PNAS-1935-Stratton-51-6

PHYSICS: J.A.STRATTON SPHEROIDAL FUNCTIONS ByJ.A.STRATTON DEPARTMENT OFPHYSICS, MASSACHUSETTS INSTITUTE OFTECHNOLOGY Communicated December 1,1934 Ifthewaveequation V2V+k2V=0 beseparated inthecoordinates oftheellipticcylinder, ortheprolateor oblatespheroids,' itisobserved thatbothradialandangular functions satisfyanequation ofthetype (1-z)w- 2(a+1)zw'+(b-c2z2)w=0. (1) Inthecaseoftheellipticcylinder, theparameter ahasthevalue-1/2 and(1)istheMathieu equation inalgebraic form,whereas inthespheroidal case,aisapositive integer. Ofthetwoindependent solutions of(1), oneatleastmustbefiniteatthepoles='=1iftheusualrequirements of aphysical problem aretobesatisfied, andthiscondition restricts the parameter btoadiscrete setofcharacteristic valuesintermsofaandc. Thesolutions of(1)corresponding tocharacteristic valuesofbwillbe termedspheroidal functions, anditistheobjectofthepresentnoteto definethesefunctions inamanner mostappropriate tophysical applica- tionsandtostatesomeoftheirmostimportant properties. Thedetails oftheproofs,furtherproperties ofthefunctions ofthesecondkindand aninvestigation oftheconditions ofconvergence willappearelsewhere atalaterdate. Equation (1)ischaracterized byanirregular singularity ofthesecond speciesatinfinityandbyregularsingularities at='=1,ateachofwhich theexponents are0and-a.Itisnotthemostgeneralrepresentative ofthistypesincetwoofthepossible fiveirreducible constants havebeen placedequaltozero.Intheneighborhood ofthepointatinfinity itis desirable toestablish solutions whichinthelimitreducetospherical waves. Suchfunctions, however, arenotvalidintheregionofsmall valuesofzanditisnecessary todefinesolutions appropriate tothisdomain andtodetermine theiranalytic connection withthefunctions suitable totheneighborhood ofinfinity. Inwhatfollows aandcmayhaveany realorcomplex valuesprovided R(a)>-1. Ifin(1)wesetc=0theequation admitsofpolynomial solutions when b=1(1+2a+1)andIisanypositiveintegerinclusive ofzero.These polynomials aretheGegenbauer functions Cl+l/2(z), butsincetheir numerical valueshavenotbeentabulated, itappearsadvantageous to defineaslightlymodified function whichforintegral valuesofaissimply related totheassociated Legendre functions. LetVOL.21,1935 51 PHYSICS: J.A.STRATTON r(2a) .C / T7'(z)=2G(a) (z), (2) orasahypergeometric function T,a(Z)=2ar+lr)1)F2a +I+l1,-1,a+1,1z) 2ar~(a+1)r'(+1) Thenifa=m,aninteger, )dmp(Z)=(1-z2)-m/2Pl+(Z). (4)dzm Onehasfurthermore (21+2a+1)zTI'=(I+2a)Tj¶_.+(I+1)Tj+,, (5) (21+2a +1) Thesolution of(1)whichshallbecalledofthefirstkind,validinthe neighborhood ofthecordinary pointz=0,isnowobtained asanexpan- sionintermsofthefunctions T.Ifcd0thecharacteristic valueb isoftheformb1(1+2a+1)+el(c),wheree(c)vanishes withc andistobedetermined suchthattheexpansion converges atz=1. Let wz(a,c;z)=EdlT.(z) beasolution ofthetyperequired. Thecoefficients d,,satisfythere- cursionformula (n+2a+2)(n+2a+1)c2d'+ n(n-1) (2n+2a+5)(2n +2a+3) n+2(2n+2a-1)(2n+2a-3) C2d'-2+22+2n(2a+1)+2a-12 nn+2a 1-b n(2+2a-1)(2n+2a+3)1) d'=0,(7) andsincethereisbothanevenandanoddseries,nmayhavethevalues n=0,2,4...orn=1,3,5....Itwillbeobserved thatascvanishes, thereremainsonly [n-(n+2a+1)-1(1+2a+1)]d4=0, andhenceallcoefficients mustvanishwithcwiththeexception ofdi whichremains finite. Itisevident,furthermore, thattheevenseriesin nisassociated withevenvaluesoftheindex1,andtheoddseriesinn withoddvaluesof1.Wedefine:52 PROC.N.A.S. PHYSICS: J.A.STRATTON Sea,1(c,z)=Z'd,T.(z), (8)n wherein itisassumed thattherehasbeenassigned tobacharacteristic valuesuchthattheexpansions converge atz=='=1.Theprimeindicates thatthesummation isoverallevenvaluesofnifIisevenandoverall oddvaluesifIisodd.Thequestion ofnormalization isdeferred until certainotherfunctions havebeendefined asfollows. Ifz>>1,(1)issatisfiedapproximately by w(cz)--/2Z(CZ),[p2=b+(a+1/2)2], whereZp(cz)isanysolution oftheBesselequation. Oneisledtherefore toseekexpansions validintheneighborhood ofz=0oofthetype wC(a,c;z)=(cz)1/2Zt4Zn+a+i/2(CZ) w Itmaybeverifiedthatthecoefficients a.satisfytherecursion formula (n+1)(n+2) 2+(n+2a-1)(n +2a) (2n+2a+3)(2n+2a+5)cat2(2n+2a-3)(2n +2a-1) C2a'-2+b-n(a+2a+1)-2n2+2n(2a+1)+2a-121L-2+[(++)(2n+2a-1)(2n+2a+3)Cj al=0,(9) andthatagainthereisanevenandanoddseriesinnassociated respec- tivelywithevenandoddvaluesof1.Wedefine: Re'j(c,z)(CZ)a1/2 I.4IJn+a+i12(CZ) (10)n Re1(C,Z)=(cz)-a-/2 Sta'Nn+a++12(cz), (11)an whereNistheNeumann function. Rea,j=Re',,+iRe';Re4,j=Re'I-iRe2,1. (12) Thenormalization ofthecoefficients isnowfixedonthebasisofthe asymptotic behavior of(10)and(11).Forlargevaluesoftheargument onehasasymptotically fortheevenseries J2n+a+1/2 (CZ) COS - ¼) =(-1)'Sn/"-sin cz-21+a)7r Letusnormalize suchthat X ~~~~~~I_ E3(-1)'4a2=, (n,I=0,1,2...). (13)n=O2VOL.21,1935 53 PHYSICS: J.A.STRATTON Theasymptotic expression forRe4aszapproaches infinityisthen Re,,(cz)2a-ls -21+a7)(I=0,1,2...).(14) similarly fortheoddseries 0(-laa21+1 i(n,Op01,2 .. (15)n=O2 Rel (cz)-a-Isin(cz_21+21+a)(IOp0,o12..).(16) Forthefunctions ofthesecondkind Reaj -(cz)a1 sin(cz +i+a)(I=0,1,2...).(17) Inordertodeducetheanalytic connections between thevariousfunc- tions,useismadeofcertainintegralrepresentations ofthesolution of(1). IfaLaplacetransformation of-thetype w(z)=J"eist(i _t2)au(t)d1 (18) beintroduced into(1),itisfoundthatif(18)istobeasolution, u(t) mustitselfsatisfy(1)andthecontour mustbesuchthatthebilinear con- comitant |es'£(l_.t2)a+l[iczu -dt] c (19) vanishes identically. Inparticular onemaytakeu(t)=Sea,(t)and choosethesectionoftherealaxisbetween -1and+1asthepathof integration. Thenwiththeaidoftheintegral i(-ir)(n+1)fcst(cz)~a1/2J~~+1i2(~) =V2r(2a +n+1)J-(1-t)T(d (20) itisreadilyshownaftermultiplying bothsidesof(20)byda"andsumming overnthat k1Rela,l=es(l-t2)aSea,.(t)dt, (21)-1 wherek1isaproportionality factor,and dix =1r(n+i) -a.k (22)(i)'Ni\/2irr'(n +2a+1),k.(254 PRoc.N.A.S. PHYSICS: J.A.STRATTON Thecoefficients cd,arenownormalized suchthatthefunctions Se4,1(z) areofunitmagnitude atz=1.Thisisaccomplished withtheaidof (3),andonefindsthatiftheproportionality factork1befixedsothat k1=(i)L2a+1r(a +1), (23) then Se4,'(i) =1,Sel,1(-1) =(-1) (24) andinvirtueof(22),(13)and(15)also Zr(n+1) (25) SinceSe',,andRe,'arebothintegralfunctions ofz,andbothsolutions ofthesameequation constructed withthesameparameters, theycan differonlybyafactorindependent ofz. Se1,j=X1kRe4, (26) or Sejl(z)=X1feis(l-t2)aSeal(t)dt (27) Ofthemanyothercontour integral expressions forthesolutions of(1) onlythefollowing willbementioned atthistime. rl k1Rea31=2feics((-P)aSd,1(t)dt, (28) io~ k1Re4,=2Je"(1-2)aS41(t)dt, (29)/-1 provided R(z)>1. ThefunctionsSe',aarebynomeanstheonlysolutions of(1)which maybeconstructed withthepolynomials T,provided onewithdraws therequirements offiniteness andsingle-valuedness atz= =1.Forif oneplacew=(1-Z2)aU(Z)itmaybeverifiedthatu(z)mustsatisfy (1-z2)u- 2(1-a)zu'+(b+2a-c2z2)u =0,(30) andifc=O,(30)admitsofpolynomial solutions ofthetypeT-a(z)when bisgivenoneofthecharacteristic valuesb=1(1-2a+1)-2a.In analogywithwhathaspreceded, wedefineanewsetoffunctions S =(1-Z2)-aEfJ%Txa(Z) (31) ThecoefficientsfAsatisfytherecursion formula(7)ifwereplacetherea by-aandbbyb+2a.ThenatureofthefunctionsSeajandSo',Iis clarified ifonemakesthetransformation z=cos0.Thecorresponding Fourierexpansions arethenVOL.21,1935 55 PHYSICS: P.M.MORSE Sea,1(c,0)=D'cosnO, (32)U Sol,&(c,0)=(sin0)2a-1f,F41sin(n+1)0. (33) Associated withtheSol,larethefunctionsRol,,whichbyanalogy withtheRe4,1aremostreadilydefinedby (-l1)'kRo',,(c, z)=(z2-1)-afeictSo4i(t)dt, (34) Rol,, =(Z21)-a(CZ)a-1/2 'g',J"-a112(cz), (35) (i)x/2rr(n -2a+1)g, (36) r(n+1) .=ki.(6 Thedetermination ofthecharacteristic valuesbandtheexpansion coefficients asfunctions oftheparameters aandcisessential tothecom- pletedefinition ofthefunctions, andadetailed account ofthisportionof theinvestigation willbegivenelsewhere. lBateman, PartialDifferential Equations ofMathematical Physics, p.440etseq. Forarathercomplete account ofprevious workonthesubjectandabibliography, see Strutt,Lame'sche-Mathieusche-und verwandte Funktionen inPhysikundTechnik, in thecollection Ergebnisse derMathematik, Springer, 1932. ADDITION FORMULAE FORSPHEROIDAL FUNCTIONS ByPHILIP M.MORSE DEPARTMENT OFPHYSICS, MASSACHUSETTS INSTITUTE OFTECHNOLOGY Communicated December 1,1934 Thefunctions developed byStratton' inthepreceding paperareof considerable importance inthestudyofwavemotioninellipticcylinder andinspheroidal coordinates. Bytheirmeansalargenumber ofdiffrac- tionproblems canbestudied: thescattering ofwavesfromathinstrip, fromarodoradisc,thediffraction ofwavesthrough aslitorthrough a circular aperture, thescattering ofelectron wavesfromadiatomic mole- cule,etc.Beforetheseproblems canbesolved,however, anumber of addition formulae mustbeobtained, relating thespheroidal functions to theotherknownsolutions ofthewaveequation. Someoftheseformulae aredeveloped below. 1.EllipticCylinder Co6rdinates.-In theellipticcylinderco6rdinates, x=(d/2)cosq' cosh,u,y=(d/2)sinq, sinhj,,thesolutions ofthewave equation whichareeverywhere finiteare,56 PROC.N.A.S.