {x), Yy,.i{x); with zeros, Sn,dn,ofJ_i,3 {X)+J„3 {X),J_i,3 (X-)- e/i;3 {X) [Note. The lasttwofunctions arceciualto v'3-Wvi{x)iiOfiZO°- l\,3{x)smio''\, ^3. :./i,,3(.r)cos 120°- I'j3(.r)8in i20°[ respectively.] )l i,n M::ui> I 8 9 ID II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3^ 37 38 39 402-9025862 6-0327471 9-1705067 12-3101938 15-4506490 18-5914863 21-7325412 24-8737314 28-0150117 3I-I563549 34-2977437 37-4391666 40-5806158 43-7220857 46-8635719 50-0050715 53-1465821 56-2881019 59-4296294 62-5711634 65-7x27030 68-8542475 71-9957961 75-1373484 78-2789040 81-4204625 84-5620234 87-7035867 90-8451519 93-9867191 97-1282878 100-2698581 103-4114297 106-5530025 109-6945765 112-83615x6 115-9777275 1x9-1x93044 122-2608821 125-40246051-3530196 4-4657883 7-6012412 16-7402x28 13-8803575 17-02x0330 20-1619929 23-303x228 20-4443623 29-5856767 32-7270444 35-8684514 39-0098884 42-15x3485 45-2928269 48-4343202 51-5758256 54-7173410 57-8588648 61-0003956 64-1419325 67-2834747 70-42502x3 73-56^5718 76-7081259 79-8496829 82-99x2426 86-1328048 89-2743691 92-4159353 95-5575032 98-6990728 101-8406437 104-9822160 108-1237894 111-2653639 114-4069394 117-5485159 120-6900931 123-83x67122-3834466 5-610x956 8-6473577 11-7868429 14-9272068 18-0679953 21-2090210 24-3501925 27-4914601 30-632794X 33-7741762 36-9155941 40-0570394 43-1985061 46-3399899 49-48x4874 52-0229964 55-7645147 58-90604x0 62-0475740 65-1891x27 68-3306564 71-4722044 74-6137562 77-7553112 80-8968692 84-0384298 87-X799926 90-3215576 93-4631244 96-6046929 99-7462629 102-8878343 106-0294070 109-X709808 112-3125557 115-4541315 iiS-5957082 121-7372858 124-8788641d., 0-8477186 3-9441020 7-0782997 10-2x69407 13-3569532 16-4975630 19-6384856 22-7795923 25-9208x65 29-0621201 32-2034801 35-34488x3 38-4863138 41-6277704 44-7692461 47-9I0737X 51-0522406 54-1937545 57-3352769 60-4768067 63-6183^27 66-7598840 69-9014299 73-0429798 76-1845333 79-3260899 82-4676492 85-6092109 88-7507749 91-8923408 95-0339085 98-1754777 101-3170485 104-4586205 107-6001938 IXO-741768X 113-8833435 1x7-0249x97 120-1664969 123-308074S/ 8 9 10 II X2 13 14 15 16 17 18 19 20 21 -yy 24 26 27 28 29 30 31 32 33 34 35 36 37 38 39 -I" 752 TABLES OFBESSEL FUNCTIONS Table VIII. 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INDEX OFSYMBOLS [^Thenumbers refer tothepagesoninhich thesymbolsaredefined.^ J„,,243,244,263j>F,{a„oi,,... ,oi^\p„p,,...,J.(^),308 An,Q; 584...,p,; ^),1003).(^),61 ^,,,(0,283 F(*-). 639 j,507 S4o{cc, t);598F,(a,-), 646 j.n,576 S^:,.n{z),529 §M, /9;7,V;IV),•'^71>,./;,>", 485 ^n..(0,571 /.(^),55 i.,n,479 a,6 /„(z),483 /(^)j^, 526 K,559 ^„,6 f{2)R, 536 A^n(^),78[seealsop.65] Bm(€2), 247 K,(z), 78[seealsop.65] Bn; M,.(t),293 (?,211 ;232 /C,,(z),46 ^0(^),597 Gn(z),64 :65 K(2t)n, 288 ^''^..m(^),529 a(a,-,a),640 K,(^),78 ber(x),bei(a;),81 G, (a;,a),648 ker(z),kei(^),81 ber^(a;), bei^(a;),81gn(t),276 5'n,.(0284 Ln(z),71 0,,554^^,,(^),303 Ii.(4 329 C„(^), 658 Z,^,,111 C{n(a),S20 H,o77 Cn"(z),50 ir,w (^), i7.'=' (z),73 il/,6;505 •^.(^X 82 H,(^),328 M{a;),mi W(zip),129 her(^),hei(z),81 ii\471,475,514 c,508 her, (^),hei^ (z),81 ..iV_,,,.,„,324 i),,598 //,13 iV^(^), 661 i)„(2),74 /.(^),77 0,197 E,6 J,(z), 10,16 On(t), 271,272 En{z), 56 Jn{z\ 6,14,19 0_,(0,276 Bin (a),320 J-n{z\16 <©«(0-569 E;5), 308 /^,„,10 e,e,,111 J'^rC^), 30 7^547 J,(^), 38 P.,227 i^(6'), 258 J(/i, a?),49 Pn(^), 156 J?'(a,0,90 J{z\p),129 P,„'^-' (^),295 F{e, x),253 ^n.fc(^),326 P.-'^ (^),51 P,(a;),77 J{z; v,k),327 F,Acoscf>p), 129 790 THEORY OFBESSEL FUNCTIONS Pn{r; «i,ao,...,a„), 420 LISTOFAUTHORS QUOTED [Thenumbersrefertothepages. Referencesarenotgiventoentries inthebibliography/, pp.753—788.] Abel, N.H.,68,616,621 Adamoff, A.,196 Aichi, K.,80 Airey,J.R.,65,142,214,247,319,502,505, 516, 659,660 Airy,SirGeorge B.,188, 189,229, 249, 320, 321, 322,654,659 Aldis,W.S.,65,655,656,657,658,662,663, 664 Alexander, P.,579 Anding, E.,313,657 Anger,C.T.,21,22,308, 309,310, 311,312 Ani^imov, V.A.,92 Appell,P.E.,146,371 Autonue, L.,94 Bach, D.,96,102,110 Bachmann, P.,197 Baehr, G.F.W.,479 Ball,L.de,157 Barnes, E.W., 83,100, 102, 104, 105, 156, 190, 192, 195, 196,220, 221,340,351,357, 367, 383, 387,402,409 Basset, A.B.,76,77,78,80,172, 173, ISO, 385, 388, 395, 425, 426,454 Bateman, H.,130, 131, 367,370, 372, 373, 376, 379, 380,389, 406, 417,437,456,530, 533,535 Bauer, G.,50,128,368,370 Beltrami, E.,51,358,361,374,386,389,390, 391, 500, 579,621 Berne ulli,Daniel (1700-1782), 2,3,4,9,85, 86,87,88,111, 123,478,576 Bernoulli, James (1654-1705), 1,2,3,88,90, 92 Bernoulli, John (1667-1748), 1,2,3 ^louUi,Nicholas, (1687-1759),2 Bernoulli, Nicholas, (1695-1726),2 Bessel,F.W., 1,3,4,5,9,13,14,15,18,19, 21,24,25,38,84,140, 148, 153, 160,295, 308, 478, 551, 554,654 Binet, J.P.M.,183 Bocher, M.,46,57,64,376,494, 495,517 Bohmer, P.E.,142 Boole, G.,27,47, 110,627Borel, E.,8,281,536 Bourget, J.,6,324, 325, 326, 484, 485, 517, 659,660 Brajtzew,J.Ti.,169 Brassinne, E.,91 Brenke, W.C,23 Bridgeman,P.W.,597 Bromwich,T.J.Fa., 8,11,44,68,156, 187, 189, 191,202, 203, 214, 230,231,234,279, 302, 349, 360, 385,391, 393,399,574,575, 601 Bruns, H.,327 Bryan,G.H.,127,480 Burkhardt, H.F.K.L.,236,280 Burnside, W.S.,305 Cailler, C.,44,149,386, 395, 415, 437, 455, 536,537 Callandreau, 0.,196,208,387 Cantor, G.F.L.P.,637 Carhni, F.,6,7,194, 225, 226,227,249,255, 268,572 Carslaw, H.S.,177,366, 395,499, 500,507, 509,583 Catalan, E.C,21,22,27,96,173,188 Cauchy, (Baron) A.L., 7,15,16,21,150, 183,230, 231, 232,233,247,249, 259,309, 319, 324,449, 545, 554, 557,579 Cayley, A.,88,90,96,102, 103, 109, 188, 502*^ Challis,IT.W.,91 Chapman, S.,621 Chessin, A.S.,135, 175,325,346,382 Chree, G,6,597 Christofiel, E.B.,154 Chrystal, G.,102,288,295 Cinelh, M.,633 Clebsch, R.F.A.,359,363 Cliftbrd, W.K.,90,91 Coates, C.v.,173, 180,313,622 Cotter,J.R.,41 Crawford, L.,27 Crelier, L.,286,287, 288, 295,300,301,302 Curtia, A.H.,96,110 (hirzon,11.E.J.,395 792 THEORY OFBESSEL FUNCTIONS D'Alembert,J.leEond,3 Dandelin, G.P.,503 Dai-boux,J.G.,233 Darwin,C.G.,437 Debye, P.,225, 235, 237, 240, 241, 247,249, 250, 251, 255, 262,263, 268, 513,516 DelaValine Poussin, Ch. J.,53,160,189 DeMorgan, A.,188,190 Dendv, A.,107 Dini, U.,10,577, 578,597, 600,616,651 Dinnik, A.,579, 659,660 Dirichlet, P.G.Lejeune, 157, 230, 406, 581, 623 Dixon, A.C,35,480, 481,482 Donkin, W.F.,109 Dougall, J.,65,411 DuBoisReymond,P.D.G.,183, 455,470 Duhamel,J.M.C,38,49,59,68,227 Earnshaw, S.,108 Ellis, R.L.,95,109, 110,173 Emde, F.,248 Encke,J.F.,342 Enestrom, G.,92 Enneper, A.,173 Epstein,S.S.,145,290 ErmakoflF, W.,455 Escherich, G.von,165 Euler, L.,3,4,5,6,24,49,53,60,62,87,88, 92,93,123, 133, 183, 410, 498, 500, 501, 503, 576,659 Falkenhagen,J.H.M.,94 Fejer, L.,610 Feldblum, M.,92 Fields, J.C.,110 Filon, L.N.G.,51,578, 622,623, 625,629 Ford,W.B.,578,605 Forsyth,A.R.,42,57,107, 109, 117, 346, 358, 400,499 Fourier, (Baron)J.B.Joseph, 4,9,10,13, 22,84,135, 449, 450, 454, 455, 456, 478, 482,483, 501,576, 577, 578,616 Freeman, A.,501 Frenet, F.,27 Fresnel, A.J.,544,545 Frobenius,F.G.,57 Frullani, G.,14,19 Gallop,E.G.,405,421,422 Gaskin, T,,109 Gasser, A.,509,517 Gauss, C.F.(Johann FriedrichCarl), 191,506Gegenbauer,L.von, 50,51,129, 138, 151, 274,283, 284, 290, 293,351,362,363,365, 366,367,368, 369, 370,373,378,379,383, 384,385,386, 389, 390, 391,393,395,396, 398, 406, 407,413, 414, 415,418,426,430, 438,439,480, 508, 517, 522,524, 525,579 Genocchi, A.,119 Gibson, G.A.,197 Gifford, E.,662 Gilbert,L.P.,545,548,549 Giuliani, G.,155, 156,324, 326,327 Glaisher, J.W.L.,89,96,102, 103,108,109, 140, 171, 173, 183,664 Gordan, P.A.,55 Goursat, Edouard J.B.,120 Graeffe, C.H.,502,503 Graf,J.H.,32,64,75,145, 153, 160, 165, 175, 197, 215, 227,286, 287,290,295,296, 299,301, 302,303, 341, 344,345,359,360, 362,398,498, 502,583 Gray, A.,64,65,78,194,206, 454,480,655 656,657,658,660 Green, G.,124 Greenhill, SirA.George, 91,96 Gregory, DuncanFarquharson, 391 Gregory, Walter, 224 Grunert, J.A.,27 Gubler, E.,32,64,145, 160, 165, 177, 197, 215,227, 286, 287,301,329,341,351,398, 408,410, 426, 498, 502,583 Gunther, S.,153 Gvvyther,R.F.,621,636 Hadamard, J.,204, 205,527 Haentzschel, E.,71,96,159 Hafen, M.,389 Hague, B.,656 Hall, A.,15 Hamilton,SirWilliam Rowan, 12,195,655 Hankel, H.,10,38,57,58,61,62,63,65,73, 75,76,77,160, 163, 164, 165, 167, 175, 195, 196,203, 206, 208, 211,384,386,390, 393, 395, 424,427,428, 429,430,434,453, 454, 456, 457, 458,459, 462,464,465,471, 488,513,514, 577, 579, 581, 582,633 Hansen,P.A., 14,20,30,31,37,152, 154, 155, 158, 195,292,406,655,656 Hanumanta Rao, C.V.,437 Hardy,G.H.,8,111,180,183,188, 189,200, 309,320,321, 322, 324,373,382,386,395, 406, 421,422, 437,441,442,463,464,542, 546,547,573,575, 579,606,615,621 Hargreave,C.J.,88,170,171 LISTOFAUTHORS QUOTED 793 Hargreavcs, 11.,538 Harnack, A.,577 Harris, J.A.,15 Havelc.ck, T.H.,125,171 Hayashi, T.,165 Heaviside, 0.,64,65,203,367,385,387,388, 393,395, 410,426 Heine, H.E.,4,56,65,66,84,154, 155,156, 157, 181,358, 363,365 Hermite, C,55,477 Hertz, H.,80,81 Herz, N.,554,555 Hill, C.J.D.,94 Hobson, E.W., 10,33,54,58,125,128,129, 149, 172, 174,280, 353,363,369,385,386, 387, 480, 485, 578, 586, 591,602 Hopf, L.,178,406 Horn, J.,225,526 Hurwitz, A.,9,297, 302, 303,304, 305, 306, 307, 483,484 Ignatowsky, W.von,365 Isherwood,J.(>.,657, 658,664 Jackson, Dunham, 343 Jackson, Frank Hilton, 43,44 Jackson, William Hartas, 177 Jacobi, C.G.J.,6,8,14,21,22,25,26,27, 28,29,84,195,379, 555,572 Jamet,E.Y.,94 Johnson, W.W.,92 Jollifte, A.E.,528,529 Julius, Y.A.,65,200 Kalahne, A.,505, 507,660 Kapteyu, W., 35,183, 200, 268, 279, 281, 282, 292, 351,370, 373,376,380,386,404, 413, 498, 499, 531,532,533,535,536,538, 551, 559, 560, 562, 565, 566, 568, 569, 570 Kelvin(SirWilliam Thomson), Lord, 81, 124, 203, 225, 229, 230, 233, 248, 654, 658 Kepinski, S.,99 Kepler, J.,551,552 i^rchhoff, G.,107, 196,203, 389, 578,616 Kluyver,J.C,367, 419,420 Kneser,J.C.C.A.,499,578,583 Knockenhauer, K.W.,545 Konig, J.,354,523 Koppe, M.,247 Kummer, E.E.,49,90,101, 102,104, 105, 148, 185, 190, 191, 196,203, 394,409Lacroix,S.F.,27 Lagrange,J.L.de,6,27,28,551 Lamb, H.,56,96,385,416, 475,502 Lambert, J.H,485 Lamd, G.,96,159 Landau, E.G.H.,197 Laplace, P.S.de,6,7,8,9,53,280, 395, 421, 450 Largeteau, C.L.,501 Laurent, PaulMathieu Hermann, 157 Lam'cnt, Pierre Alphonse,100 Lebedeff, WeraMyller-, 99 Lebesgue, Henri, 457 Lebesgue, Victor Amddee, 110,123 Lefort, F.,6 Legendre, A.M.,52,90,183,204, 485,557 I'Hospital,G.F.A.(Marquis deStMesme), 134 Leibniz, G.W., 1,2,3 LePaige, C.,96 Lerch, M.,382, 393,433,434,617 Lindner, P.,484 Lindstedt, A.,545,661 Liouville, J.,27,28,87,111, 112, 116, 117, 119, 120,123 Lipschitz,R.O.S.,11,12,195,200,206,331, 339, 384, 386, 390,633 Lobatsclievsky, N.,503 Lobatto, R.,49,90 Lodge, A.,224,229 Lommel, E.C.J.von, 13,21,23,25,30,34, 38,43,45,46,47,49,53,59,62,65,66, 71,73,76,77,96,97,98,99,106,107, 132, 133,135, 136, 140,142, 143, 145, 148, 151, 152, 154,196,200, 294, 295, 296, 297, 298, 299,300, 303, 308, 315, 328,341,345,348, 350,364,374,406,478,479,482, 529, 531, 537,538,539,540, 542, 543, 544, 545,546, 548, 549,550,576,654,655,656,658,660, 661,664 Lorenz, L.,57,96,224, 229,500 Love, A.E.H.,56,226, 417,449, 654,662 Macdonald, H.I\l.,78,79,80,158, 170, 171, 225,229,233,365, 377,385,386, 389,395, 396, 412,413, 439,440, 482, 509,511 Maclean, j\L,658 McMahon, J.,64,195,200, 505, 507,581 MacRobert,T.M.,197 Maggi, G.A.,13 Mabnst^i, C.J.,99,173, IS.J,187, 188, 196, 203 Manfredius, G.,92 50—5 794 THEORY OFBESSEL FUNCTIONS March, H.W.,56,225,449 Marcolongo, R.,135 Marshall, W.,505,506 Mathews, G.B.,64,65,78,194, 206, 454, 480,655,656, 657,658,660 Maxwell,J.Clerk, 125 Mayall,R.H.D.,550 Mehler, F.G.,65,155, 157, 169, 170, 180, 183,425, 431, 455,475,476 Meissel, D.F.E.,7,145,204,226, 227, 229, 232, 233, 234,247, 391, 521,557,558,561, 564, 572,655, 656,658,660,662 Mellin,R.Hj., 190,196 Mittag-Leffler,M.G.,83,497 Molins, H.,106 Moore,C.N.,479, 578,579, 597,649 Morton, W.B.,65,66 Murphy, R.,91,156,157 Myller-Lebedeff,W.{seeLebedeflF) Nagaoka, H.,340, 633,634 Neumann, Carl Gottfried, 16,19,22,23,30, 31,32,33,34,36,37,46,59,60,65,66,67, 68,69,70,71,73,128, 143, 150, 151, 155, 271, 273,274,276, 277,278, 280, 281,284, 286,290, 291, 292,345, 358,359,361,363, 365,386,418,424, 440, 441, 453, 455,456, 470, 471, 473, 474, 475,476,522,523,524, 525 Neumann, FriedrichE.,154 Newman,F.W.,663,664 Newton, SirIsaac, 120 Nicholson,J.W., 107, 108, 145, 146, 149, 150,189,226,229, 231, 247, 248, 249, 250, 252,262,329,332,413, 415, 425,426, 431, 440, 441,446,448, 505,656 Nicolas, J.,77,84 Nielsen, N.,24,44,49,64,73,74,77,82,83, 132,142, 145, 148, 149, 154, 169,224, 297, 298,299,315,350, 355, 357,359,392, 405, 455,465,522, 523, 525,526, 527,528, 571, 572, 574, 597,622,629,636 NiemoUer, F.,57,68,195 Olbricht, R.,158,481 Oltraraare, G.,173 Orr,W.McF., 145,146,206,224,454,455, 579 Otti, H.,71,274,286,341 Panton, A.W.,305 Paoli, P.,53,95,186 Parseval, M.A.,9,21,24,68,105, 229,358, 359,384Pearson, Karl, 98,99,419,421 Peirce, B.0.,501,660,664 Perfes, J.,44 Perron, O.,154 Petzval, J.,49 Phragm^n, E.,358 Picard,C.E.,93,94 Pincherle, S.,190,196,271,274,386,526,528 Plana, G.A.A.,10,38,42,45,49,53,95,96, 99,195,554 Plummer, H.C,270, 552,555 Pochhammer, L.,100, 101,297, 346,410 Pocklington, H.C,537 Poincare, J.Henri, 236 Poisson,S.D.,6,9,10,11,12,13,24,25,38, 47,49,52,67,68,69,73,95,96,160, 173, 183, 185,186,187, 194, 195,308, 369, 477, 501 Porter, M.B.,299, 477,480, 485, 515,517 Preece, C.T.,27 Puiseux, v.,559 Raffy, L.,94 Ramanujan, S.,382,449 Rawson, R.,91 Rayleigh (J.W.Strutt), Lord, 50,55,56,74, 95,137, 155, 157, 189,230, 231, 233,331, 333,374,389,395, 419, 421, 477, 502,510, 511, 616,618,660 Riccati, (Count)J.F.,1,2,3,85,86,87,88,94 Riemann, G.F.B.,80,158, 172, 203, 229, 235,427, 457,486,623,637,649 Riesz, M.,606,614 Rodrigues, O.,27 Rohrs, J.H.,10 Rudski, P.,477,508 Russell, A.,81,82,204 Rutgers,J.G.,373, 374,375, 376,380,579 Rybczynski, W.von, 56,225,449 Sasaki, S.,507 Savidge, H.G.,82,204,658 Schafheitlin, P.,64,137, 142, 168, 169,207, 215,373,391,392, 398,401,402,406,408, 421, 447,477, 479,482, 485,487,489,490, 491, 492,493, 494, 508, 510,543 Scheibner, W.,6 Schlafli, L.,10,14,27,28,30,32,33,63,64, 65,67,72,79,90,91,143, 145, 151,160, 171,174, 175, 176, 178, 179, 181,185,195, 196,215, 216, 228,253,274,276,278, 284, 285, 286, 288, 289, 290,341,342,344,345, 508, 577, 579, 581, 582, 583,585 LISTOFAUTHORS QUOTED 795 Schlomilch, O.X.,14,18,33,34,35,36,153, 173,183,617,618,619,021,022,628,655,050 Schonholzer,J.J.,145 Schott, G.A.,551, 556, 572,573 Schwarz, K.H.A.,358,643 Schwarzschild, K.,361 Schwerd, F.M.,477,654 Searle,J.H.C,199 Segar, H.W.,483 Serret,J.A.,171, 173,188 Sharpe, H.J.,105,157 Sheiipard, W.F.,199,454,579, 595,615 Siacci, F.,92 Siemon, P.,328,398 Smith, Bernard A.,655 Smith, ClaraE.,621 Smith, OttoAndreas, 50 Sommerfeld, A.J.W.,56,57,178,361,389, 395, 406, 417, 464,499 Sonine, N.J.,82,83,132,137, 139,143, 169, 170, 171, 175, 176, 177, 180,279, 280, 281, 290,353,354,362, 363, 367, 373,374,375, 376,377, 378, 383, 386,391,394,395,398, 401, 411,415,417,418, 431,432, 433, 434, 439,454 Spitzer, S.,68,71,153 Stearn, H.,482,621 Steiner, L.,655,661 Steinthal, A.E.,171,387 Stephenson, A.,579 Stern, M.A.,500 Stieltjes,T.J.,195, 196, 207,208, 209, 213, 214,464 Stirling, James, 7,8,214 Stokes, SirGeorge Gabriel, 8,12,53,55,68, 69,70,80,95,97,188, 189, 195,201, 202, 225, 229,238,320,324, 336, 391, 405, 503, 505, 507,605,659 Strutt,J.W.{seeRayleigh) Struvo, H.,328,329, 333,337, 392, 396, 397, 417,661 Sturm,J.C.F.,304, 477,479, 517, 518,521 Suchar, P.J.,90 Svanberg,A.F.,173 Takeuchi, T.,313 Tannery, J.,11,156,302 Theisinger, L.,184, 185,338Thomson, SirJoseph John, 65,173 Thomson,SirWilliam{seeKelvin) Tisserand, F.,371 Todhunter, I.,27,157,199 Turriure, E.,15 Unferdinger, F.,310 Valewink, G. (J.A.,196 Vandermonde, A.,102 vanVleck, E.B.,480 Verdet, E.,477 Vessiot, E.,94 Volterra, V.,579,621 Voronoi, G.,200 Voss, A.,406 Wagner, C.,13,142 Walker, Gilbert Thomas, 360,361 Walker, James, 328,331, 333, 537,544 Wallenburg, G.,94 Waring, E.,503 Watson,(J.N., 11,105, 125, 158, 226, 231, 249, 268,355,444,483, 485,513,519,566, 575 Weblj, H.A.,351, 523, 533,536 Weber, Heinrich, 63,64,67,75,165,167,195, 196,210, 211, 212, 386,391,392,393,394, 395, 396,398,402,405,406,408,421, 450, 451,452, 453,454, 455, 468, 469, 470,495 Weber, Heinrich Friedrich, 308,309,310,311, 312, 315,320 Weierstrass, C.T.W.,358 Wendt, Cacilie, 363 Weyl, H.,189,454 Weyr, E.,93,94 Whewell, W.,479 Whipple,F.J.W.,177,313,387 AVhitehead, C.S.,81,82,132, 148,203 Whittaker, E.T.,44,50,124, 125, 173, 197, 339, 503, 632,633 Wigert,(\S.,200 Williamson, B.,110 Willson,R.W.,501, 660,064 Wilton,J.R.,154 Young, W.H., 10,351, 406, 578, 579, 580, 582, 586, 596, 616,617 GENERAL INDEX [7^henumbersrefertothejmges.] Addition theorems, 358-372 (Chapter xi);forBessel coefficients oforder zero, 128,359; forBessel coefficients oforder «,29;forBessel functions ofthe firstkind(Gegenbauer's type), 362,367; forBessel functions ofthe firstkind(Graf's type), 130,143,359;forBessel functions orcylinder functions ofanykind(Gegenbauer's type), 363;forBessel functions orcylinder functions ofany kind(Graf's type), 143,361;forhemi-cylindrical functions, 354; forLommel's functions oftwo variables, 543; forScblafli's functionr„(;), 344; forSchlafli's ijolynomial, 289; integrals de- rived from, 367;phvsical significance of,128, 130, 361, 363,366; special anddegenerate forms of,366,368 Airy's integral, 188;expressedinterms ofBessel functions oforder one-third, 192;generalised by Hardy, 320;Hardy's expressionsforthegeneralised integralinterms ofthefunctions ofBessel, Anger andWeber, 321;references totables of,659 Analytic theory ofnumbers associated withasymptotic expansionsofBessel functions, 200 Anger's functionJvi"), 308;connexion withWeber's function, 310; differential equationsatisfied by,312; integrals expressedinterms of,312; recurrence formulae for,311; representationof Airy's integi-al (generalised) by,321; with large argument, asymptotic expansion of,313; with largeargument andorder, asymptotic expansion of,316 Approximations toBessel coefficients oforder zerowithlargeargument, 10,12;toBessel functions oflarge order(Carlini), 6,7;(extensions duetoMeissel), 226,227,232,247,521;(intransitional regions), 248; tofunctions oflargenumbers(Darboux), 233; (Laplace), 421; toLegendre func- tions oflarge degree, 65,155,157,158;toremainders inasymptotic expansions, 213;tothesum ofaseries ofpositive terms, 8.SeealsoAsymptotic expansions, Method ofstationary phase and Method ofsteepest descents Arbitrary functions, expansions of,seeNeumann series (indKapteyn series (forcomplex variables); Dini series, Fourier-Bessel series, Neumann series </*;</Schlomilch series (forrealvariables) Argument ofaBessel function defined, 40 Asymptotic expansions, approximationstoremainders in,213; conversion intoconvergent series, 204; forBessel coefficients oforder zerowith large argument, 10,12,194; forBessel functions ofarbitrary order withlarge argument, 194-224 (Chapter vii);(functionsofthe firstandsecond kinds), 199;(functionsofthethirdkind), 196; (functions ofthethird kindbyBarnes' methods), 220;(functionsofthethird kindbySchlafli's methods), 215; (functions withimaginary argu- ment), 202; forBessel functions with order andargument bothlarge, 225-270 (Chapter viii); (order greater thanargument),241;(orderlessthanargument), 244;(order nearly equal toargu- ment), 245; (ordernotnearly equal toargument, bothbeing complex), 262; forcombinations of squares andproducts ofBessel functions oflarge argument, 221,448; forFresnel's integrals, 545;forfunctions ofAnger andWeber(ofarbitrary order withlargeargument),313;(withorder andargument bothlarge), 316; forI,omniel's functions, 351; forLommel's functions oftwo variables, 549; forStruve's function(ofarbitrary order withlarge argument), 332; (withorder andargument bothlarge), 333;forThomson's functions, ber(,:)andbei(z),203;forWhittaker's function, 340; magnitude ofremainders in,206, 211, 213, 236, 314, 332, 352, 449; signof remainders in,206, 207,209, 215,315, 333,449. SeealsoApproximations Basicnumbers appliedtoBessel functions, 43 Bateman's typeofdefiniteintegral, 379,382 Bei(-:),Ber(:).SeeThomson's functions Bernoullian polynomials associated with Poisson'sintegral, 49 Bernoulli's (Daniel) solution ofEiccati's equation, 85,89,123 Bessel coefficient oforder zero, Jy(z),3,4;differential equation satisfiedby,4,5;(generalsolu- tionof), 5,12,59,60;expressed aslimit ofaLegendre function, 65,155,157; oscillations ofa uniform heavy chain and, 3,4;Parseval'sintegral representing, 9;withlarge argim^ent, asymp- toticexpansion of,10,12,194; zeros of,4,5.SeealsoBessel coefficients, Bessel functions and Bessels differential equation Bessel coefficients J„(z),5,6,13,14-37 (Chapter n);addition theorem for,29;Bessel'sintegi-al for,19;expansioninpower series of,15;generating function of,14,22,23;inequalitiessatisfied by,16,31,268; notations for,13,14;order of,14;(negative), 16;recurrence formulae for,17; square of,32;tables of(oforders and1) ,662,666-697;(ofordern),664,730-732;(with equal order andargument), 664, 746; tables of(references to),654, 655, 656, 658. SeealsoBessel coefficient oforder zero, Bessel's differential equation andBessel functions Bessel functions, 38-84(Chapter in) ;argument of,defined, 40; differential equations oforder higher than thesecond satisfied by,106;expressed aslimits ofLam^ functions, 159;expressedaslimits ofP-functions, 158;history of,1-13(Chapter i) ;(compiled byMaggi andbyWagner),13;indefiniteintegrals containing, 132-138;order of,defined, 38,58,63,67,70;rank of,de- fined, 129;relations between thevarious kinds of,74;representationofcylinder functions in terms of,82;solutions ofdifference equations interms of,83,355; solutions ofLaplace's GENERAL INDEX 797 equation containing, 83,124;solutions oftheequation ofwavemotions containing, 123;tbree- temi relations connecting, 300;with negative argument, 75. Seealso thetwopreceding<iud ten foUou-int) nitri,.<, andCylinder functions Bessel functions ofthefirstkind, -/,.(:),38;addition theorems for,143, 359,362, 363,367,368; Barnes' typeofintegral representing, 190;Bessel's typeofintegial representing, 176;cutin plane torender uniform, 45;differential equation (Bessel's)satisfied by,38;expansion of,in ascending series, 40;expansion of,indescending series, xeeAsymptotic expansions ;expressed asageneralised hypcrgeometric function, 100,101;expressed asthelimit ofahypergeometric function, 154;expressedasthelimit ofaLegendre function, 156; (physical significance of),155; expressed asthelimit ofaLommel i)o]ynomial, 302;functional])roperties of,45;generalisations of,43,44,308-357;Inequalitiessatisfiedby,49,255,259,270,406;infiniteintogi-als containing. Chapter xiii,paxdm ;ofcomplex order, 46;oforder u\-\,41;(expressedinfinitetei-ms), 52, 55;(notations for), 55,80;Poisson's integral representing, 47,48;(modifications of),161,163, 164, 169,170;products of,seeProducts ofBessel functions;quotientoftwo, expressedasa continued fraction, 153, 154,303;recurrence formulae for,45,294;relations withLommel's polynomial, 297; represented bvintegrals containing Legendre functions, 173,174; symbolic formulae for,170;tables of(oforders and1),662,666-697;(ofordern),664,730-732;(of ordern-\-\). 664,740-745; (oforder^),664.714-729; (oforder -^,method ofcomputing), 664; (with equal orderandargument), 664,746; (zeros of),664,748-751; tables of(references to), 654,655. 656, 658,659,660;Weierstrassian product representing, 497;withlarge argument, xee Asymptotic expansions ;zeros of,iteeZeros ofBessel functions Bessel functions ofthesecond kind, ¥„(-) (after Hankel), 57,63;G„{z)(after Heine), 65;r''"(s) (after Neumann), 67;ir(~) (after Weber-Hchlatli, thecanonicalform), 63;addition theorems for,144,361, 365,368; Bessel's typeofintegral representing, 177;component parts of,71,72, 840; continuityof[quafunction oftheirorder), 63;differential equation (Bessel's)satisfiedby, 59,63;expansion of,inascending series, 59,60,61,69,72;expansion of,indescending series, neeAsymptotic expansions ;expressed asanintegral containing functions ofthe firstkind, 5, 133,382,433;infiniteintegrals containing, 385,387,393,394,424,425,426,428,429,430,433; Poisson's typeofintegral representing, 68,73,165; (modifications of),169,170; products of, 149; (rej)reRented byinfiniteintegrals), 221. 441,446; (asymptotic expansions of),221,448; recurrence formulae for,66,71;represented byintegi-als containing Legendre functions, 174; symbolic formulae for,170;tables of(oforders and1),662,666-697;(oforder«),664,732- 735;(oforderh),664,714-729;(oforder -J,method ofcomputing), 664;(with equal order andargument), 664,747 ;(zeros of),748-751;tables of,referencesto,655,656,658;withlarge argument,xeeAsymptotic expansions ;with negative argument, 75;zeros of,t^eeZeros of Bessel functions. SeealsoNeumann's polsoiomial Bessel functions ofthethird kind, Hv'^'(z),Hi.'-'(z),73;Barnes'integi'als representing, 192 Bessel's typeofintegiul representing, 178; Poisson's typeofintegral representing, 166; (modi- ficationsof),168, 169. 170; represented byintegrals containing Legendre functions, 174; symbolic formulae for,170;tables of(oforders and1),662,666-697;(oforder^,),664,714- 729; tables of(references to),657; withlarge argument, asymptotic expansions of,199,210, 215; with largeargument andorder, asymptotic expansions of,244, 245,262; withnegative argument, 75 Bessel functions whose order andargument areequal, approximations to,229,231,232, 259,260, 448,515;asymptotic expansions of,245;integrals representing, 258;tables of,658, 664, 746, 747;tables of(references to),658 Bessel functions whose order isafraction. Oforders ±i(and Airy's integi-al), 190; (andthe sta]iilityofavertical pole), 96;tables of,664,714-729;tables of(references to),659;zeros of, 751. Oforders ±?,tables of(references to),659. Oforders ±^, ±1/,tables of(references to), 659 Ofsmall fractional orders, tables ofzeros of(references to),502, 660. SeealsoBessel functions whose order is±(n+^) Bessel functions whose order islarge, 225-270 (Chapter viii);asymptotic expansions of,241,244, 245,262;Carlini's approximation to,6,7;(extended byMeissel), 226,227;Horn's(elementary) approximation to,225;Laplace's approximation to,7,8,9;method ofstationary phase applied to,232;method ofsteepest descents applied to,237;miscellaneousproperties of,252-261; talDles of(reference to),658; transitional formulaefor,248; zeros of,513, 516, 517, 518. See ^Iso Bessel functions wnose orderandargument areequal Bessel functions whose order is^n,-^),10,52,80;expressibleinfinite terms, 52;notations for, 55,80;tables of.664,740-745; tables of(references to),658,659 Bessel functions withimaginary argument, I,.{^), Ki>{z), Ki(c), 77,78; differential equation satisfiedby,77;integrals representing (ofBessel'stype),181;(ofPoisson'stype), 79,171.172; (proofofequivalenceofvarioustypes),185-188;monotonic property of,446;oforder=" [iifA), 80;recun-ence formulae, 79;tables ofloforders and 1),663,698-713;(oforder i),664,714- 729;(ofvarious integral orders), 664, 736,737-739; tables of(references to),657,658; withlarge argument, asvmptotic expansions of,202; zeros of,511;(computation of),512; (references to), 660 Bessel's differential equation, 1,19;(generalised), 38;forfunctions oforder zero, 5,12,59,60; forfunctions withimaginary argument, 77;fundamental systemofsolutions of.42,75 ;hasno 798 THEORY OFBESSEL FUNCTIONS algebraic integral, 117;soluble infinite termswhen andonlywhen thefunctions satisfyingit areoforder n+^,52,119; solution of.inascending series, 39,40,57,59-61;solution of,in descendingseries" seeAsymptotic expansions ;symbolicsolution of,41;transfoi-mations of,94, 97.SeealsoBessel coefl&cients andBessel functions Bessel's integral representing Bessel coefficients, 19,21;generalisations andextensions of,see Anger's function, Bourget's function, Bruns' function andWeber's function; modifications of, torepresent Bessel functions ofarbitrary order, 175,176,177,178,181;Theisinger's transforma- tion of,184;used intheoryofdiffraction, 177;used toobtain asymptotic expansions,215. See alsoParseval's integral Bounds, upper,seeInequalities Bourget's function J„ ^.(z),326;differential equationsatisfiedby,327;recurrence formulae for, 326 Bruns' function J(z; v,k),327 Carlini's approximation forBessel functions oflarge order, 6,7;extended byMeissel, 226,227 Cauchy's numbersjV_„_i. ,„,324; recurrence formulae for,325 Cayley's solution ofRiccati's equation, 88 Chain, oscillations ofauniform heavy, 3,4,576 Cognate Eiccati equations, 91 Complex variables, expansions ofarbitrary functions of,seeKapteyn series andNeumann series Complex zeros ofBessel functions, 483;ofBessel functions with imaginary argument, 511;of Lommerspolynomials, 306 Composition ofBessel functions ofthesecond kind ofintegral order, 340 Computation ofzeros ofBessel functions byvarious methods(Graeffe's), 500,502; (Stokes'), 503; (Sturm's, forthesmallest zero), 516. SeealsoZeros ofBessel functions Constant phase, Sehlafli's method of,216 Constants, discontinuityofarbitrary (Stokes' phenomenon), 201, 203, 238,336 Continuants, connected with Sehlafli's polynomial, 288 Continued fractionsrepresenting quotientsofBessel functions, 153;convergence of,154,303 Convergent series, Hadamard's conversion ofasymptotic expansions into,204 Crelier's integral forSehlafli's polynomial,288. SeealsoNeumann's integralforNeumann's polynomial Cross-ratio ofsolutions ofEiccati's equation, 94 Cube ofaBessel function, expansion of,149 Cutnecessaryfordefinition ofBessel functions, 45,77 Cylinder (circular), motion ofheat in,9,10,576,577 Cylinder functions, '^^(^), 4,82,480; addition theorems, 143, 361,365; connexion with Bessel functions, 83;originofthename, 83;rank of,129; solutions ofdifferential equationsoforder higher than thesecondby,106;three-teiTn relations connecting, 300. SeealsoBessel functions andHemi-cylindrical functions Darboux' method ofapproximatingtofunctions oflarge numbers, 233 Definite integrals, containing Bessel functions under theintegral sign, 373-382 (Chapter xii) ; evaluated bygeometrical methods, 374, 376,378;theRamanujan-Hardy method ofevaluation, 382. SeealsoInfinite integrals Definite integrals representing special functions, seeBessel functions andIntegrals Determinants, representing Lommel's polynomials, 294;Wronskian, 42,76,77 Difference equations (linear with linear coefficients) solved bymeans ofBessel functions, 83.See alsoFunctional equations andRecurrence formulae Differentiability ofFourier-Besselexpansions, 605;ofspecial Schlomilch series, 635 Differential coefficients, fractional, 107,125 Differential equations (ordinary),linear ofthesecond order, equivalenttothegeneralisedRiccati equation, 92;oforder higher than thesecond solved byBessel functions, 106;oscillation of solutions of,518;satisfied bytheproductoftwoBessel functions, 145,146;solved byelemen- tarytranscendants, 112; symbolic solutionsof,41,108. Seealsounder thenames ofspecial equations, such asBessel's differential equation, andunder thenames ofvarious functions a7id polynomials satisfying differential equations, such asAnger's function Differential equations (partial),solution ofbyanintegi-al containing Bessel functions, 99;seealso Laplace's equation andWave-motions, equation of Diffraction, theory of,connected withAiry's integral, 188;with Bessel's typeofintegral, 177;with Schlomilchseries, 633;with Struve's functions, 417 Diffusion ofsalts inaliquid, andinfiniteintegi-als containing Bessel functions, 437 Diniexpansion, 580. SeealsoDini series GENERAL INDEX 799 Dlni series, 577, 580,596-005, 615-617 (Chapter xviii), 651-653;expansionofanarbitrary func- tion ofarealvariable into. 580,600;methods oftheory offunctions ofcomplex variables applied to,596,602;Kiemann-Lcbesgue lemma, analogue of,599;Riemann's theorem, analogue of, (>49;summability of,601,615;uniformityofconvergence of,601,604;uniqueness of,616,651; value atendofrange, 602 Dirichlefs discontinuous factor, 406 Discontinuityof:ul)itniry constants(Stokes' phenomenon), 201, 203, 238,336 Discontinuous factor(Dirichlet's), 406; (Weber's),405 Discontinuous integrals, 398,402, 406, 408, 411, 415,421 Domain A'(Kapteyn's), 559;diagram of,270 DuBoisRaymond's integrals withoscillatoiy integrands expressedinterais ofBessel functions, 183 Electric waves, 56,226,449 Electromagnetic radiation, 551,556 Elementary transcendants, definition of.111;order of,111;solution ofdifferential equations by, 112 Equal order andargument, Bessel functions with, 231,232, 258,260;tables of,746,747 ;tables of(references to),658,664 Eulers solution ofEiccati's equation, 87 Exponential function, tables of,698-713;tables referred to,663,064 Factors, discontinuous(Dirichlet's), 406;(Weber's), 405;Neumann'se„ (=1or2),22;expression ofBessel functions asproductsofWeievstrassian, 497 Fej^r's theorem, analogue of,forFourier-Bessel expansions, 610 Finite terms, Bessel functions oforder±(n+i)expressed in,52;Bessel functions ofother orders notsoexpressible, 119;solutions ofEiccati's equation in,85,86,89;thesolution ofEiccati's equation in,notpossible exceptinDaniel Bernoulli's casesandtheir limit, 123 Flights, problemofrandom, 419 Fourier-Bessel expansion, 580. SeealaoFourier-Bessel series Fourier-Bessel functions, 4,84 Fourier-Bessel integrals,^^eeMultiple infinite integrals Fourier-Bessel series, 576-617 (Chapter xviii), 649-651;expansionofanarbitrary function ofa realvariable into, 576,580; Fejer's theorem, analogue of,610; Kneser-Sommerfeld expansion ofacombination ofBessel functions into,499;methods oftheoryoffunctions ofcomplexvari- ables applied to,582,607;order ofmagnitude ofterms in(Sheppard's theorem), 595;Eiemann- Lebesgue lemma, analogue of,589;Eiemann's theorem, analogue of,649;summability of,578, 606,613;term-by-term differentiationof,578,605;uniformityofconvergence of,593,594; (near origin), 015;uniformityofsummability of,612;uniqueness of,616,649;value atendof range, 594,603 Fractional differential coeCacients, 107,125 Fresnel's integrals, 544;asymptotic expansion of,545;tables of,744, 745;tables ofmaxima andminima of,745;tables of(references to),660, 661,664 Functional equations delining cylinder functions, 82;generalised byNielsen, 355 Functions oflarge numbers, approximations duetoDarboux, 233;approximations due toLaplace, 8,421. Seeahi>Approximations, Asymptotic expansions, Method ofstationary phase andMethod ofsteepest descents Fundc^mental system ofsolutions ofBessel's differential equation, 42,75,78 Gallop's discontinuous infinite integrals, 421 Gamma functions, representation ofBessel functions byintegrals containing, 190,192,221;appli- cations todetermination ofasymptotic expansions, 220,223;applicationstoevaluation ofinfinite integrals, 383, 434,436 jBamma functions, representationofLommel's functions byintegrals containing, 351;applications todetermination ofasymptotic expansions, 352 Gegenbauers addition theorem forBessel functions, 362, 363,367 Gegenbauer's discontinuous infinite integrals, 415,418 Gegenbauer's function Cj {z),50,129,363,365, 367, 308, 369,378,407 Gegenbauer's polynomial A„^v{t), 283;contour integrals containing, 284,524; differential equa- tion satistiedby,283;equivalence withspecial forms ofLommel's function, 351;recurrence fornxulae for,283 Gegenbauer's polynomial I'n.n.i' (t),293,525 Gegenbauer's representation ofJt,(z)byadouble integral resembling Poisson's integral. 51 Gegenbauer's typeotdelinite integral, 378 Generalised hypergeometric functions, -feeHypergeometric functions (generalised) 800 THEORY OFBESSEL FUNCTIONS Generalised integrals (with implied exponential factor), 188,441, 463,464 Generating function ofBessel coefficients, 14,22,23;ofNeumann's polynomials, 281,282 Gilbert's integrals, 548,549 Giuliani's function,sr-rBourget's function GraefiFe's method ofcalculating zeros, 500,502 Grafs addition theorem forBessel functions, 359,361 Group velocity, 229 Growth ofzeros ofBessel functions, 485 Hankers infinite integrals, 384, 386, 389, 390, 393, 395, 424, 427, 428,434 Hansen's upper hound forJ^i^),31;generalised,406 Hardy's functionsCi,^{a), Si,^(a), Ei,^{a), (generalisationsofAiiy's integral), 320; expressedin terms offunctions ofBessel, Anger andWeber, 321,322 Hardy's integrals representing Lommel's functions oftwovariables, 546 Hardy's method ofevaluatingdefinite integrals, 382 Heat, conduction of,9,10,450, 576, 577,616 Hemi-cylindrical functions S„(~),defined, 353;expressedinterms ofthefunction oforder zero, 353;addition theorem for,354 Hypergeometric functions, limiting forms expressed asBessel functions, 154 Hypergeometric functions (generalised), 90,100; Bessel functions expi-essedinterms of,100,101; notations for,100;relations between (Rummer's formulae), 101,102;Sharpe'sdifferential equa- tionsolvedby,105 Imaginary argument, Bessel functions with,.'ieeBessel functions with imaginary argument ; Struve's functions with, 329,332 Indefinite integrals containing Bessel functions under theintegral sign, 132-138, 350,581;tables of,744, 745,752;tables of(references to),660, 661,664 Inequalities satisfied byBessel functions, 16,31,49,255, 259, 268,406; byNeumann'spoly- nomial, 273,282;byStruve's function, 328, 337,417;byzeros ofBessel functions, 485,489, 490, 492,494, 515,516,521 Infinite integrals containing Bessel functions under theintegral sign,383-449 (Chapter xrn);dis- continuous, 398,402,406,408,411,415,421;generalised,441;methods ofevaluating, described, 383;Eamanujan's type (integralsofBessel functions with respecttotheiroi'der), 449. Seealso under thenames ofvarious integrals, e.p.Lipschitz-Hankel infinite integral Infinity ofthenumber ofzeros ofBessel functions andcylinder functions, 4,478,481, 494,495 Integrals, expressedinterms ofLommel's functions oftwovariables, 540; expressedinterms of thefunctions ofAnger andWeber, 312; Fresnel's, 544, 545, 660, 661,664,744,745; Gilbert's, 548,549;values of,deduced from addition theorems, 367;withoscillatory integrands, 183; with thepolynomialsofNeumann andGegenbauer under theintegral sign, 277, 285. Seealso Definite integrals (uulInfinite integrals Interference, 229 Interlacing ofzeros ofBessel functions andofcylinder functions, 479,480,481 Irrationality oftt,90,485 Jacobi's transformation connecting sin»j5 with the{n-l)thdififerential coefficient ofsin2"-i^ with respect tocos6,26;erroneously attributed toEodrigues, 27;various proofs of,27,28 Kapteyn's domain K,559;diagram of,270 Kapteyn series, 6,13,551-575 (Chapter xvii);connexion with Kepler's problem, 551;expansions into, derived from Kepler's problem, 554,555;expansion ofanarbitrary analytic function into, 570;fundamental expansions into, 557,559, 561, 564,566, 568,571;Kapteyn's domain A',of convergence of,559;(diagram of),270;nature ofconvergence outside andontheboundaryof K,574;second kind of,572 Kapteyn's polynomial ©„ (t),568; expressedinterms ofNeumann's polynomial, 569 Kapteyn's type ofdefiniteintegral, 380 Kepler's problem, 6,551,554;Bessel's solution of,13;Lagi-ange's solution of,6 Kinds ofBessel functions, (first) 40;(second) 58,63,64,65,67;(third) 73 Kneser-Sommerfeld expansion ofacombination ofBessel functions asaFourier-Bessel series, 499 Kummer's formulae connecting generalised hypergeometric functions, 101,102 Lam6 functions, limiting forms expressed asBessel functions, 159 Laplace's equation, general solution duetoParseval, 9;general solution due toWhittaker, 124; solutionsinvolving Bessel functions, 83,124; used toobtain addition theorems forBessel func- tions, 127 GENERAL INDEX 801 Laplace's methods ofapproximatingtofunctions oflargenumbers, 8,421 Laplace's transformation, 280,395 Large numbers, mctliods ofapproximationtofunctions of(Darboux), 283;(Laplace), 8,421. See (thoApproximations"'(*/Asymptotic expansions Large order, mvAnger's function, Bessel fimctions whose order islarge, Struve's function mid Weber's function Lebesgue's lemma,-feeRiemann-Lebesgue lemma Legendre functions, Ikunes' notation for, 1.56; integials containing, 50,173,174,339,475; limits of,expressedasBusscl functions, 65,155,157 ;(physical significance of),155;oflarge degree, approximations to,15H;relation between twokinds of,174;Whipple's transformation of,387. SeealsoGegenbauer's functionC'„'' {z) LipscMtz-Hankelinfinite integral, 384;generalised, 389 Lommel's functions'^^,v(z), S^,i,(z), 345,347; cases ofexpressioninfinite terms, 350; integrals representing, 346,350;recurrence formulae, 348;specialcases expressible bythepolynomials ofGegenbauer, Neumann and Schliitii, 350; specialcases with^±canoddnegative integer, 348;withlarge argument, asymptotic expansion of,351 Lommel's functions oftwovariables, ?v(w, z),V:,{w, z),537,538; addition formulae for,543; integi-als representing, 540,546;reciprocationformulae, 542;recurrence formulae, 539;special ease of,581,752; tables of,752; tables referred to,660;with large argument, asymptotic expan- sions of,549 Lommel's polynomial Rm, v[z),294,295;differential equationsatisfied by,297;Hurwitz' notation (]m,v{z),303;limit of,expressedasaBessel function, 302;ofnegative order,R-m,v [z],299; recurrence formulae, 298; recurrence formulae inHurwitz' notation, 303; relations with Bessel functions, 295, 297,302; three-term relations connecting, 300,301; zeros of,304, 305,306 Magnitudes ofremainders inasymptotic expansions, 206,211,213, 236,314, 332,352,449 Maxima ofBessel functions, 488; ofFresnel's integrals,table of,745; ofintegralsofBessel func- tions, table of,752 Mean anomaly, expansionsofelements ofanorbit intrigonometricalseries of,6,13,552,554,556 Mebler-DiricMet integral representing Legendre functions, limiting formexpressedasPoisson's integral, 157 Mehler-Sonine integrals representingBessel functions, 169,170 Meissel's approximationstoBessel functions oflarge order, 226, 227, 232, 247,521;typesof Kapteyn series, 557,561, 564,566 Membrane, vibrations ofacircular, 5,576,618;vibrations ofasectorial, 510 Method ofconstant phase (Schlalli's),216 Method ofstationary phase, 225,229;appliedtoBessel functions, 231,233 Method ofsteepest descents, 235;appliedtoBessel functions, 237,241,244,245,262;appliedto functions ofAnger andWeber, 316;appliedtoStruve's function, 333;connexion with Laplace's method ofapproximation,421 Minima ofBessel functions, 488;ofFresnel's integrals,table of,745;ofintegi-alsofBessel func- tions, 752 Monotonic propertiesofJ^{vx)lJi, {v),257;ofJv{v)andJJ{v),260;ofA'^ (.i),446 Multiple infinite integrals, 450-476 (Chapter xiv);investigated byNeumann, 453,470;(generalised byHankel), 453, 456,465; (generalised byOrr), 455; (modified byWeber), 468;Riemann- Lebesgue lemmas, analogues of,457,471;Weber's type of,450 Neumann series, 522-537 (Chapter xvi); expansionofanarbitrary analytic function into, 523; generalised, 525;(special series), 30,31,36,69,71,151;Laurent's expansion, analogue of,524; Pincherle's theorem onthesingularities of,526; special series, 18,23,25,33,34,35,12.^. 130, 138,139,140,527,581;Webb-Kapteyn (real variable) theory of,533. SeealsoAddition theorems andLommel's functions oftwovariables Neumann's factorf,i(=1or2),22 ^x-Neumann's integral forJJ{z),32;forNeumann's polynomial, 278,280 Neumann's polynomial 0„{t),271,272,273;connected withKapteyn's polynomial. 569;connected withNeumann's polynomial t2„{t),292;connected with Schliifli's polynomial, 285,286 ;contour integrals containing, 277;differential equationsatisfiedby,276;expressedinterms ofLommel's functions, 350;formerlycalled aBessel function ofthesecond kind, 67,273;generalised by Gegenbauer,seeGegenbauer's polynomial .(«,i/(0; generating function of,281,282; inequali- tiessatisfied by,273,282;infinite integrals containing, 433;Neumann's integral representing, 278,280;ofnegative order defined, 276;recurrence formulae for,274 Neumann's polynomial (2,^(f),290,291; expressedasintegral containing Neumami's polynomial 0„(t),292;Gegenbauer's generalisation of,seeGegenbauer's polynomial Ihi.^. y(t);recurrence formula for,292 Nicholson's infinite integrals, 431,441 802 THEORY OFBESSEL FUNCTIONS Nielsen-Hankel functions,st'fBessel functions ofthetliird kind Null-functions, Lerch's theorem onintegrals representing, 382;represented bySchlomilch series, 634, 636, 642,647 Numbers, analytic theory of,associated withasymptotic expansionsofBessel functions, 200 Numbers, Cauchy's, 324;recurrence fonnulae for,325 Order ofaBessel function defined, 38,58,63,67,70;integrals with regard to,449 Ordinary differential equations,seeDifferential equations Oscillation ofsolutions oflinear differential equations, 518 Oscillations ofmembranes, 5,510,576,618;ofuniform heavy chains, 3,4,576 Oscillatory integrands, DuBoisKeymond's integrals with, expressedinterms ofBessel functions, 183 P-functions, limiting forms expressed asBessel functions, 158 Parseval's integral representing Jo(^)i9i21;modifications of,21 Partial differential equations, seeDifferential equations Phase, method ofstationary, general principles of,225,229; appliedtoBessel functions, 231,233 Phase, Schlafli's method ofconstant, 216 Pincherle's theorem onsingularitiesoffunctions defined byNeumann series, 526 Poisson's integral forBessel coefficients, 12,24,25;forBessel functions, 47,48,49;(generalised by Gegenbauer), 50;(symbolic formof),50;forBessel functions ofimaginary argument, 80;for Bessel functions ofthesecond kind, 68,73;limit oftheMehler-DirichletintegralforLegendre functions as,157;transformation intocontour integralstorepresent Bessel functions ofanyorder (ofthe firstkind), 161,163,164;(ofthesecondkind),165;(ofthethirdkind), 166,167; (with imaginary argument), 171,172; transformations ofthecontourintegrals, 168, 169, 170. See alsoParseval's integral nmJStruve's function Polar coordinates, changeofaxes of,used toobtain transformations ofintegi'als, 51,374,376,378; used toexpress Bessel functions aslimits ofLegendre functions, 155 Probl^me demoments ofStieltjes, 464 Products ofBessel functions, 30,31,32,82,146,147,148,149;Bateman's expansion of,130,370; expansionsofarbitrary functions into series of,525,572; integrals representing, 31,150,221,438, 439, 440,441,445,446,448;series of,30,151,152;with largeargument, asymptotic expansions of,221,448 Products ofWeierstrassian factors, Bessel functions expressed as,497 Quotient ofBessel functionsexpressed asacontinued fraction, 153, 154,303 Radius vector ofanorbit, expansionas trigonometricalseries ofthemean anomaly, 6,13,552,553,554 Ramanujan's integrals ofBessel functions with respecttotheir order, 449 Ramanujan's method ofevaluatingdefinite integrals, 382 Random flights, problem of,419 Rank ofBessel functions andcylinder functions, 129 Real variables, expansions ofarbitrary functions of,seeDini series, Fourier-Beasel series, Neumann series (Webb-Kapteyn theory), andSchlomilch series Reality ofzeros ofBessel functions, 482, 483,511 Reciprocation formulae forLommel's functions oftwovariables, 542 Recurrence formulae forAnger's functions, 311;forBessel coefficients, 17;forBessel functions ofthe firstkind, 45;forBessel functions ofthesecond kind, 66,71;forBessel functions ofthe third kind, 74;forBessel functions withimaginary argument, 79;forBourget's functions, 326; forCauchy's numbers, 325;forcylinder functions, 82;forGegenbauer's polynomials, 283;for Lommel's functions, 348;forLommel's functions oftwovariables, 539;forLommel'spoly- nomials, 298,303;forNeumann's polynomials 0,^(t),274;forNeumann's polynomials ii,^(t), 283; forSchlafli's functions, 71,342,343; forSchlafli'spolynomials, 285; forStruve's func- tions, 329;forWeber's functions, 311;forWhittaker's functions, 339. SeealsoFunctional equations, Hemi-cylindrical functions midThree-term relations Reduced functions, Cailler's, 536 Remainders inasymptotic expansions, magnitudes of,206,211,236,314,332,352;signs of,206, 207, 209, 215, 315,333;Stieltjes' approximations to,213 Repetition ofzeros ofBessel functions andcylinder functions, impossibility of,479 Riccati's differential equation, 1,2,85-94; connexion with Bessel's equation, 1,90;equation cognate to,91;limiting form of,86;soluble cases of(D.Bernoulli's), 85;soluble cases of, exhausted byD.Bernoulli's formula and itslimit, 123;solutions byvarious mathematicians (D.Bernoulli), 2,85,89;(Cayley),88;(Euler), 87;(Schlafli), 90;solved bymeans ofinfinite series byJames Bernoulli, 1;transformations of,86 Riccati's differential equation generalised, 3,92,94;cross-ratio ofsolutions, 94;equivalence GENERAL INDEX 803 with thelinear equationofthesecond order, 92;singularities of,94;solublebyvarious num- bers(two, oneornone)ofquadratures, 3,93 Riemann-LebesgTie lemma, analogues ofthe,457, 471, 589,599 Riemanu's theorem ontrigonometrical series, analoguesforSchlomilch series, 642,647;analogues forseries ofFourier-Bessel andDini, 649 Rodrlgues' transformation,set'Jacobis transformation Schafheitlin's discontinuous infinite integral, 398, 402,405, 406, 408,411 Schafheitlin's integrals representing Bcssel functions andcylinder functions, 168,169,490,491,493 ScMafli's functions7'„{z)andU,^(z), 71,340,343; addition theorems for,344,345; differential equations satisfiedby,342,343; ofnegative order, 343; recurrence formulae for,71,342,343 ScMaflis hypergeometric function, 90 Schlafli's polynomial &„(<), 284,286; addition theorem for,289; connexion withNeumann's polynomial 0,^(t}, 285,286; Crelier's integral representation of,288; differential equation satisfiedby,285;expression bymeans ofBessel functions, 287;expressioninterms ofLomnicl's function, 350;integrals evaluated interms of,350;recurrence formulae for,285 Schlafli's solution ofRiccati's equation, 90 Schlomilch series, 618-649(Chapter xix) ;definition of,621;definition ofgeneralised, 623;ex- pansionofanarbitrary function ofarealvariable into, 619,623,629;nature ofconvergence of, 637,645;null-functions expressed by,634;Riemann's theorem ontrigonometrical series(ana- logue of),642,647; special cases of,632; symbolic operatorsinthetheory of,626,627; theory offunctions ofcomplex variables connected with, 623;uniqueness of,643,647 Series containing Bessel functions, seeDini series, Fourier-Bessel series, Kapteyn series, Neumann series andSchlomilch series Series ofBessel functions, definition of,580 Series ofpositive terms, approximationtothesum of(greatest termmethod),8 Sharpes differential eoLuation, 105;solution bygeneralised hypergeometric functions, 105 Sign ofremainders inasymptotic exj)ansions, 206, 207, 209, 215, 315, 333,449;ofStruve's func- tion, 337,417 Sine-integral expressedasaseries ofsquaresofBessel coefficients, 152 Singularities offunctions defined byNeumann series (Pincherle's theorem), 526;ofthegeneralised Riccati equation, 94 Smallest zeros ofBessel functions, 5,500, ")16 Sommerfeld's expansion, seeKneser-Sommerfeld expansion Sonine-Mehler integrals representing Bessel functions, 169,170 Sonine's definite integral, 373;generalised, 382 Sonine's discontinuous infinite integrals, 415 Sonine's infinite integrals, 432 Spherical geometry used toobtain transformations ofintegrals, 51,374,876,378; used toexpress Bessel functions aslimits ofLegendre functions, 155 Sound, Sharpe'sdifferential equationinthetheory of,105 Squares ofBessel functions,seeProducts ofBessel functions Stability ofavertical poleassociated with Bessel functions oforder one-third, 96 Stationary phase, method of,225,229;appliedtoBessel functions, 231,233 Steepest descents, method of,235;appliedtoBessel functions. 237, 241, 244, 245,262;applied tofunc^tions ofAnger andWeber, 316; appliedtoStruve's function, 333; connexion w'ith Laplace's method ofapproximation,421 Stokes' method ofcomputingzeros ofBessel functions andcylinderfunctions. 503, 505,507 Stokes' phenomenon ofthediscontinuityofarbitrary constants, 201, 203, 238,336 Struve's function R„(z),328;connexion withWeber's function, 336;differential equation satisfied by,329;inequalities connected with, 328;infinite integrals containing, 392,397,417,425,436; integi-al representations of,328,330;occurrence ingeneralisedSclililrailch series, 622, 623,631, 64^646, 647; oforder±("H-|),333;recurrence formulae for,329;sign of,337,417;tables of, 663,666-697; Theisinger's integral for,338; withimaginary argument, 329,332; withlarge argument, asymptotic expansions of,332;withlargeargument and oi-der, asymptotic expan- sions of,333;zeros of,479 Struve's infinite integrals, 396, 397,421 Sturm's methods appliedtodetcnnine thereality ofzeros ofBessel functions, 483 :ofLommel's polynomials, 304,305,306;appliedtoestimate thevalue ofthesmallest zero ofBessel functions andcylinder functions, 517,518 Symbolic operators inexpi-essions representingBessel functions. 50,170; inexpressions repre- senting solutions ofvarious differential equations, 41,51,108; inthetheoryofSchlomilch series, 627 804 THEORY OFBESSEL FUNCTIONS Tables ofBessel coefficients(oforders and1),662,666-697; (ofordern),664,730-732; (with equal orderandargument), 664,746;ofBessel functions ofthetirstkind(oforders n+i,-n-i), 664,740-741;(oforder^),664,714-729;ofBessel functions ofthesecond kind(oforders and 1),662,666-697;(oforder?(),664,732-735;(oforderA),664,714-729;(with equal orderand argument), 664,747;ofBessel functions ofthethird kind(oforders and1),662,666-697; (of order^),664,714-729; ofBessel functions withimaginary argument (oforders and1),663, 698-713;(oforder)i),664,736,737-739;(oforderi),664,714-729;ofc^',663,698-713;ofFresnel's integrals, 664,744-745;ofintegralsofBessel functions oforder zero, 664,752;ofStruve's func- tions(oforders and1),663,666-697;ofzeros ofBessel coetJicients andfunctions ofintegral order nandoforderi,664,748-751 Tables(references to)ofAiry's integral, 659; ofBessel coefficients andfunctions derivable from them, 654, 655, 656,658;ofBessel functions(oforders «+i,-n-i),658,659;(oforders ±i, ±1),659; (oforders ±i,±|), 659; ofBessel functions ofthesecond kind, 655,656,658; of Bessel functions ofthethird kind, 657;ofBessel functions withimaginary argument, 657,658;of Fresnel's integrals, 661;ofintegralsofBessel functions andStruve's functions, 661;ofLommel's functions oftwovariables, 660;ofThomson's functions ber .rand beix,etc.,658;ofzeros of Bessel coefficients, functions andassociated functions, 659,660 Theisinger's integral representationofBessel functions, 184; ofStruve's andWeber's functions, 338 Thomson's(SirWiiliam) functions, berz, bei£, 81;connexion withBessel functions, 81;generali- sations, 81;references totables of,658;squares andproducts of,82,148;withlargeargument, asymptotic expansions of,203 Three-term relations connecting Bessel functions, cylinder functions andLommel's polynomials, 300,301 Transcendants, elementary,definitionof.111;order of,111;solutions ofdifferential equations by,112 Transitional regions associated with Bessel functions oflarge order, 248 Uniformity ofconvergence ofDini series, 601;ofFourier-Bessel series, 593,594;ofKapteyn series, 575;ofSchlomilch series, 632 Uniqueness ofFourier-Bessel andDini series, 616, 649,651;ofSchlomilch series, 643,647 Upper bounds, seeInequalities Viscous fluid, motion of,associated with Airy's integi-al, 189 Wave-motions, equation of,general solutions, 125;generalisedtopdimensions, 128;used to obtain addition theorems forBessel functions, 129 Waves, electric, 56,226,446;onwater, andthemethod ofstationary phase, 229 Weber's(H.)discontinuous factor, 405 Weber's (H.)infinite integrals, 391, 393, 395,396; (discontinuous types of),398, 402,405, 406, 408,411 Weber's (H.F.)function E,/(z),308;connexion withAnger's function, 310;connexion with Struve's function, 336;differential equation satisfiedby,312;integrals expressedinterms of,312;re- currence formi^lae for,311;representation ofAiry's integral (generalised) by,321;tables of,see Struve's function;Theisinger's integi-al for,338;withlargeargument, asymptotic expansion of, 313;withlargeargument andorder, asymptotic expansion of,316 Weierstrassian products, expression forBessel functions as,497 Whipple's transformation ofLegendre functions, 387 Whittaker's function W^{z),339;differential equation satisfiedby,339;recurrence foi-mulae for, 339;withlarge argument, asymptotic expansion of,340 Wronskian determinant, 42,76,77 Zeros ofBessel functions, 477-521(Chapter xv);computationof(various methodsof),142,500,502, 503,516;inequalities connected with, limits of,rates ofgrowth of,485,489,490,491,494,507, 513, 516,518;infinity of,4,478;interlacing of,479, 480,481;non-coincidence of(Bourget's hypothesis),484;non-repetition of,479;number of,inastripofarbitrary width, 495; reality of, 482,483; tables of,664,748-751;tables of(references to),659;values of,4,5,512,516;with imaginary argument, 511;withunrestrictedly large order, 513,516 Zeros ofLommel's polynomials (reality of),304, 305,306 Zeros ofStruve's function, 479 PRINTED INENGLAND BYJ.B.PEACE, M.A.,ATTHECAMBRIDGE UNIVERSITY PRESS ai)Dh:aHT| a HtHh i \Oq i |aiHih}Mi||iihii ||qna“