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Lebedev Special Functions and their Applications

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Textbook (Prentice-Hall, 1965, revised English edition translated and edited by Richard A. Silverman) by N.N. Lebedev. It covers the gamma function, probability and exponential integrals, orthogonal polynomials (Legendre, Hermite, Laguerre), cylinder (Bessel) functions, Airy functions and spherical harmonics, with applications to heat conduction, vibrations, potential theory and electromagnetism. Each chapter ends with problems. It sits in Phil's folder of downloaded math methods books and is not his own work.

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SPECIAL FUNCTIONS AND THEIR APPLICATIONS LEBEDEV SPECIAL FUNCTIONS AND THEIR APPLICATIONS N.N.LEBEDEV Physico-Technical Institute Academy OfSciences, U.S.S.R. Revised English Edition Translated andEdited by Richard A.Silverman @ PRENTICE~HALL, INC. Englewood Cliffs, N.J. SELECTED RUSSIAN PUBLICATIONS INTHE MATHEMATICAL SCIENCES Richard A.Silverman, Editor ©1965 by PRENTICE-HALL, INC. Englewood Cliffs, N.J. A11rights reserved. Nopart ofthisbook may bereproduced inany form, by mimeograph oranyother means, without permission inwriting from thepublisher. PRENTICE-HALL INTERNATIONAL, n~1c., London PRENTICE-HALL orAUSTRALIA, P'1'Y., LTD., Sydney PRENTICE-HALL orCANADA, 1.rn., Toronto PRENTICE-HALL orINDIA (PRIVATE) 1:rn., New Delhi PRENTICE-HALL orJAPAN, lNC., Tokyo Library ofCongress Catalog Card Number 65-16942 Printed intheUnited States ofAmerica 82652 C ALITHOR’S PREFACE This book deals with abranch ofmathematics ofutmost importance toscientists andengineers concerned with actual mathematical calculations. Here thereader will ﬁnd asystematic treatment ofthebasic theory ofthe more important special functions, aswellasapplications ofthis theory tospeciﬁc problems ofphysics and engineering. Inthechoice oftopics, Ihave been guided bythegoal ofgiving asuﬁiciently detailed exposition ofthose problems which areofgreatest practical interest. This hasnaturally ledtoacertain curtailment ofthe purely theoretical part ofthebook. Inthisregard, it should benoted that various useful properties ofthe special functions which donotappear inthetextproper, willbefound intheproblems attheendoftheappro- priate chapters. Thebook presupposes that thereader isfamiliar with theelements ofthetheory offunctions ofacomplex variable, without which onecannot govery farinthe study ofspecial functions. However, inorder tomake thebook more accessible tonon-mathematicians, I have made aserious attempt tokeep toaminimum the required background incomplex variable theory. In particular, thishascompelled metodepart from the order ofpresentation found inother treatments ofthe subject, where thespecial functions areﬁrstdeﬁned by certain convenient representations interms ofcontour integrals. Theusual elementary course incomplex variable theory isadequate foranunderstanding ofmost ofthematerial presented here. Itisalsodesirable, butnotnecessary, toknow something about theanalytic theory oflinear ivAU'moR’s PREFACE differential equations. Ioccasionally draw upon other branches ofmathematics andphysics, butonlyincon- nection with certain speciﬁc examples, sothatlack of familiarity with therelevant information isnoobstacle toreading thebook. Itisassumed thatthereader already appreciates, from hisown experience, theneed forusing special functions. Therefore, Ihave notmade aspecial point ofmotivating theintroduction ofvarious functions. Bythesame token, Ihave always sought thesimplest way ofdeﬁning the special functions andderiving their properties, without concern forhistorical orother considerations. Thearrangement ofthematerial intheseparate chapters isdictated bythedesire tomake thedifferent parts of thebook independent ofeach other, atleast toacertain extent, sothatonecanstudy thesimplest classes offunc- tions without becoming involved with functions ofa more general type. Forexample, Ihave separated the theory oftheLegendre polynomials andBessel functions ofintegral order from thegeneral theory ofspherical harmonics andcylinder functions, andIhave alsocon- structed thetheory ofspherical harmonics without re- course totheproperties ofthehypergeometric function. The applications ofthetheory were selected with the aimofillustrating thedifferent ways inwhich special functions areused inproblems ofphysics andengineer- ing.Noattempt hasbeen made togive adetailed treat- ment ofthecorresponding branches ofmathematical physics. Inthisregard, most space hasbeen devoted to theapplication ofcylinder functions, andparticularly, ofspherical harmonics. Inpreparing thepresent second edition ofthebook I have revised anearlier edition invarious ways: Chapter 4now contains anewversion ofthetheorem onexpan- sions inseries ofHermite polynomials, which extends theprevious theorem toalarger class offunctions. I have alsoincreased thenumber ofexamples illustrating thetechnique ofexpanding functions inseries ofHermite andLaguerre polynomials. InChapter 5there isanew section dealing with thetheory ofAiry functions, which areoften encountered inmathematical physics andplay Auri-roR’s PREFACE v animportant roleinderiving asymptotic representations ofvarious special functions. Chapter 9,devoted tothe theory ofthehypergeometric function, hasbeen com- pletely revised, andIhope thatinitspresent form, this chapter willbeuseful totheoretical physicists andothers concerned with theapplication ofthehypergeometric function, thereby partially ﬁlling agapintheliterature onthesubject. Ihave added many newproblems, which serve both asexercise material andasasource ofsupple- mentary information nottobefound inthetextitself. Atthesame time, Ihave removed afewproblems ofno particular interest. Finally, thereferences have been brought up-to-date. Iwould liketotake thisopportunity tothank I.P. Skalskaya forhelp inpreparing thepresent edition of mybook. N.N.L. TRANSLATOR’S PREFACE Forthemost part, thisedition adheres closely tothe revised Russian edition (Moscow, 1963). However, as always with thevolumes ofthisseries, Ihave nothesi- tated tointroduce whatever improvements occurred to meinthecourse ofworking through thebook. Inthe present case, twodepartures from theoriginal textmerit special mention: 1.The Bibliography andthereferences cited inthe footnotes have been slanted towards books available inEnglish ortheWest European languages. 2.Chapters 6and8have been equipped with prob- lems, most ofthem taken from theexcellent collection byLebedev, Skalskaya andUﬁyand (Moscow, 1955). Finally, itwasdeemed impractical tobuild insuﬁiciently detailed references tonumerical tables ofthespecial functions. Here allroads eventually lead toaconsulta- tionoftheexhaustive bibliography compiled byFletcher, Miller, Rosenhead andComrie, oritsRussian counter- partbyLebedev andFedorova. R.A.S. vi CONTENTS THE GAMMA FUNCTION, Page 1. 1.1. Deﬁnition oftheGamma Function, 1. 1.2. Some Relations Satisﬁed bytheGamma Func- tion, 3. 1.3. The Logarithmic Derivative oftheGamma Function, 5. 1.4. Asymptotic Representation ofthe Gamma Function forLarge Izl,8. 1.5. Deﬁnite Integrals Related tothe Gamma Function, 13. Problems, 14. THE PROBABILITY INTEGRAL AND RELATED FUNCTIONS, Page 16. 2.1. The Probability Integral and ItsBasic Prop- erties, 16. 2.2. Asymptotic Representation OftheProb- ability Integral forLarge |z|,18. 2.3. TheProbability Integral ofImaginary Argu- ment. TheFunction F(z), 19. 2.4. The Probability Integral ofArgument \/Tx. TheFresnel Integrals, 21. 2.5. Application toProbability Theory, 23. 2.6. Application totheTheory ofHeat Conduc- tion. Cooling ofthe Surface ofaHeated Object, 24. 2.7. Application tothe Theory ofVibrations. Transverse Vibrations ofanInﬁnite Rod under theAction ofaSuddenly Applied Con- centrated Force, 26. Problems, 28. VII viiiconrrurs THE EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS, Page 30. 3.1. The Exponential Integral anditsBasic Prop- erties, 30. 3.2. Asymptotic Representation oftheExponential Integral forLarge |z|,32. 3.3. The Exponential Integral ofImaginary Argu- ment. TheSine andCosine Integrals, 33. 3.4. TheLogarithmic Integral, 37. 3.5. Application toElectromagnetic Theory, Ra- diation ofaLinear Half-Wave Oscillator, 39. Problems, 41. ORTHOGONAL POLYNOMIALS, Page 43. 4.1. Introductory Remarks, 43. 4.2. Deﬁnition and Generating Function ofthe Legendre Polynomials, 44. 4.3. Recurrence Relations and Differential Equa- tionfortheLegendre Polynomials, 46. 4.4. Integral Representations ofthe Legendre Polynomials, 48. 4.5. Orthogonality oftheLegendre Polynomials, 50. 4.6. Asymptotic Representation oftheLegendre Polynomials forLarge n,51. 4.7. Expansion ofFunctions inSeries ofLegendre Polynomials, 53. 4.8. Examples ofExpansions inSeries ofLegendre Polynomials, 58. 4.9. Deﬁnition and Generating Function ofthe Hermite Polynomials, 60. 4.10. Recurrence Relations and Differential Equa- tionfortheHermite Polynomials, 61. 4.11. Integral Representations oftheHermite Poly- nomials, 63. 4.12. Integral Equations Satisﬁed bytheHermite Polynomials, 64. 4.13. Orthogonality oftheHermite Polynomials, 65. 4.14. Asymptotic Representation ofthe Hermite Polynomials forLarge n,66. 4.15. Expansion ofFunctions inSeries ofHermite Polynomials, 68. 4.16. Examples ofExpansions inSeries ofHermite Polynomials, 73. 4.17. 4.18 4.19 4.20. 4.21 4.22 4.23 4.24 4.25CONTENTS ix Deﬁnition and Generating Function ofthe Laguerre Polynomials, 76. Recurrence Relations and Differential Equa- tionfortheLaguerre Polynomials, 78. AnIntegral Representation oftheLaguerre Polynomials. Relation between theLaguerre andHermite Polynomials, 80. AnIntegral Equation Satisﬁed bytheLaguerre Polynomials, 82. Orthogonality oftheLaguerre Polynomials, 83. Asymptotic Representation oftheLaguerre Polynomials forLarge n,85. Expansion ofFunctions inSeries ofLaguerre Polynomials, 88. Examples ofExpansions inSeries ofLaguerre Polynomials, 88. Application totheTheory ofPropagation of Electromagnetic Waves. Reﬂection from the End ofaLong Transmission Line Terminated byaLumped Inductance, 91. Problems, 93. CYLINDER FUNCTIONS: THEORY, Page 98. 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10. 5.11 5.12.Introductory Remarks, 98. Bessel Functions ofNonnegative Integral Or- der,99. Bessel Functions ofArbitrary Order, 102. General Cylinder Functions. Bessel Functions oftheSecond Kind, 104. Series Expansion oftheFunction Y,,(z), 106. Bessel Functions oftheThird Kind, 107. Bessel Functions ofImaginary Argument, 108. Cylinder Functions ofHalf-Integral Order, 111. Wronskians ofPairs ofSolutions ofBessel’s Equation, 112. Integral Representations oftheCylinder Func- tions, 113. Asymptotic Representations oftheCylinder Functions forLarge |z|,120. Addition Theorems fortheCylinder Func- tions, 124. XCONTENTS 5.13. Zeros oftheCylinder Functions, 126. 5.14. Expansions inSeries and Integrals Involving Cylinder Functions, 128. 5.15. Deﬁnite Integrals Involving Cylinder Func- tions, 131. 5.16. Cylinder Functions ofNonnegative Argument andOrder, 134. 5.17. Airy Functions, 136. Problems, 139. CYLINDER FUNCTIONS: APPLICATIONS, Page 143. 6.1. Introductory Remarks, 143. 6.2. Separation ofVariables inCylindrical Co- ordinates, 143. 6.3. The Boundary Value Problems ofPotential Theory. TheDirichlet Problem foraCylinder, 146. 6.4. The Dirichlet Problem for aDomain Bounded byTwo Parallel Planes, 149. 6.5. TheDirichlet Problem foraWedge, 150. 6.6. The Field ofaPoint Charge near theEdge ofaConducting Sheet, 153. 6.7. Cooling ofaHeated Cylinder, 155. 6.8. Diffraction byaCylinder, 156. Problems, 158. SPHERICAL HARMONICS: THEORY, Page 161. 7.1. Introductory Remarks, 161. 7.2. The Hypergeometric Equation andItsSeries Solution, 162. 7.3. Legendre Functions, 164. 7.4. Integral Representations ofthe Legendre Functions, 171. 7.5. Some Relations Satisﬁed bythe Legendre Functions, 174. 7.6. Series Representations oftheLegendre Func- tions, 176. 7.7. Wronskians ofPairs ofSolutions ofLegend- re’sEquation, 181. 7.8. Recurrence Relations fortheLegendre Func- tions, 183. CONTENTS X1 7.9. Legendre Functions ofNonnegative Integral Degree andTheir Relation toLegendre Poly- nomials, 184. 7.10. Legendre Functions ofHalf-Integral Degree, 186. 7.11. Asymptotic Representations oftheLegendre Functions forLarge |v|,189. 7.12. Associated Legendre Functions, 192. Problems, 199. SPHERICAL HARMONICS: APPLICATIONS, Page 204. 8.1. Introductory Remarks, 204. 8.2. Solution ofLaplace’s Equation inSpherical Coordinates, 205. 8.3. TheDirichlet Problem foraSphere, 206. 8.4. The Field ofaPoint Charge Inside aHollow Conducting Sphere, 208. 8.5. TheDirichlet Problem foraCone, 210. 8.6. Solution ofLap1ace’s Equation inSpheroidal Coordinates, 213. 8.7. TheDirichlet Problem foraSpheroid, 215. 8.8. The Gravitational Attraction ofaHomo- geneous Solid Spheroid, 218. 8.9. The Dirichlet Problem foraHyperboloid of Revolution, 220. 8.10. Solution ofLaplace’s Equation inToroidal Coordinates, 221. 8.11. TheDirichlet Problem foraTorus, 224. 8.12. TheDirichlet Problem foraDomain Bounded byTwo Intersecting Spheres, 227. 8.13. Solution ofLaplace’s Equation inBipolar Coordinates, 230. 8.14. Solution ofHelmholtz’s Equation inSpherical Coordinates, 234. Problems, 235. HYPERGEOMETRIC FUNCTIONS, Page 238. 9.1. The Hypergeometric Series and ItsAnalytic Continuation, 238. 9.2. Elementary Properties oftheHypergeometric Function, 241. 9.3. Evaluation oflimF(ot,f3;y;z) forz»1- Re(y—u— B)>0,243. xiiconraurs 9.4. F(<X, 13;Y;Z)asaFunction ofitsParameters, 245. 9.5. Linear Transformations of the Hyper- geometric Function, 246. 9.6. Quadratic Transformations ofthe Hyper- geometric Function, 250. 9.7. Formulas forAnalytic Continuation of F(<x, 13;Y;2)inExceptional Cases, 256. 9.8. Representation ofVarious Functions inTerms oftheHypergeometric Function, 258. 9.9. TheConﬂuent Hypergeometric Function, 260. 9.10. The Differential Equation fortheConﬂuent Hypergeometric Function and ItsSolution. The Conﬂuent Hypergeometric Function of theSecond Kind, 262. 9.11. Integral Representations ofthe Conﬂuent Hypergeometric Functions, 266. 9.12. Asymptotic Representations oftheConﬂuent Hypergeometric Functions forLarge lzl.268. 9.13. Representation ofVarious Functions inTerms oftheConﬂuent Hypergeometric Functions, 271. 9.14. Generalized Hypergeometric Functions, 275. Problems, 276. PARABOLIC CYLINDER FUNCTIONS, Page 281. 10.1. Separation ofVariables inLap1ace’s Equation inParabolic Coordinates, 281. 10.2. Hermite Functions, 283. 10.3. Some Relations Satisﬁed bythe Hermite Functions, 287. 10.4. Recurrence Relations fortheHermite Func- tions, 288. 10.5. Integral Representations ofthe Hermite Functions, 290. 10.6. Asymptotic Representations oftheHermite Functions forLarge |z|,291. 10.7. The Dirichlet Problem foraParabolic Cylin- der,293. 10.8. Application toQuantum Mechanics, 297. Problems, 297. BIBLIOGRAPHY, Page300. INDEX, Page304. THE GAMMA FUNCTION l.I.Deﬁnition oftheGamma Function Oneofthesimplest andmost important special functions isthegamma function, knowledge ofwhose properties isaprerequisite forthestudy of many other special functions, notably thecylinder functions andthehyper- geometric function. Since thegamma function isusually studied incourses oncomplex variable theory, andeven inadvanced calculus} thetreatment given herewillbedeliberately brief. Thegamma function isdeﬁned bytheformula P(z)=jwe"t"1dt, Rez>0, (l.1.1)0 whenever thecomplex variable zhasapositive realpartRez.Wecanwrite (1.1.1) asasumoftwointegrals, i.e., 1 co I‘(z)=j e-‘F-1 dz'l'J~ e“t"‘1 dt, (1.I.2)O 1 where itcaneasily beshownz thattheﬁrstintegral deﬁnes afunction P(z) 1SeeD.V.Widder, Advanced Calculus. second edition, Prentice-Hall, Inc.,Englewood Cliffs, N.J.(1961), Chap. 11. “SeeE.C.Titchmarsh, TheTheory ofFunctions, second edition, Oxford University Press, London (1939), p.100,noting thattheintegrand e"t=-1 isanalytic inzandcon- tinuous inzandIforRez>0,0<t<oo,while theﬁrst integral isuniformly convergent forRez28>0and thesecond integral isuniformly convergent for Rez<A<oo,since then 1 1 no no e"'t"1dt sj. e“‘t°‘1dt <00 e-'z=-1 dzlsj e“t"'1dt< co.0 1 1 2TI-IEGAMMA FUNCTION CI-IAP. 1 which isanalytic inthehalf-plane Rez>0,while thesecond integral deﬁnes anentire function. Itfollows thatthefunction P(z) =P(z) +Q(z)isanaly- ticinthehalf-plane Rez>0. The values ofI‘(z) intherestofthecomplex plane canbefound by analytic continuation ofthefunction deﬁned by(1.l.1). First wereplace the exponential intheintegral forP(z) byitspower series expansion, andthen weintegrate term byterm, obtaining 1 °° k °° R; 1 _ (—1) (—1)I _P=frZ1dr -iz"= i "*=1dr(Z) 0 ago k! 20 k! 0I 1.1.3 =3<—_1>t;, ‘’k=o k!2+k where itispermissible toreverse theorder ofintegration andsummation since“ 1 1 IC 1 _ (— I _ 1 I -1 zld = xld _: tx dLl: |tkZ0 kr or rkzokl oet z<oo (thelastintegral converges forx=Rez>O).Theterms oftheseries (1.1.3) areanalytic functions ofz,ifzaé0,-1,-2,...Moreover, intheregion‘ |z+k|>8>0, k=0,1,2,..., (1.1.3) ismajorized bytheconvergent series °°l andhence isuniformly convergent inthisregion. Using Weierstrass’ theorem andthearbitrariness of8,weconclude thatthesumoftheseries (1.1.3) isa meromorphic function with simple poles atthepoints z=0,—1,-2,... ForRez>0thisfunction coincides with theintegral P(z), andhence isthe analytic continuation ofP(z). Thefunction P(z)differs from P(z)bytheterm Q(z), which, asjustshown, isanentire function. Therefore I‘(z)isameromorphic function ofthecom- plex variable z,with simple poles atthepoints z=0,—1,——2,...An5 3E.C.Titchmarsh, op.cir.,p.45. 4Byaregion wemean anopen connected point set(oftwoormore dimensions) together withsome, all,orpossibly none ofitsboundary points. Inthelatter case, we often speak ofanopenregion ordomain, intheformer case, ofaclosed region orclosed domain. 5SeeA.I.Markushevich, Theory ofFunctions ofaComplex Variable, Vol. I (translated byR.A.Silverman), Prentice-Hall, Inc., Englewood Cliffs, N.J. (1965), Theorem 15.6, p.326. sac1.2 THEGAMMA ruucrrou 3 analytic expression forI‘(z), suitable fordeﬁning 1"(z)inthewhole complex plane, isgiven by °°—1" 1 0°I‘(z)= ZQT)-;7c+f1 we-14¢, z;é0,—l,—2,... (1.1.4),,=0 k. z Itfollows from (1.1.4) thatP(z)hastherepresentation re)= +Q(z+n) (1.1.s) inaneighborhood ofthepole z=—n(n=0,1,2,...),with regular part Q(z+n). 1.2.Some Relations Satisﬁed bytheGamma Function Wenowprove three basic relations satisﬁed bythegamma function: F(z+1)=zF(z), (1.2.l) r(z)r(1 -Z)= (1.2.2) 22=-1r(z)r(z +I.)=\/;¥r(2z). (1.2.3) These formulas playanimportant roleinvarious transformations andcalcula- tions involving P(z). Toprove (1.2.l), weassume that Rez >0andusetheintegral repre- sentation (1.1.l). Anintegration byparts gives F(z+1)=J:e"t”dt =—e"l‘ :+zj:e-'¢=-1 dz=zI‘(z) Thevalidity ofthisresult forarbitrary complex zaé0,—1,-2,...isan immediate consequence oftheprinciple ofanalytic continuation,“ since both sides oftheformula areanalytic everywhere except atthepoints z=0,-1, -2,... Toderive (l.2.2), wetemporarily assume that0<Rez<1andagain use(1.l.1), obtaining I‘(z)F(l —z)=‘LaoI:e““‘)s""t”“ dsdt. °According tothisprinciple, which wewilluserepeatedly, iff(z) andq>(z) are analytic inadomain Dandiff(z) =q>(z) forallzinasmaller domain D*contained in D,thenf(z)=q>(z)forallzinD.Thesame istrueiff(z) =<p(z)forallzinanysetof points ofDwithalimit point inD,say,alinesegment. SeeA.I.Markushevich, op.cit., Theorem 17.1, p.369. 4rueGAMMA ruucrrou CI-IAP. 1 Introducing thenewvariables lu=s+t, v=—»s weﬁndthat” no no dd co z—1 Using theprinciple ofanalytic continuation, weseethatthisformula remains valid everywhere inthecomplex plane except atthepoints z=0,1*1,J;2,... Toprove (l.2.3), known astheduplication formula, weassume that Rez>0andthen use(1.l.1) again, obtaining 22“‘1I‘(z)I“(z + =J00Jug €'(‘+‘)(2\/.;)2Z‘1t'1'2 dsdt 00 =4J1: Jr e_(°‘2*°2)(2ot[3)2z‘1ot dotdB, where wehave introduced newvariables at=1/E,[5=\/i.Tothisformula weaddthesimilar formula obtained bypermuting ozandI5.This gives the more symmetric representation 2“-1P<z>r<z +-I)-2fwfwe-<~“+“’><2<=e>e-1<¢ +B)d<1dB O O =4ffe'(°‘2*°2)(2otf5)2"1(<x +pa)dotdis, where thelastintegral isover thesector 0:0<at<oo,0<<oz.Intro- ducing newvariables/‘co u=<x2+l32, v=2otf3, weﬁndthat 22”‘1I‘(z)F(z +%)=foo022*‘ dujw \/éduo o u—v =2fa)e‘”o2z‘1 dofooe‘“’2 dw=\/;I‘(2z). o o Asbefore, thisresult canbeextended toarbitrary complex values zaé0,—%, —1,—%,...,byusing theprinciple ofanalytic continuation. Wenowuseformula (1.2.1)tocalculate P(z)forsome special values ofthe variable z.Applying (1.2.l) andnoting that F(l) =1,weﬁndbymathe- matical induction that I‘(n+1)=nl, n=0,1, 2,... (1.2.4) 7Fortheevaluation oftheintegral inthelaststep, seeE.C.Titchmarsh, op.cit., p.105. sac.1.3 THEGAMMA runcrrou 5 Moreover, setting z=Iin(l.1.1), weobtain I‘(%) =fooe"t“'2 dt=ZJW e‘“2 du=\/E, (l.2.5) 0 O andthen(1.2.l) implies F(n+1)= n=1,2,... (12.6) Finally weuse(1.2.2) toprove thatthefunction P(z)hasnozeros inthe complex plane’. First wenote that thepoints z=n(n=0,il,i2,...) cannot bezeros ofP(z), since F(n) =(n—1)!ifn=1,2,..., while F(n) =ooifn=0,-1,-2,...Thefactthatnoother value ofzcanbea zero ofP(z) isanimmediate consequence of(l.2.2), since ifanonintegral value ofzwere azeroofP(z)itwould have tobeapole ofF(l—z),which is impossible. Itfollows atonce that[F(z)]'1 isanentire function. 1.3.The Logarithmic Derivative oftheGamma Function Thetheory ofthegamma function isintimately related tothetheory of another special function, i.e.,thelogarithmic derivative ofP(z): ¢(z)=PI’) (1.3.1)P(z) Since P(z)isameromorphic function withnozeros, 1l»(z)canhave nosingular points other than thepoles z=—n(n=0,1,2,...)ofP(z). Itfollows from (1.1.5) that111(2)hastherepresentation” 41(2)=— +Q(z+n) (l.3.2) inaneighborhood ofthepoint z=—n,andhence 111(2), likeP(z), isamero- morphic function with simple poles atthepoints z=0,—1,——2,... Thefunction 111(2)satisﬁes relations obtained from formulas (1.2.1-3)” by taking logarithmic derivatives. Inthisway, weﬁndthat in+1)=é+41(2), (1.13) t]»(l—z)—111(2) =Tccotrcz, (l.3.4) tl/(z)+\l»(z+i)+2log2 =2¢(2z). (l.3.5) °Ofcourse, theregular part Q(z+n)in(1.3.2) isnotthesame asin(1.1.5). 9By(l.2.1—3) wemean formulas (1.2.1) through (1.2.3). Similarly, (1.2.1, 4,6)means formulas (1.2.1), (1.2.4) and(1.2.6), etc. 6tTHE GAMMA FUNCTION CHAP. 1 These formulas canbeused tocalculate 41(2) forspecial values ofz.For example, writing - 41(1)=P’(l) =—y, (l.3.6) where Y=0.5772l566 ...isEuler’s constant, andusing (l.3.3), weobtain §~I1—AtI1(n+l)= —-Y+ 2:127 n=1,2,... (l.3.7) Moreover, substituting z=Iinto(l.3.5), weﬁndthat <l1({-)=—y—-2log2, (1.3.8) andthen (l.3.3) gives " 1\]1(n+1)_—Y-2log2 +2kZ1§k__T, n_1,2,... (1.3.9) The function 111(2) hassimple representations intheform ofdeﬁnite integrals involving thevariable zasaparameter. Toderive these representa- tions, weﬁrstnote that(1.1.l) implies 1° F’(z)=fooe“‘t2“1logtdt, Rez>0. (1.310)D Ifwereplace thelogarithm intheintegrand byitsexpression interms ofthe Frullani integral“ 00e—x _e—xt logt =f idx, Ret >0, (l.3.ll)0 X weﬁndthat" 1"’(z)=fw dljw (e"‘-e-*')e-*t=-1 dt0x0 =Jim dic[e—x1“(z) _Jlw e—t(x+1)tz—1 0X o Introducing thenewvariable ofintegration u=t(x+1),weﬁndthatthe integral inbrackets equals (x+1)"1"(z). This leads tothefollowing integral representation of111(2): 41(2)=jaw[e""- ﬁg‘. Rez>0. (1.3.12) 1°Tojustify differentiating behind theintegral sign, seeE.C.Titchmarsh, op.ct't., pp.99-100. 1‘SeeH.Jeffreys andB.S.Jeffreys, Methods ofMathematical Physics, third edition, Cambridge University Press, London (1956), p.406,andD.V.Widder, op.cit.,p.357. 12Here, aselsewhere inthischapter, weomit detailed justiﬁcation ofthereversal of order ofintegration. Anappropriate argument canalways besupplied, usually byprov- ingtheabsolute convergence ofthedouble integral andthen using Fubini’s theorem. SeeH.Kestelman, Modern Theories ofIntegration, second revised edition, Dover Pub- lications, Inc., New York (1960), Chap. 8,esp.Theorems 279and280. sac.1.3 THEGAMMA ruucrrou 7 Toobtain another integral representation of41(2), wewrite (l.3.12) inthe form . °°_,, 1 dx_. °°e"‘ °° dx ff’)‘ l“(x+1)-rlx"159?. x“"‘ (x+1rxl’ andchange thevariable ofintegration inthesecond integral, bysetting x+1=e’.This gives we—t no e—tz 11.O0 —t —tz 5 —¢ =1imU (5---"i_,)dr-‘I L111].6-*0 1og(1+o> I 1_9 1og(1+6) I andtherefore, since thesecond integral approaches zero as8—->0, Q et etz 41(2)= _-id dt, Rez>0. (13.13)0 t 1—€ Setting 2=1andsubtracting theresult from (1.3.13), weﬁndthat we—t _e—tz 41(2)=—y+J‘Adi, Rez >0, (1.3.l4)0 __ or 11 ___xz—1 41(2)=-Y+4. qidx, Rez >0, (1.3.l5)0 —x where wehave introduced thevariable ofintegration x=e". From formula (l.3.15) wecandeduce animportant representation of41(2) asananalytic expression valid forall2aé0,—1,-2,...,i.e.,inthewhole domain ofdeﬁnition of41(2). Toobtain thisrepresentation, wesubstitute the power series expansion (1—x)'1=l+x+x2+---+x"+---, 0<x<l into(1.3.l5) andintegrate term byterm (this operation iseasily justiﬁed). Theresult is 41(z)= -Y+ - (13.16),,=n+ n 2 Theseries (l.3.l6), whose terms areanalytic functions for2aé0,—1,-2,..., isuniformly convergent intheregion deﬁned bytheinequalities |z+n| 23>0,n=0, 1,2,... and [2|<a, since l_ 1 < a+l n+1 n+2 (n+l)(n-a) 8-meGAMMA FUNCTION CHAP. 1 forn2N>a,andtheseries °° a+1 11;:(""1" 1)("—l1) converges. Therefore, since 8isarbitrarily small andaarbitrarily large, both sides of(l.3.l6) areanalytic functions except atthepoles 2=0,—-1,-2,..., andhence, according totheprinciple ofanalytic continuation, theoriginal restriction Re2>0used toprove thisformula canbedropped. Ifwereplace 2by2+1in(l.3.l6), integrate theresulting series between thelimits 0and2, andthen take exponentials ofboth sides, weﬁndthefollowing inﬁnite pro- duct representation ofthegamma function: 1 V°° _,(1 2)———= = Z"+-- 1.3.17I‘(z+l) 941‘? n () This formula canbemade thestarting point forthetheory ofthegamma function, instead oftheintegral representation (1.l.1). Finally wederive some formulas forEuler’s constant Y.Setting z=1in (l.3.l2—l3), weobtain °° 1 _xdx_°° l __1 _Y-—41(1)-L —e)-; -L(-———1_ e_, t)e‘dt.(l.3.18) Moreover, (l.3.10) implies Y=-Le“logtdt, (13.19) which, when integrated byparts, gives 1 11> 11_e-t uoe-t Y=j1<>gtd(e-1-1)+j logtd(e“) =j—idt -I-111.0 1 0 t 1 t Replacing tby1/tinthelastintegral ontheright, weﬁndthat 1_-t_-1/t Y=J01i"tie- dt. (13.20) 1.4.Asymptotic Representation oftheGamma Function for Large |z| Todescribe thebehavior ofagiven function f(z) as|2|—->oowithin a sector at<arg2<15,itisinmany cases sufﬁcient toderive anexpression of theform f(z)=<P(Z)l1 +f(z)]. (1-4-1) where <p(z)isafunction ofasimpler structure thanf(z), andr(2)converges uniformly tozeroas]2|->oowithin thegiven sector. Formulas ofthistype arecalled asymptotic representations off(z) forlarge Itfollows from SEC. 1.4 TI-IE GAMMA FUNCTION 9 (1.4.l)thattheratiof(2)/<p(z) converges tounity as|2|—->oo,i.e.,thetwofunc- tions f(z)and<p(z)are“asymptotically equal,” afactweindicate bywriting f(z)z41(2), |z[—>oo, on<arg2<13. (l.4.2) Anestimate of|r(2)] gives thesizeoftheerror committed when f(2)isreplaced byq>(z)forlarge butﬁnite |2|. Wenow look foradescription ofthebehavior ofthefunction f(z) as I2]—>oowhich ismore exact than thatgiven by(l.4.l). Suppose wesucceed inderiving theformula N f(2)=q>(2)[Z2,2-"+r,,(2)], 2,=1,N=1, (14.3)n=0 where z”r~(z) converges uniformly tozeroas|2|—>oo,on<arg2<13. [Note that(l.4.3) reduces to(l.4.1) forN=0.]Then wewrite Q f(2)z<p(Z)20a,,z‘", |2|->oo,.2<argz<ta, (1.4.4) andtheright-hand sideiscalled anasymptotic series orasymptotic expansion off(z)forlarge |2|.Itshould benoted thatthisdeﬁnition does notstipulate thatthegiven series converge intheordinary sense, andonthecontrary, the series willusually diverge. Nevertheless, asymptotic series arevery useful, since, bytaking aﬁnite number ofterms, wecanobtain anarbitrarily good approximation tothefunction f(z) forsufﬁciently large |2|.Inthisbook, thereader willﬁndmany examples ofasymptotic representations andasymp- totic series (seeSecs. 1.4,2.2,3.2,4.6,4.14, 4.22, 5.11, etc.). Forthegeneral theory ofasymptotic series, werefer tothereferences cited intheBibliography onp.300. Toobtain anasymptotic representation ofthegamma function P(z), itis convenient toﬁrstderive anasymptotic representation oflogP(z). Tothis end, letRez>0,andconsider theintegral representation (1.3.13), with 2 replaced byz+1,i.e., F’(z+1) ‘ ‘ ""<”1>=mIn= l€TfTe%—t2—1)dt _ me_i_ Em —tz _ w(l_i 1 —tz-40-_t dr+2joe dtfa2t+ie,_1)e dt, r'(2+1)_ i_ ~1_1 1_,, i-‘°gZ+22 (2lift)“ d’, where wehave used (l.3.1 1).Integrating thelastequation between thelimits 1and2,andbearing inmind thatNeCM8 or logF(z+1)=logP(z) +log2, I0 TI-IEGAMMA FUNCTION CHAP. 1 weﬁndthat“ logP(2) =(2— logz —2+1 (1.4.5)°°11 l e"*—e“ ii,(2';+e__—1)—‘;*"” where Re2>0.Itshould benoted thatthefunction 11 1 1f(t) = —E‘+ ;-9 appearing intheintegrand in(l.4.5), iscontinuous fort 20,withf(0) =T12-, ascaneasily beveriﬁed byexpanding f(t)inapower series inaneighbor- hood ofthepoint t=0. Tosimplify (l.4.5), weevaluate theintegral J=Lf(t)e" dt. (1.4.?) This canbedone byusing thefollowing trick: If 2=jomf(t)@-"2 dt, (1.4.s) 2-2-1:~1421111-1:-F'l—_i)Itfollows that e'”2 —e" e" dtthen 0+0+88de“’2 —e“ e“—e'”2 F [TE(—ti_) +'T__l dt e'”2—e"°° 1°°e"‘—e“/2 111 =";1‘_.+il.,"'._""’-2+2‘°g2'(l.4.9) Ontheother hand, substituting 2=4into(l.4.5), weﬁndthat ,7—J=glogrc —=}, (1.4.10) andhence J=1-§log 21:. (l.4.l1) Using thisresult, wecanwrite (l.4.5) intheform logP(z) =(2—5)logz —2+}log 21:+w(2), (l.4.12) 1“Thechoice ofthepathofintegration isunimportant. Tojustify integration behind theintegral sign, weuseanabsolute convergence argument (cf.footnote 12,p.6). sec.1.4 TI-IEGAMMA FUNCTION II where 11(2)=I[(1)2-*2 dt, R¢2>0. (14.13)O Since f(t)decreases monotonically astincreases,“ theintegral (l.4.l3) also converges forRe2=0,Im2aé0.15 Using (l.4.12) and(l.4.l3), wecaneasily derive anasymptotic representa- tionofP(z). First let[arg2|<rc/2,andintegrate (l.4.l3) byparts, obtaining 11(2)=é[f(0)+J:of'(t)e"" dt]. (1.4.14) Since f’(t)<0,|f'(t)]=—f’(t),wehave 1 °°, 2f(0) 14(2)!<E[/(0)-/(1)44-—|,|—- i.e., 1|o(z)1<El-Z-I. Iarg2|< (1.4.15) Then, taking exponentials ofboth sides of(l.4.l2), weﬁndthat f(z) =e<z—V2)1us2-2+1/2 1o821r[l +r(z)], |al.gZ!slg’ (1_4_16) where r(z) =em“) —1. According to(l.4.15), |r(2)| <lg? (l.4.l7) where Cisanabsolute constant (weassume that2isbounded away from zero, i.e.,|2|2a>0).Thus r(z)isoforder |2|'1 as|z[—>oo,afactindi- cated bywriting“ r(z)=0(|2|‘1), (l.4.l8) andhence (l.4.l6) isanasymptotic representation ofP(z) intheindicated sector. Toderive anasymptotic representation ofP(z) which isvalid inother 1*Thisfollows atonce from theexpansion “O 1 1'“)= SeeK.Knopp, Theory andApplications ofInﬁnite Series (translated byR.C.H.Young), Blackie andSon, Ltd., London (1963), p.378. 15E.C.Titchmarsh, op.cit.,p.21. ‘°Wesaythatf(z) isoforder q>(2) as2-—>20,and write f(z) =O(q>(z)) as2—>20if theinequality 1f(2)|sAlcp(z)| holds inaneighborhood of2°,where Aissome con- stant. If20isnotexplicitly mentioned, then2°=oo. I2 THEGAMMA FUNCTION CHAP. 1 sectors ofthecomplex plane, weproceed asfollows: Let8beanarbitrarily small ﬁxed positive number, andlet géargz <Tc-8. (1.4.19) Since arg(-2) =arg2—1:,thisimplies -g<arg(-2) <-8. Itfollows from (l.2.l-2) that r(2)= (1.4.20) where, according to(l.4.l6) and(l.4.18), F(_z) =e—(z+ 1/2)(1oz z—1ri)+z+ V210: 21v[l +0(|2|- (L421) Ontheother hand, inthesector (l.4.19), I eniz __e-niz e-M2 2l S1n1rz=-?-— = —em’) e_m 1 e_m (1.4.22) =—T(1— E292”) =—Tl-11+ 0(lZl_1)l, since 2e2"1" isbounded inthissector. Substituting (l.4.2l-22) into(l.4.20), weagain arrive atformula (l.4.l6). Asimilar result isobtained forthesector -(1:— 8)<arg2< —g- Finally, therefore, inanysector Iarg2|<1:-8, wehave theasymptotic representation I‘(z) =e"""/=>1°g""“‘/21°’? 2"[1+0(|z| -1)]. (1.4.23) Considerations resembling those justgiven, butmuch more complicated," leadtothemore exact formula ._O, 1 1 139 _ 11(z)=e<2—/2)1os2 2+V21g2 [1+m+m_ +0(jZ| 4)]. (1.4.24) If2=xisapositive realnumber, then (1.4.16) becomes Stirling’s formula I‘(x)=\/ﬁxx“/1e"‘[l +r(x)], (1.4.2s) 17SeeG.N.Watson, Anexpansion related toStirling‘s formula, derived bythemethod ofsteepest descents, Quart. J.Pure andAppl. Math., 48,1(1920). s1~:c.1.5 Tl-IEGAMMA FUNCTION I3 where forr(x)wehave asharper estimate than thatgiven by(1.4.l7). Infact, if2=x> 0,then "° 1 —xt _i,|...(x)|<f(0)L6dz_12x (1.426) sothat [r(x)] <e1’12" —1. (l.4.27) Finally, wenote that(1.2.4) and(l.4.25) imply thefollowing asymptotic representation ofthefactorial: n!z\/ﬁzzirtze-", n—><2». (1.4.2s) l.5.Deﬁnite Integrals Related totheGamma Function Theclass ofintegrals which canbeexpressed interms ofthegamma func- tionisvery large. Here weconsider only afewexamples, mainly with the intent ofderiving some formulas thatwillbeneeded later. Ourﬁrstresult istheformula fooe‘1’1t“""1 dt=-I-1-Pg, Rep >O,Rez >0, (l.5.l) 0 which iseasily proved forpositive realpbymaking thechange ofvariables s=pt,andthen using theintegral representation (l.l.1). Theextension of (l.5.l) toarbitrary complex pwith Rep>0isaccomplished byusing the principle ofanalytic continuation. Next consider theintegral B(x,y)=f1t""1(1 —t)1"1dt, Rex >0,Rey >0,(l.5.2)O known asthebetafunction. Itiseasytoseethat(1.5.2) represents ananalytic function ineach ofthecomplex variables xandy.Ifweintroduce thenew variable ofintegration u=t/(1-t),then (1.5.2) becomes a0 ux-I B(X, =J; ' du, RCX >0, Rﬁy > Settingp =1+u,2=x+yin(l.5.l), weﬁndthat 0 andsubstituting theresult into(l.5.3), weobtain B(x,y)=Tﬁiw e“t""”'1 dtLaoe'“‘u""1 du 1.5.5 =Po) ,-.,._.d,=P(x>Po>_ 111“(X+y)0 Ffx+y) I4 THEGAMMA FUNCTION CHAP. 1 Thus wehave derived theformula B(x,y)= (15.6) relating thebetafunction tothegamma function, which canbeused toderive alltheproperties ofthebeta function. PROBLEMS 1.Prove that _ _ Tr , _ = Tc |1"(1y)l’ —yi—-sinh ﬂy IPG+1y)|’ ﬁshﬂy forrealy. 2.Using (1.5.6), verify theidentity °°cosh 2yt _x_F(x+y)I‘(x ——y)L(—icoSht)2x dt-222 Fax) , Rex >0, Rex >|Re yl. 3.Prove that I,(v +1) 1:/2 1:/2 \/" 2 I cos“0d6=J sin"6d6=%i, Rev>-1, ° ° 1“(;+ 1) L1-+1 v+1 "/2 . 1F( 2)F(Z)cos“6s1n"0d6=———a-—, Reu >-1, Rev >—l. ° 2F(——“+V+1)2 4.Verify theformula 32- F(3z) =3?? P(z)F(z +§)I‘(z +§-). (i) 5.Derive theformula 3¢(3z) =v.l»(z) +tl»(z+%)+¢(z+3;)+3log3. Hint. Calculate thelogarithmic derivatives ofboth sides of(i). 6.Derive thefollowing integral representation ofthesquare ofthegamma function, where K0(t) isMacdonald’s function (deﬁned inSec.5.7): I‘2(z) =22*“ foot“-1Ko(z) dr, Rez>0.O Hint. Useformulas (5.10.23), (l.5.l) andtheintegral inProblem 2. PROBLEMS THEGAMMA FUNCTION I5 7.Derive theasymptotic formulas 1“'(z +O‘)=e(z+n¢—1/2)losz—z+1/z1og21:[l +0([Z|—1)]’ 1“(Z+<X)_ ,,_ (<1—B)(<>=+B—1) _ire+B)-z"[1+-i2~z—————— +0(|z| 2), where onandBarearbitrary constants, and|argz]<1r-8. Hint. Usetheresults ofSec. 1.4. 8.Derive theasymptotic formula |P<x+iy>|=~/2%-1/2"'~'|y1*-I/211 +r<x.y>1.where asItl->oo,r(x,y)—>0uniformly inthestrip [xi<at(ozisaconstant). 9.Show thattheintegral representation 1 1 t_-f(z) —EL?! ad! holds forarbitrary complex z,where t"=e"1°‘‘,|argt[<1:,andCisthe contour shown inFigure 13,p.117. 10.The incomplete gamma function y(z, cc)and itscomplement I‘(z,oz)are deﬁned bytheformulas M y(z,oi)=f e“t"‘1 dt, Rez >0,|argcx|<Tr,0 I‘(z,<x) =J‘e“t”‘1 dt, |arg ot[<Tr, G sothat Y(Z,11)+F(z,<1)=P(z)- Prove thatforﬁxed ct,1"(z,oz)isanentire function ofz,while y(z,<1)isamero- morphic function ofz,withpoles atthepoints z=0,-1,—2,... 11.Derive theformulas Y(Z+1,11)=zY(z,<1)—ewe‘, F(z +1,¢x) =zI‘(z, on)+e‘°‘o¢=. 12.Derive thefollowing representation ofY(z,ac): w(_1)k:ak+z = —_- o-1,-2,...Y(”°‘) ,2,k!(k+2) 2*‘' 2 THE PROBABILITY INTEGRAL AND RELATED FUNCTIONS 2.|. The Probability Integral andltsBasic Properties Bytheprobability integral ismeant thefunction deﬁned foranycomplex zbytheintegral (z)=Y3?loge“2dt, (2.1.1) evaluated along anarbitrary path joining theorigin tothepoint t=z.The form ofthispath does notmatter, since theintegrand isanentire function of thecomplex variable t,andinfactwecanassume thattheintegration isalong thelinesegment joining thepoints t=0andt=z.According toafamiliar theorem ofcomplex variable theory) <D(z) isanentire function andhence canbeexpanded inaconvergent power series foranyvalue ofz.Toﬁndthis expansion, weneed only replace e"2byitspower series in(2.l.1), andthen integrate term byterm (thisisalways permissible forpower series 2),obtaining __1k2k _1lc2k+1 (D(z)= §o( kl!’dz= g§d(2])(z+ 1), |2|<00.(2.1.2)§~P5‘ o 1Iff(t)isanalytic inasimply connected domain D,thentheintegral ‘P(Z)=Z/todt, evaluated along anyrectiﬁable path contained inD,deﬁnes ananalytic function inD. SeeA.I.Markushevich, op.cit.,Theorem 13.5, p.282. Thetheorem remains trueif f(a) =ooora=oo,provided thattheimproper integral exists. 2Ibid., Theorems 16.3and15.4, pp.348and325. I6 SEC. 2.1 TI-IE PROBABILITY INTEGRAL AND RELATED FUNCTIONS Itfollows from (2.l.2) that(z)isanoddfunction ofz.Forrealvalues ofits argument, €D(z)isarealmonotonically increasing function, whose graph is shown inFigure 1.Atzero wehave <D(0) =0,andaszincreases, <D(z) rapidly approaches thelimiting value (oo) =1,since fme"2dz=ﬁ- (2.13)0 2 Thedifference between <D(z) andthislimit canbewritten intheform 2 °° 2 2 z_2 2 °°_€_l 6't €tadl. ﬂblxl 10 0.5 l : I I I 1,; O 05 1.0 1.5 2.0 2.5 3.0 FIGURE] The probability integral isencountered inmany branches ofapplied mathematics, e.g.,probability theory, thetheory oferrors, thetheory ofheat conduction, andvarious branches ofmathematical physics (seeSecs. 2.5-2.7). Intheliterature, oneoften ﬁnds twofunctions related totheprobability integral, i.e.,theerror function Z vErfz =J e“’dt=%(z), (2.1.5)0 anditscomplement Erfcz=fwe-t“dt=%[1-(z)]. (2.1.6) Many more complicated integrals canbeexpressed interms oftheprobability integral. Forexample, bydiﬁerentiation oftheparameter zitcanbeshown that e_ 7%low%dr =e"[l-<1>(\/5)]. (2.1.?) l8 THE PROBABILITY INTEGRAL AND RELATED FUNCTIONS CHAP. 2 2.2. Asymptotic Representation oftheProbability lntegral for Large |2| Toﬁndanasymptotic representation ofthefunction (D(z) forlarge |2|, weapply repeated integration byparts totheintegral in(2.l.4), obtaining 0042 1 co1 _t2 e—z2 1 doe—t2 ze d[=—‘5 ZTdl e"2 e‘2z l3°°e“2 =Y'ﬁ+Z—2l. Td’ _ l l 1-3 l-3~5 nl-3---(2n—l) =8z’lz“zTZ='1+2?"E1-+""+<-1) 0 nun 1 +<—1)""%‘:l'Z——+‘llz %md1- Itfollows that _.22 11, 1-<I)(z)=$711 +kZ1(-1)'¢ 1% +r,,(z)]. (2.2.1) where 0° _2 no)=(-1)"+1 Ifgdt (2.21) Now let |argz|<1;—8, where 8isanarbitrarily small positive number, andchoose thepath ofin- tegration in(2.2.2) tobetheinﬁnite linesegment beginning atthepoint t=z andparallel totherealaxis. Ifz=x+iy=rem’, then thissegment has theequation t=u+iy(x<u<oo),andonthesegment wehave |e22—t2| =ex’-1:2, |t|—(2n+3) s|zl—(2n-+3), It] <usec Therefore 1-3~~~2 l °° l-3---2 +1|r,,(z)| s sec¢L e"2‘“2u du= sec cp, which implies 1-3---(2n+1) l-3---(2n+l)_ll'n(Z)l < T— SCC Q< 2|Zl2)n+1 Sin 8 SEC. 2.3 Tl-IE PROBABILITY INTEGRAL AND RELATED FUNCTIONS Itfollows from (2.2.3) thatas|z|—>ootheproduct z2"r,,(z) converges uni- formly tozerointheindicated sector, i.e., °° l-3---(2n—— 1) (Z)./.2 .2.‘)(212)(2.2.4) |z|—>oo, |argz| <g—8. Thus theseries ontheright istheasymptotic series (seeSec.1.4)ofthefunc- tion 1—(z), and abound ontheerror committed inapproximating l—(z) bythesum ofaﬁnite number ofterms oftheseries isgiven by (2.2.3). Forpositive realzthiserror does notexceed theﬁrstneglected term inabsolute value. Anasymptotic representation oftheprobability integral inthesector 3g+8<argz<§—8 canbeobtained from (2.2.l) byusing therelation (z) =—<D(—-z), butthe construction ofanasymptotic representation inthesector g—8<argz<;+8 requires aseparate argument [cf.(2.3.5)]. 2.3.The Probability Integral ofImaginary Argument. TheFunction F(z) Intheapplications, oneoften encounters thecasewhere theargument of theprobability integral isacomplex number. Wenow examine theparti- cularly simple casewhere z=ixisapure imaginary. Choosing asegment of theimaginary axisasthepath ofintegration, andmaking thesubstitution t=iu,weﬁndfrom (2.l.1) that <1>" ~E = e“2du. (2.3.l) 1 \/1': 0 Theintegral intheright increases without limit asx——>oo,andtherefore itis more convenient toconsider thefunction F(z)=e—=’fie“du, (2.3.2)0 which remains bounded forallrealz.Inthegeneral caseofcomplex z,F(z) isanentire function, andthechoice ofthepath ofintegration in(2.3.2) is completely arbitrary. 20 THE PROBABILITY INTEGRAL AND RELATED FUNCTIONS CHAP. 2 Toexpand F(z)inpower series, wenote thatF(z)satisﬁes thelinear dif- ferential equation F’(z) +2zF(z) =l, (23.3) with initial condition F(0) =0.Substituting theseries F(z) =Za,,z" lc=0 into(2.3.3), andcomparing coefficients ofidentical powers ofz,weobtain the recurrence relation ao=0, a1=1» (k+1)a1¢+1 'l'2%-1 =O- After some simple calculations, thisleads totheexpansion °<> (_1 k2kz2k+1_ )F(z) —kg‘) IZI <Q). Fix) 0.6 0.4 0.2 I I I I I |_, ; O 0.2 0.4 0.6 0.8 1.0 2.0 FIGURE 2 Tostudy thebehavior ofF(z)asz—>oo forrealz,weapply L’Hospital’s ruletwice totheratio Z 22Ie"’du ( e22 ’ andthen use(2.3.2) todeduce that lim2zF(z) =l, i.e., F(z)z Z_>Q). (23.5) InFigure 2weshow thegraph ofthefunction F(z)forrealz2O.Themaxi- mum ofthefunction occurs atz=0.924... andequals F,,,,,, =0.541_. ... Thefunction F(z)comes upinthetheory ofpropagation ofelectromagnetic SEC. 2.4 THE PROBABILITY INTEGRAL AND RELATED FUNCTIONS waves along theearth’s surface, andinother problems ofmathematical physics. 2.4.TheProbability Integral ofArgument \/ix. TheFresnel Integrals Another interesting case from thestandpoint oftheapplications occurs when theargument oftheprobability integral isthecomplex number z=\/'ix=%§(1+i), where xisreal. Inthiscase, wechoose thepath ofintegration in(2.l.l) tobe asegment ofthebisector oftheangle between therealandimaginary axes. Then, using theformula t=\/iutointroduce thenewrealvariable u,weﬁnd from (l.1.l) that LQ2-X) =%J:ce"‘“” du=72:J‘oxcosu2du —iéjtsinuzdu. (2.4.l) Theintegrals ontheright canbeexpressed interms ofthefunctions 2 1rt2 Z.rrtzC(z)=cos-dt, S(z)=J.S111_dt, (2.4.2)O 2 0 2 where theintegration isalong anypath joining theorigin tothepoint t=z. Thefunctions C(z) andS(z)areknown astheFresnel integrals. Since the integrands in(2.4.2) areentire functions ofthecomplex variable t,thechoice ofthepath ofintegration does notmatter, andboth C(z)andS(z)areentire functions ofz. Forrealz=x,theFresnel integrals arereal, with thegraphs shown in Figure 3.Both C(x)andS(x)vanish forx=0,andhave anoscillatory char- acter, asfollows from theformulas 2 2 C’(x)=cos S’(x)=sin which show thatC(x)hasextrema atx=iVTFL while S(x)hasextrema atx=i\/5(n =0,1,2,...).Thelargest maxima areC(l) =0.779893. .. andS(\/2) =0.713972 ...,respectively. Asx~> oo,each ofthefunctions approaches thelimit C(00) =5(°°) =%, asimplied bythefamiliar formula“ facos12at=losin:2at= (24.3) 3D.V.Widder, op.cit.,p.382. 22 TI-IE PROBABILITY INTEGRAL AND RELATED FUNCTIONS CHAP. 2 .Cl C(x) 0.5- S(x) l l l >1 O 1.0 20 3.0 FIGURE 3 Replacing thetrigonometric functions intheintegrands in(2.4.2) bytheir power series expansions, andintegrating term byterm, weobtain thefollowing series expansions fortheFresnel integrals, which converge forarbitrary z: M M (2.4.4) The relation between theFresnel integrals andtheprobability integral is given bytheformula Z 2 Mm 29=i=III/4 C(z) i_I-S(z) =f eimﬁ/2 dt=A/gent/4Jl e-1121114 o o _ (2.4.5)=L_e¢1t1/4(D(A/lizeent/4), V2 2 which implies C(z) =T1/5 [e"”‘*(I>(,/5 ze""”'*) +e"‘”4(A/g ze"”4)], _ _ (24.6) S2 =L'_ em/4(1) fze-m/4 _e-m/4(1) Ezeni/4 _ (2n/2 2 2 Using (2.4.6), wecanderive theproperties ofC(z) andS(z)from thecor- responding properties oftheprobability integral. Inparticular, theresults of SEC. 2.5 THE PROBABILITY INTEGRAL AND RELATED FUNCTIONS Sec. 2.2lead tothefollowing asymptotic representations oftheFresnel integrals, valid forlarge |z|inthesector |argz|<in-8, 2 2 C(z)=é-é[B(z)cos%-A(z)sin 1 1 TCZ2 1:22 (2-4'7)S(z)=5——[A(z) cos? +B(z)sin7],TCZ where /1(2) =§0 +0(lZ|-4N—4)’ R‘ N Z Ba)= +0(lzl'*”‘°). ...,,=1-3---(2/< -1), at,=1. TheFresnel integrals come upinvarious branches ofphysics andengineer- ing,e.g., diffraction theory, theory ofvibrations (seeSec. 2.7), etc.Many integrals ofamore complicated type canbeexpressed interms ofthefunc- tions C(z)andS(z). 2.5. Application toProbability Theory Byanormal (orGaussian) random variable with mean mandstandard deviation <1ismeant arandom variable Zsuch that theprobability of Elying intheinterval [x,x+dx]isgiven bytheexpression ‘-5 1 -<->2/22d —: "'"°. 2.5.1 \/2110' e x ( ) Then theprobability P{a<E—m<b} (2.5.2) thatE—mliesintheinterval [a,b]isjusttheintegral 11+». b/‘/201I_,_,.,,,. 1I _.ii xm Gd =i: tdt \/21:0 11+». e X \/rt ,1)./5., e ﬁ =%l@(%)=@(%...)l’‘Asusual, [a,b]denotes theclosed interval a<xsb,and (a,b)theopen interval a<x<b. 5SeeW.Feller, AnIntroduction toProbability Theory andItsApplications, Vol. I, second edition, John Wiley andSons, Inc., New York (1957). Ifx1,.. .,x,.arethe results ofmeasurements ofE,where nislarge, then 1" 2 1" 2mz— xk, oz— (x—m).n”1¢=1 1¢=1(2.5.3) 24 Tl-IEPROBABILITY INTEGRAL ANDRELATED FUNCTIONS CHAP. 2 where (x) istheprobability integral. Asonewould expect, (2.5.2) equals 1 ifa=—oo, b=oo. Setting a=—8,b=8,weobtain theprobability that IE—m|does not exceed 8: 3P-<s=<1>(__-)- 2.5.4 {l5ml } X/26 () Then theprobability that [E—mlexceeds 8isjust 8P- s=1-(_T)- 2.5.5 {l5 ml>} X/26 () Thevalue 8=8,,forwhich (2.5.4) and(2.5.5) areequal iscalled theprobable error, andclearly satisﬁes theequation 8 1(_:) =--\/20" 2 Using atable ofthefunction <D(x) tosolve thisequation,6 weﬁndthat 8,,=0.674496. Example. With standard deviation lmm, amachine produces parts of average length 10cm.Find theprobability thatapart isoflength 10cmto within atolerance oflmm. Therequired probability is P{|£-101<0.1}=<1>(\/L5) z0.683, i.e.,some 68percent oftheparts satisfy thespeciﬁed tolerance. Inthiscase, theprobable error isapproximately 0.7mm. 2.6. Application totheTheory ofHeat Conduction. Cooling of theSurface ofaHeated Object Consider thefollowing problem inthetheory ofheatconduction: Anobject occupying thehalf-space x20isinitially heated totemperature T0.Itthen cools oﬂ"byradiating heat through itssurface x=0intothesurrounding medium which isatzero temperature. Wewant toﬁnd thetemperature T(x,t)oftheobject asafunction ofposition xandtime t. Lettheobject have thermal conductivity k,heatcapacity c,density pand 6SeeE.Jahnke andF.Emde, Tables ofHigher Functions, sixth edition, revised byF. Losch, McGraw-Hill Book Co.,New York (1960), p.31. SEC. 2.6 THE PROBABILITY INTEGRAL AND RELATED FUNCTIONS emissivity A,andlet-r=kt/cp. Then ourproblem reduces tothesolution of theequation ofheatconduction arair8_T=-8-F (2.6.1) subject totheinitial condition T|T=0 = T0 andtheboundary conditions” er(5-/1T)M_0,T|,.-... ~T0, (2.63) where h=A/k>0. Tosolve theproblem, weintroduce theLaplace transform T=T(x,p)of T=T(x,1),deﬁned bytheformula T=fooe""Td-r, Rep>0. (2.64)0 Asystem ofequations determining Tcanbeobtained from (2.6.l—3) ifwe multiply theﬁrstandthird equations bye""andintegrate from 0tooo,taking thesecond equation intoaccount. Theresult is d2T - F = _T09 - (2.6.5)dT - - ToZ: T‘ hTiI=O _ 09 Tix»OO _ 7' Thesystem (2.6.5) hasthesolution _T h _T=7°(1-ﬁe“/P"): Rep>0,Re\/Z>0.(2.6.e) Wecannowsolve forTbyinverting (2.6.4). This canbedone either byusing atable ofLaplace transforms,“ orbyapplying theFourier-Mellin inversion the0rem,9 which states that 1 a+ioo _ =_ PrT2m,L“)eTdp, (2.6.?) where aisaconstant greater than therealpartofallthesingular points ofT. "Forthederivation ofequations (2.6.1, 3),seeG.P.Tolstov, Fourier Series (trans- lated byR.A.Silverman), Prentice-Hall, Inc., Englewood Cliffs, N.J. (1962), Chap. 9, Secs. 20and24. 8SeeA.Erdélyi, W.Magnus, F.Oberhettinger andF.G.Tricomi, Tables ofIntegral Transforms, Volume I(oftwo volumes), Chaps. 4—5, McGraw-Hill Book Co., New York (1954). This two-volume set(based, inpart, onnotes leftbyHarry Bateman) willhenceforth bereferred toastheBateman Manuscript Project, Tables ofIntegral Transforms. 9H.S.Carslaw and J.C.Jaeger, Operational Methods inApplied Mathematics, second edition, Oxford University Press, London (1953), Chap. 4,Secs. 28-31. 26 THE PROBABILITY INTEGRAL AND RELATED FUNCTIONS CHAP. 2 Thequantity ofgreatest interest isthesurface temperature oftheobject. Setting x=0in(2.6.6), weﬁndthat —_i_= 1_hL.T'*=°‘~/,:(\/,?+/1) T°lp—h2 1»-hwgl <2“) Thesimplest waytosolve (2.6.8) fortheoriginal function T|x=(,istousethe convolution theorem,” which states thatifflandf2aretheLaplace trans- forms offlandf2,then =flfz istheLaplace transform ofthefunction f(e)=I1f1(l)f2('r _t)dt. (2.6.9)0 Since itiseasily veriﬁed that h 1ﬂ—-V-5’ /<2-—W aretheLaplace transforms of _ h __ I121 fl f2 _e a (2.6.9) implies .hO".d . 2“Y.T'6=<>=Tole"‘“vii.8"vi)=Toe"‘ll-ﬁl. dsl’i.e., :r|,,=.,=T0e"2‘[l -(h\/1)], (2.6.10) where (D(x) istheprobability integral. Itfollows from theasymptotic formula (2.2.l) thatforlarge -rthesurface temperature fallsofflikel/\/1-1 TT|x=0zﬁ, 1—>0O. (2.6.ll) Thetemperature inside theobject (xaé0)canalso beexpressed inclosed form interms oftheprobability integral. 2.7. Application totheTheory ofVibrations. Transverse Vibra- tions ofanlnﬁnite Rod under theAction ofaSuddenly Applied Concentrated Force Consider aninﬁnite rodoflinear density pandYoung’s modulus E,lying along thepositive x-axis. LetIbethemoment ofinertia ofacross section of therodabout ahorizontal axisthrough thecenter ofmass ofthesection, and let1=\/E1/pl. Suppose theendx=0satisﬁes asliding condition, while 1°H.S.Carslaw andJ.C.Jaeger, op.cit.,Chap. 4,Sec.33. SEC. 2.7 TI-IE PROBABILITY INTEGRAL AND RELATED FUNCTIONS theendx=ooisclamped, andsuppose aconstant force Qissuddenly applied attheendx=O.Then thedisplacement u=u(x,t)atanarbitrary point x20oftherodisdescribed bythesystem ofequations 1‘ 82u 6‘*u w+w=Q 6u].=.,=521:0 =0, (27.1) 8u Q 6uQ-x=0 — 0, 3:0 — FI! uixwm i O, '5;-xaw — O- Tosolve thissystem, weusetheLaplace transform, asinthepreceding section. Writing a=liee'l"ud1, Rep>0, (27.2)0 weobtain thefollowing equations for12: d‘*12 _w +1721! =0’ dd dad _Qam _0,who _E11], (2.7.3) - d‘u|,s..,=0,-dgxw =0. Simple calculations then show that _ Q e—‘/tﬁx e-—*/ﬁx “=iWﬁ(T7i"TE)’ Rep>0, Re\/ipi>0. (2.7.4) Toﬁndu,weagain usetheconvolution theorem. Since" _~/:51,‘ _~/Ex -_Q -_I(e _e ) aretheLaplace transforms of 1 . 2 2 fl=E2I1, f2=ﬁ(s1n%+cos-%_)> (2.6.9) implies u= fo1 (sin +cos Tgitdt =gitf(2;¥)’ (2.7.5) 11SeeR.E.D.Bishop andD.C.Johnson, TheMechanics ofVibration, Cambridge University Press, London (1960), p.285. 1”Bateman Manuscript Project, Tables ofIntegral Transforms, Vol. I,formula (27), p.146orformula (6),p.246. 28 THEPROBABILITY INTEGRAL ANDRELATED FUNCTIONS CHAP. 2 where f(x)=X/LE m(siny”+cosye)Ea/J3) dy. (2.7.6) Thefunction f(x) canbeexpressed interms oftheFresnel integrals C(z)and S(z), introduced inSec.2.4.Infact, integrating (2.7.6) byparts twice, weﬁnd that /<e>=1~e~>1;~@< »>1~<=%X2)l%—S(/ixll..%X 2 sinx2 cosx2+W-i[(1+J62)? +(1-X2) (2.7.7) PROBLEMS 1.Show thatthefunctions <1>(Z)=%@”<1’(Z) satisﬁes thedifferential equation q>’—Zzqz=1,andusethisfacttoderive the expansion 2z _2w (2z2)"(D =*_ Z -is . (Z)t/we ,.-Z01-3---(2/< +1) lzl<°° 2.Using formula (2.4.5) and theresult ofProblem 1,derive thefollowing expansions oftheFresnel integrals C(x) =x|:<x(x) cos7%+l3(x)sin7%?’ S(x) =xl:oc(x) sin7%? —l3(x)cos%c2:l, where _ ‘>9 (___ x2)2lc — °° __1)Ic(.n.x2)2k+1 I °‘(")'Z01-e...(Ik +1)’ B0‘)“Z -(4k+3) Ft‘op-ir\Q) 3.Useintegration byparts toshow that f(x) dx=x(I>(x) +L_e"‘2 +C. \/7r 4.Let<1)betheLaplace transform oftheprobability integral, i.e., 6(p) =fooe"”‘ <I1(x) dx.0 e~@l%ll~Prove that PROBLEMS THEPROBABILITY INTEGRAL ANDRELATED FUNCTIONS 29 5.Derive theintegral representations F(z)=looe"2sin2ztdt, (I>(z)=Zre-“Mat.O 7rO t Hint. Replace sin2ztbyitspower series expansion andintegrate term by term. 6.Derive thefollowing integral representations forthesquare oftheprob- ability integral: 41e-z2<1+t2) 2(Z)=1— ale dt! 4 ane—z2(1+t2) n. [1—‘1>(Z)l2 =all pd’, largZl<Z‘ Hint. Represent 2(z) asadouble integral over theregion 0ss<2, 0<t<z,andtransform topolar coordinates. 7.Derive theformulas 2 00 1_ (D =__ ~z2J\ —t2—2ztd’(z) \/we 0e t [1-<1>(p)]2 =—i;e‘2Z2 looe-*2-2“'~‘=*<1>(¢)d1.\/1: 0 Hint. Thesecond formula isobtained from theﬁrstafter introducing new variables ct=s+t,B=stinthedouble integral over theregion 0<s<oo, 0<t<s. 8.Prove that 2 21sin%(l ¥t2) C2(z) iS2(z) =ZLg dt. 9.Prove that me?/2 nxz/2 C(x) =Jo -I-1/2(1) dt, =J0 J1/2(t) dt, where J.,(x) istheBessel function oforder v(seeSec. 5.8). 3 THE EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS 3.l.TheExponential Integral andItsBasic Properties Theexponential integral isdeﬁned by Ei(z)= Q11, |arg(—z)| <TC, (3.1.1) where theintegration isalong anypath Linthet-plane with acutalong the positive realaxis(seeFigure 4).Since theintegrand isananalytic function in Z / 0 FIGURE 4 theresulting simply connected domain, theintegral ispath-independent and Ei(z) isananalytic function ofz(cf.footnote 1,p.16).Apossible choice of thepath ofintegration istheinﬁnite linesegment —oo <Ret< Rez, Imt=Imz, (3.l.2) passing through thepoint zandparallel totherealaxis. 30 sac.3.1 THEEXPONENTIAL INTEGRAL ANDRELATED FUNCTIONS 3| IfWereplace zby—zandtby—t,formula (3.l.l) becomes ane—t _Ei(—z)f Tdt, |argzl<1:, (3_1_3) where thefunction —Ei(—z) isanalytic intheplane with acutalong the negative realaxis. The graph ofthisfunction forz=x>0isshown in Figure 5.Itwill benoted that —Ei(—x) ,. -F/(-xldecreases monotonically from the value ‘ -—Ei(0) =+00 tothevalue —Ei(—oo) =0, ‘ andinfact, itsderivative is - l.O- %L£Mqﬂ=—f;<0 ifx>Q Toderive aseries expansion oftheex- ponential integral, werepresent (3.1.l) in theform -12 0 _ m@=l 5a+f §—lm—=>’ -1’ 0.5-zt_ 2 +fi_lp+f Q0r _,z andobserve that thesum oftheﬁrst two integrals isanabsolute constant, which we denote byC.Setting t=—u“1 intheﬁrst integral andt=—uinthesecond, weﬁnd that 012345' 11-e'“-e-1/"C=IG du. (3.1.4) FIGURE 50 ll Comparison of(3.1.4) and (1.3.20) shows that Ccoincides with Euler’s constant: C=Y=0.5772157... Thus wehavel Ei(z) =Y+log(—z) +Jzittl dt, |arg(—z)| <7:.(3.l.5)O Theintegral ontheright, whose integrand isanentire function, isitself an entire function ofthecomplex variable z,andcantherefore beexpanded ina power series which converges inthewhole plane. Toobtain thisseries, we 1Inthisbook logzalways means thesingle~valued branch ofthelogarithm deﬁned by logz =log|z|+iargz, |argz| <71:. Similarly, z"(varbitrary) means e"1°‘=,andsoon. 32 THE EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS CHAP. 3 need only expand theintegrand inpowers oftandintegrate term byterm. Theresult is 'zet_1 2wtk—1 0°Zk J0Tdt=Lk§1T!dt=k;1]w€s |Zl<®, andtherefore thedesired expansion oftheexponential integral is . °°z"121(2)=Y+lOg(—Z) +ZW. |arg(—z)| <TC, (31.6)k=l ‘ valid everywhere intheplane cutalong thepositive realaxis. Itfollows from (3.l.6) thatthevalues ofEi(z) ontheupper andlower edges ofthecutare respectively Ei(x ii0)=Ei,(x) Tni, x>0, where Ei1(x) istherealfunction deﬁned by It Ei,(x)=§[Ei(x+10)+Ei(x-i0)]=Y+logx+Z x>0,Ic=1 - (31.7) andknown asthemodiﬁed exponential integral.’ Theexponential integral isoften encountered intheapplications, e.g.,in antenna theory andother branches ofphysics andengineering. Many inte- grals ofamore complicated typecanbeexpressed interms oftheexponential integral. Forexample, theintegral fezf(2)dz, where f(z)isanarbitrary rational function, canbewritten inﬁnite form in terms ofthefunction Ei(z) andelementary functions (seeProblem 9,p.42). 3.2.Asymptotic Representation oftheExponential Integral for Large |z] Toﬁndanasymptotic representation ofthefunction Ei(x) forlarge |z|, weapply repeated integration byparts toformula (3.l.1), obtaining z et z 1 ez z et f_w7dr=l_w;d(e*)=;+f_mFdi ez ez 2 et ——;'l';2+l~2J‘_wt—3dl ,1 11-2 1-2-~-n Ze=—€ [2"l-Z2‘-l“?§-"l"--+-7,7,-,7-:|+l-2---(n+l)J‘_mgdt. 2Since (3.1.1) does notdeﬁne Ei(z) forz=x>0,onecanformally extend thede- ﬁnition oftheexponential integral bydeﬁning Ei(x) EEi1(x) forx>0. SEC. 3.3 THE EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS Itfollows that Ei(z)=[if +r(z)]> (3.2.1)= z n where r,,(z) =(n+1)lze"'-'fz %dt, |arg(—z)| <7:. (3.2.2) Toestimate theremainder r,,(z), wechoose thelinesegment (3.l.2) asthepath ofintegration. Suppose |arg(—z)| <7-:—8,where 8isanarbitrarily small positive number, andletz=x+iy.Then along thesegment t=0'+iy (—oo <e<x)wehave [e“*| =e""‘, lt|2lz|sin8, andhence +1! " _ +1!__ __ll'n(Z)l < _]‘—w ea xdd =(g€i IZI 111= it1). (3.2.3) Therefore wehave theasymptotic representation Ei(z)=l +O(|z|'"“)]> |arg(—z)| <pt-s.(3.2.4) Itfollows from (3.2.4) thatthedivergent series ea°°k! z,2,z" istheasymptotic series forEi(z) inthesector |arg(—z)| <7:—8. Itshould benoted thatifRez<0,i.e.,inthesector larg(—z)| <71:/2,we have thesharper estimate |r.<z>|s (12.5) Inthiscase, theerror committed inapproximating Ei(z) bythesumofaﬁnite number ofterms oftheasymptotic series does notexceed theﬁrstneglected term inabsolute value. 3.3.The Exponential Integral ofImaginary Argument. TheSineandCosine Integrals Ifz=ixisapure imaginary, thefunction Ei(z) canbeexpressed interms oftworealfunctions Si(x) andCi(x), known asthesineintegral andthecosine integral, respectively. These functions, which areinteresting intheir own right, aredeﬁned forarbitrary complex zbytheintegrals Si(z)= dt, Ci(z)=fz@dz, Iargzl <1..(33.1) 34 THE EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS CI-IAP 3 Thechoice ofthepath ofintegration intheﬁrstintegral isentirely arbitrary, butinthesecond integral itisrequired thatthepathofintegration LheIn theplane cutalong thenegative realaxis, asshown schematically InFigure 6 Fortheusual reason (cf.footnote 1,p.16),Si(z) isanentire function, whileXM 0 FIGURE 6 Ci(z) isanalytic intheplane cutalong thenegative realaxis. Forrealz=x>0,both functions arereal, with thegraphs shown In A 2_ ]_ 5 101 4.5‘/'(x O _|.l1 _2r Figure 7.Moreover, Si(x) andCi(x) have anoscillatory character, asfollows from theformulas which show that Si(x) hasextrema atthepoints x=nrr:(n=012 While Ci(x) hasextrema atthepoints x=(n+§)1-c. Forx<0FIGURE 7 d. sinx d. cosxZ; Sl(X) —T1 Z1; Cl(.X) =-T 5i(X)=—5i(lxl),>1’ C/(x) SEC. 3.3 TI-IE EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS whereas Ci(x) isnotdeﬁned. Forlarge andsmall values oftheargument, we havethelimiting values Si(oo)= Ci(oo) =0,(3.3.2)s1(o)=0,c1(+o) =-00. Toestablish therelation between thefunctions Ei(ix), Si(x) andCi(x), we substitute z=ix(x>0)into(3.1.1).First wenote thattheintegration along theoriginal path Lcanbereplaced byintegration along theimaginary axis. Infact, consider theintegral ofthefunction e‘/talong theclosed contour consisting ofanarcCRofthecircle ofradius Rwith center attheorigin, the //P CR /x 0 FIGURE 8 arcLRofthecurveL lying inside thiscircle, andthesegment oftheimaginary axisjoining thepoints ixand iR(seeFigure 8).According toCauchy’s integral theorem, E ill ll l‘id1+l§d¢+l5dz=o.L3 t it t C13 t ButasR—> oo,theintegral along LRapproaches Ei(ix), while theintegral along CRvanishes.“ Therefore .._ ‘°°e‘ _"e"‘ _“cosu ."sinuE1(zx)- -L7211- Ludu-L-u-du+ 1L-Tdu, 3OnthearcCpwehave t=Re“, 11:/2é6<1c,andhence n M2 1:12 -R1 __e—R Sit c058 .... — in I -2 In ___Lntdtéiﬂze" d6-Jlo e'“’"d)(< 0eR‘dx-2-——-—R where weusetheinequality sinX2(ZX/71:), valid for0<)(<7:/2[seeA.I.Markushe- vich, op.cit.,formula (13.20), p.272]. Itfollows that 1 -e—dt—>0 Cut 8SR—>OO. 36 THE EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS CHAP. 3 i.e., Ei(ix)=Ci(x)--Si(x)]> x>0, (3.3.3) andsimilarly, Ei(—ix) =Ci(x)+-Si(x)], x>0. (3.3.4) Wehave proved formulas (3.3.3-4) forx>0,butitiseasily seenbyusing theprinciple ofanalytic continuation thatthey hold inalarger region, and infact, Ei(—ze"‘”2) =Cl(Z)--s1(z)]. -g<argZ<1., (33.5) Ei(—ze"”2) =Ci(z)+-s1(z)]. —TC<argZ< Toprove (3.3.5), wemerely note thatboth sides areanalytic functions ofz intheindicated sectors, andthat these functions coincide forz=x>0. From (3.3.5) wededuce theuseful formulas Ci(z) =1[Ei(—ze"”2) +Ei(—ze"‘”2)], |arg2|<g,2(3.3.6) Si(z) =T-it—%l_[Ei(—ze"”2) —Ei(—ze"‘”2)], |arg2|<go which express Ci(z) andSi(z) interms oftheexponential integral. Thefunctions Si(z) andCi(z) have simple series expansions. Theexpan- sionofSi(z) isfound bysubstituting thepower series forsintinto(3.3.l) and then integrating term byterm. Theresult is 2°° _1ict2k °° _1lcz2k+l~ () ()S =I ——i d= Li, .3.3.7‘lzl 0,,Z,(2/e +1)!',Zo(2/e +1)1(2/<+1) lzl<°°() Thederivation oftheexpansion ofCi(z) issomewhat more complicated. The simplest approach istousetherelation between thefunctions Ci(z) and Ei(—ze=*="”2), together with theexpansion (3.1.6). Inthisway, weﬁndthat‘ °°_k21¢ Ci(z) =Y+logz +kZ1 , |argzl<71:. (3.3.8) Inparticular, (3.3.8) leads tothefollowing values ofthefunction Ci(z) onthe upper andlower edges ofthecut[—00,0]:5 Ci(-—x ii0)=Ci(x) 1-Tci, x>O. (33.9) ‘The original restriction [argz]<7:/2iseasily eliminated byusing theprinciple of analytic continuation. 5Forsimplicity ofnotation, wewillalways regard inﬁnite branch cutsaspassing through thepoint atinﬁnity, asinthefamiliar representation oftheextended complex plane bytheRiemann sphere (seeA.I.Markushevich, op.cit.,Chap. 5). SEC. 3.4 THE EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS Finally, byusing (3.2.4) and(3.3.6), wecanderive asymptotic representa- tions ofthefunctions Ci(z) and -fen—Si(z) forlarge |z|inthesector [argz|<-rc/2.Itiseasily veriﬁed that 01(1)=Po)~%Q(z), 7-c cosz sinz (3'3'10)5—Si(z) =T P(z) +T Q(z), wheren _ k ' 11(2) =kZ0 _|_0([Z|-211-2)’ Q(z)= +0<|z|-2"-a 3.4.TheLogarithmic Integral Another special function which isclosely related totheexponential integral isthelogarithmic integral. This function, which plays animportant rolein analysis, isdeﬁned by li(z)=J0zl;1%, |arg2|<7:,|arg(1—z)l<7:, (3.4.l) where theintegral isalong anypath Lbelonging totheplane with twocuts along thesegments [—oo,0]and[1,oo]oftherealaxis(seeFigure 9).Bythe 2 A M . I } O l FIGURE 9 usual argument (cf.footnote 1,p.16),li(z)isananalytic function inthecut plane. Byintroducing thenew variable ofintegration u=logt,wecan easily express li(z)interms oftheexponential integral. Infact, theoriginal cutt-plane ismapped onto thestrip |Imu|<7-:intheu-plane, with acut along thepositive realaxis, and(3.4.l) istransformed intotheintegral 1082 11(2)=‘lm%du, (3.42) 38 THE EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS CHAP. 3 evaluated along anypath belonging tothisstrip. Since thestrip isapartof thedomain ofdeﬁnition oftheexponential integral (seeSec.3.1), itfollows from (3.4.2) that li(z)=Ei(logz), (3.4.3) where, asalways, logz denotes theprincipal value ofthelogarithm (cf. footnote l,p.31). Using (3.4.3), wecaneasily deduce theproperties ofthelogarithmic integral from those oftheexponential integral. Forexample, formula (3.1.6) implies theexpansion °°(logZ)”11(2)=‘Y+log(—log Z)+2W. (3.4.4)k=l - where zbelongs totheplane with cuts along thesegments [—oo, 0]and [1,oo].Inparticular, itfollows from (3.4.4) that thevalues ofli(z)onthe upper andlower edges ofthecut[1,oo]are li(xii0)=li1(x) 1ni, x>1, (3.4.5) where li1(x) denotes therealfunction . .,._ °°1rl11(x)=-)[11(x+10)+11(x-10)]=Y+loglogx+Z(-‘%,f—). x>1,k=1 ‘ (3.4.6) known asthemodiﬁed logarithmic integral.“ Itfollows from (3.l.7) and (3.4.6) thatthemodiﬁed exponential integral andthemodiﬁed logarithmic integral areconnected bytheformula li1(x) =Ei,(logx). (3.4.7) Thefunction li1(x) isfrequently encountered inanalysis, andisparticularly important innumber theory.” Finally, wenote thattheresults ofSec.3.2imply theasymptotic repre- sentation . 7' kl11(2)=é [2W +r,,(z)], 8<|argz|s-n:—8,(3.4.8) lC=0 where lF..(Z)l =0(ll0g Zl_"_1) forlarge values ofllogz|.Inparticular, (+1! |r.<z>|< forlz|<1,andinthiscasethesector isjust|argzl<7:—8. 6Since (3.4.l) does notdeﬁne li(z) forz=x>1,onecanformally extend the deﬁnition ofthelogarithmic integral bydeﬁning li(x) Eli1(x) forx>1. 7SeeA.E.Ingham, TheDistribution ofPrime Numbers, Cambridge Tracts inMathe- matics andMathematical Physics, No.30,Cambridge University Press, London (1932). SEC. 3.5 THE EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS 3.5.Application toElectromagnetic Theory. Radiation ofa Linear Half-Wave Oscillator” Asasimple example oftheapplication ofthespecial functions studied in thischapter, weconsider theelectromagnetic energy radiated byalinear oscillator oflength 2l=A/2,driven byanalternating current Ioffrequency to=21cc/A (cisthevelocity oflight and>\thewavelength), whose distribution along theconductor isgiveby N I=I0cosgcos tot, —l<<l (3.5.1) (seeFigure 10).LetE(t)andH(t) denote thetime-dependent electric and magnetic ﬁeldvectors, with complex amplitudes EandH,sothat E(t)=Re{Ee‘°"}, H(t) =Re{He‘°"}. (3.5.2) z M __ ___ 8 , 21 _ ' 0 FIGURE 10 Then thepower radiated bytheoscillator, averaged over aperiod T=>\/c, isgiven bytheformula9 P=Re L(E><I-1*)-IldS}» (3.53) where Sisanarbitrary surface surrounding theoscillator, nistheexterior normal toS,andH*isthevector whose components arethecomplex con- jugates ofthose ofH.1° Inthepresent case, thevectors EandHhave components (E,,E9,0)and (0,0,H)inaspherical coordinate system (r,6,<p)[seeFigure 10,where Mis °Thenecessary background information inelectromagnetic theory, written inthe system ofunits used here, canbefound inG.Joos, Theoretical Physics, third edition, withthecollaboration ofI.Freeman, Blackie andSon, Ltd., London (1958). °Ibid., pp.332,341. 1°Asusual invector algebra, thedotdenotes thescalar product andthecross denotes thevector product. 40 THE EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS CHAP. 3 theobservation point], andforSwecanchoose asphere r=pofarbitrarily large radius p.Then (3.5.3) becomes -——- 4 0 6 ’ .- P—Re{i"-2-J“EH*sinede}» (354) where H*isthecomplex conjugate ofH.In(3.5.4) wecanreplace theexact values ofE9andHbytheir asymptotic expressions forlarge r.Using the well-known formulas forthecomponents oftheelectromagnetic ﬁeld ofan elementary dipole,“ andintegrating with respect toz,weeasily ﬁndthat HzEez—e‘”“’sin(*) cos—~e'z°°S dz=—eIoik . J’ rcz.k 9 210i _,,wcos(%rccos6) cp _, Zl cp Sin0 forsuﬂiciently large p,where k=m/c. Itfollows that P=5In‘H2(i"°°s6)d6, (15.5)c0 Sll’l6 where weusetheformula 1_Z‘_E_1_4—2o)—2k Theintegral in(3.5.5) canbeexpressed interms ofthecosine integral Ci(x). Infact, introducing thenew variable ofintegration x=cos6,we have 1?,1l+cosrcx I§U‘11+cosrcx fl+cosrcx)=— = ——;i dPCJ; 1—X2 dx2c01—X d"+ 0l+x X _I§ 1l—c0s1-cy 2l—cosrcy _I§ 2l—cos1-cy ~a<i.—7—dy+i-—.—dy>~ai. .dyI32“l—cosz_XL _5~ dz. (35.6) Finally, using theresult ofProblem 3,p.41,weﬁndthat I3 .P=E[Y+log21¢-c1(2n)], (3.5.7) where Ci(x) istheintegral cosine andYisEuler’s constant. Thesame method canbeused tocalculate theaverage power radiated by antennas with more complicated conﬁgurations. Itisremarkable that the results canstillbeexpressed interms ofsineandcosine integrals. 11G.Joos, op.cit.,pp.338, 340. PROBLEMS TI-IEEXPONENTIAL INTEGRAL ANDRELATED FUNCTIONS 4| PROBLEMS 1.Verify theintegral representation co —zt —Ei(—z) =e":Inﬁt, |argzl< 2.Verify thefollowing integral representation forthesquare oftheex- ponential integral: [Ei(—z)]2 =2e'2” Looe'2Z‘ ) dt, |arg2|< Hint. Represent theleft-hand side asadouble integral over theregion 0<s<oo,0<tss,andintroduce thenew variables at=s+t,[3=st. 3.Prove that Ci(z)=y+logz—J‘z1-—_%)stdt, ]argz| <1r.O 4.Starting from (3.1.l) andthedeﬁnition ofthemodiﬁed exponential integral Ei1(x), show that ‘ _ —eet xet E11(x) =ling Tdt +I7dr)» x>0, i.e.,show thatEi1(x) istheCauchy principal value oftheintegral x etd _an7 2'. 5.Verify that . "e‘—1E11(x) =Y+logx +L fdt. 6.Using L’Hospital’s rule, show inturnthat lim e"‘Ei1(x) =0, lim xe‘” Ei1(x) =1,Z—~ +db I-v +tn andthen deduce theasymptotic formula .76 Ei1(x)z x—>+00. 7.Using (3.4.7) andtheresult ofthepreceding problem, deduce theasymp- totic formula li1(x) zT6-J;—Jc, x-—> +00. Comment. This formula plays animportant roleinnumber theory. 8.Prove theformula . . 1-"dr *dr “1"‘)-9% no+ ">1-Hint. Use(3.4.7) andtheresult ofProblem 4. 42 TI-IEEXPONENTIAL INTEGRAL AND RELATED FUNCTIONS CHAP. 3 9.Consider theintegral I/wedz. (0 where f(z)isanarbitrary rational function, andthepath ofintegration does notpass through anysingular points oftheintegrand. Byseparating outthe polynomial part off(z)andthen expanding theremainder inpartial fractions, theevaluation of(i)canbereduced totheevaluation ofintegrals oftheform Jz"ez dz, (ii) I dz, (iii) where nisapositive integer. Byrepeated integration byparts, (ii)canbe expressed interms ofelementary functions, andtheproblem ofevaluating (iii)canbereduced totheproblem ofevaluating theintegral J.?e_;; dZ. Then thesubstitution u=z—areduces (iv)toanexponential integral (generally with acomplex argument). Using themethod justdescribed, prove that " ‘ e” . .j_m;,(T"_—1)d¢ =3--2E1(x) +eE1(x-1), x<0. 10.Asusual, letfdenote theLaplace transform off(seep.25).Prove that Si(x) =1%arctanit —Ei(—x) =glog(1+p), where thearctangent andthelogarithm have their principal values. ORTHOGONAL POLYNOMIALS 4.I.Introductory Remarks Asystem ofrealfunctions f,,(x) (n=O,1,2,...)issaidtobeorthogonal withweight p(x)ontheinterval [a,b]if fbPo)/..<x>f..<x> dx=0 <4-1-1) forevery maén,where p(x)isaﬁxed nonnegative function which does not depend ontheindices mand n.For example, thesystem offunctions cosnx(n=0,1,2,...)isorthogonal with weight 1ontheinterval [0,1:], since 1! fcosmxcosnxdx=0 ifmaén.O Orthogonal systems playanimportant roleinanalysis, mainly because func- tions belonging toverygeneral classes canbeexpanded inseries oforthogonal functions, e.g.,Fourier series, Fourier-Bessel series, etc. Animportant class oforthogonal systems consists oforthogonal poly- n0mialsp,,(x) (n=0,1,2,...),where nisthedegree ofthepolynomial p,,(x). This class contains many special functions commonly encountered inthe applications, e.g.,Legendre, Hermite, Laguerre, Chebyshev andJacobi poly- nomials. Inaddition totheorthogonality property (4.l.1), these functions have many other general properties. Forexample, they aretheintegrals of differential equations ofasimple form, andcanbedeﬁned asthecoeﬁicients inexpansions inpowers oftofsuitably chosen functions w(x,t),called generating functions. Orthogonal polynomials areofgreat importance in 43 44 ORTHOGONAL POLYNOMIALS CHAP. 4 mathematical physics, approximation theory, thetheory ofmechanical quadratures, etc.,andarethesubject ofanenormous literature, inwhich the contributions ofRussian mathematicians likeAdamov, Akhiezer, Bernstein, Chebyshev, Sonine, Steklov andUspensky play aprominent role. This chapter isdevoted tothetheory ofLegendre, Hermite andLaguerre polynomials, which have extremely diverse applications tophysics anden- gineering. Fortheconvenience ofreaders primarily concerned with applica- tions, each ofthese three kinds ofpolynomials istreated independently. Those interested instudying thesubject from amore general point ofview arereferred tothebooks byJackson, Sansone, Szego andTricomi cited in theBibliography onp.300.1 InProblems 21-22, p.96-97, wealsotouch upon thetheory ofJacobi andChebyshev polynomials. 4.2.Definition andGenerating Function ofthe Legendre Polynomials TheLegendre polynomials aredeﬁned byRodrigues’ formula .P(x)=L£(x2-1)" n=Ol2... (4.2.1)" 2"n!dx" ’ ’’’ forarbitrary realorcomplex values ofthevariable x.Thus theﬁrstfew Legendre polynomials are P0(x) =1: P1(x) :X: P2(x) :'2‘(3-X2 _1): P3(x) =%(5x3 —3x),... Thegeneral expression forthenthLegendre polynomial isobtained from (4.2.l) byusing thefamiliar binomial expansion Pl (—l)“n! _ 2_1n: g 21!.an (X ),2,k!(n—k)!"’ which implies ‘MEOBP..(><)= ><"‘2". (4.2.2) where thesymbol [v]denotes thelargest integer <v.Itwillbeshown in Sec.4.5thattheLegendre polynomials areorthogonal with weight 1onthe 1Seealso A.Erdélyi, W.Magnus, F.Oberhettinger and F.G.Tricomi, Higher Transcendental Functions, Volume 2(ofthree volumes), Chap. 10,McGraw-Hill Book Co., New York (1953). This three-volume set(based, inpart, onnotes leftbyHarry Bateman) will henceforth bereferred toastheBateman manuscript Project, Higher Transcendental Functions. sEc.4.2 ORTHOGONAL POLYNOMIALS 45 interval [-1,l].2Asalready noted, these orthogonal polynomials play an important roleintheapplications, particularly, inmathematical physics (see Secs. 8.3-4, 8.7-8, 8.13-14). Theproperties oftheLegendre polynomials canbederived verysimply if weﬁrstprove thatthefunction w(x, t)=(1—2xt+t2)‘1’2 (where thevalue ofthesquare rootistaken tobe1fort =0)isthegenerating function oftheLegendre polynomials, i.e.,thattheexpansion w(x,t)=(1-2xt+t2)-1'2 =ZP,,(x)t" (4.23) holds forsuﬂiciently small |t|.Letr,andr,betheroots ofthequadratic equation 1—2xt+t2=0,andlet r=min{lr1l,|r2|}. (4.2.4) Then w(x,t),regarded asafunction oft, isanalytic inthedisk |t|<r.3It follows from afamiliar theorem ofcomplex variable theory‘* that (X) w(x,t)=(1—2xt+t2)‘1’2 =2c,,(x)t", ltl<r, n=O where thecoeiﬁcients c,,(x) canbewritten ascontour integrals C(X)= (1-2x!+t2)‘1’2t-""1dt, (4.2.s)" Zrcc evaluated along anyclosed contour Csurrounding thepoint t=0andlying inside thedisk [t|<r.Ifwemake thesubstitution l—ut=(1—2xt+t2)1'2, then (4.2.5) transforms into thefollowing integral ofarational function evaluated along aclosed contour C’surrounding thepoint u=x:5 c,,(x)=L du. (4.2.6) 2Thisproperty canbeproved directly, bystarting from thedeﬁnition (4.2.l), butour approach willbedifferent. Infact, itcanbeshown that ifp,,(x) (n=0,1,2,...)isan arbitrary system ofpolynomials orthogonal with weight 1ontheinterval [—1,1],then p,,(x) =Y,,P,,(x), where 7,,isindependent ofx.SeeG.E.Shilov, AnIntroduction tothe Theory ofLinear Spaces (translated byR.A.Silverman), Prentice-Hall, Inc., Englewood Cliffs, N.J.(1961), Sec.58. 3Inthecaseofgreatest practical importance, xisarealnumber belonging tothe interval [-1, 1],andthenr=1. 4A.I.Markushevich, op.cit.,Theorem 16.7, p.361. 5Thepoint u=xcorresponds tothepoint t=0,andtheclosed contour C’cor- responds totheclosed contour C,since thesquare root returns toitsoriginal value after making acircuit around C. 46 ORTHOGONAL POLYNOMIALS CHAP. 4 This integral canbeevaluated byresidue theory. Infact,using thefamiliar rule,6 weﬁndthat thereby verifying (4.2.3). Toillustrate theutility ofthegenerating function forderiving properties oftheLegendre polynomials, wesuccessively setx=1,—1,0in(4.2.3), each time expanding theleft-hand sideinpowers oft.Asaresult, weobtain the important formulas P,,(l) =1, P,,(—1)=(—l)", _ (4.2.?) P..<0>=(-1)" P....<0> =0- 4.3. Recurrence Relations andDifferential Equation forthe Legendre Polynomials Wefurther illustrate theuseoftheexpansion (4.2.3) byderiving some recurrence relations satisﬁed bytheLegendre polynomials. First wesub- stitute theseries (4.2.3) intotheidentity (1-2xt+t2)%;'+(z—x)w=0. Since power series canbedifferentiated term byterm, thisgives (1—2xt+I2)3:n1’(x)t"‘1 +(t—x)EP,,(x)t" =0. n=O n=0 Setting thecoeﬁicient oft"equal tozero, weﬁndthat ('1+1)P..+1(X) —2nXP..(X) +(H—1)P..-r(x) +P1.-r(x) —XP..(X) =0. or (n+l)P,,,1(x) —(2n+l)xP,,(x) +nP,,_,(x) =0,n=1,2,...,(4.3.l) which isarecurrence relation connecting three Legendre polynomials with consecutive indices. Onecanusethisrelation tocalculate theLegendre poly- nomials stepbystep, starting from P0(x) =1,P1(x) =x. Similarly, theidentity 8w(l——2xt+t2)5;—tw—0 6See F.B.Hildebrand, Advanced Calculus forApplications, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1962), p.548. sEc.4.3 ORTHOGONAL POLYNOMIALS 47 leads to” (1-2x!+1”)ZP,’,(x)t" -ZP,,(x)t"*1 =0,n=0 n=0 which implies P,’,,1(x) —2xP,§(x) +P,§_1(x) —P,,(x) =0, n=1,2,... (4.3.2) Differentiating (4.3.l), weﬁrsteliminate P,’,_1(x) andthen P,’,.,1(x) from the resulting equation and(4.3.2). This gives twofurther recurrence relations” P{,.,1(x) —xP,’,(x) =(n+l)P,,(x), n=0,1,2,..., (4.3.3) xP{,(x) —P,’,_1(x) =nP,,(x), n=1,2,... (4.3.4) Adding (4.3.3) and(4.3.4), weobtain themore symmetric formula P§,,.,(x) —P,’,_1(x) =(2n+l)P,,(x), n=1,2,... (4.3.5) Finally, replacing nbyn—1in(4.3.3), andeliminating P,’,_, (x)from the resulting equation and(4.3.4), weﬁndthat (1—x2)P,’,(x) =nP,,_1(x) —nxP,,(x), n=1,2,... (4.3.6) Thislastformula allows ustoexpress thederivative ofaLegendre polynomial interms ofLegendre polynomials. Ifwedifferentiate (4.3.6) with respect to xandagain use(4.3.4) toeliminate P,’,_1(x), wearrive attheformula [(1—x2)P,’,(x)]’ +n(n+l)P,,(x) =0, n=0,1,2,...,(4.3.7) which shows thattheLegendre polynomial u=P,,(x) isaparticular integral ofthesecond-order linear differential equation [(1—x2)u’]’ +n(n+l)u=0. (4.3.8) This equation isoften encountered inmathematical physics, andplays an 7Tojustify differentiating (4.2.3) term byterm with respect tox,itissufﬁcient to prove that (4.2.3) converges uniformly inthedomain |x|<a,forarbitrary ﬁnite a>0 and sufﬁciently small |t|.(Here werely onWeierstrass’ theorem, cited infootnote 5, p.2.)LetItl<b,where b=\/a2 +1—a.Then, according to(4.2.3), theseries 21.1»n=0 I converges to(1—2a|t| —|t|2)‘1'2. The uniform convergence of(4.2.3) forlxl<a, ]t|<bnow follows from theinequality |P..<x>r"|< 1:1". implied by(4.2.2). 8Insome cases, thevalidity ofarecurrence relation forsmall ndoes notfollow from thegeneral argument, butthen onecanalways verify therelation bydirect substitution ofP(,(x) =1,P1(x) =x,... 48 ORTHOGONAL POLYNOMIALS CHAP. 4 important roleinthetheory ofLegendre polynomials. Bymaking changes of variables in(4.3.8), wecaneasily derive many other equations whose integrals canbeexpressed interms ofLegendre polynomials. Thus, forexample, the equation 57:3a-%(sine6%)+n(n+1)..=0 (43.9) issatisﬁed bythefunction u=P,,(cos 9),theequation d2 1ﬁg 'l‘ 'l'%)2 ‘l’ =0 issatisﬁed bythefunction u=\/sin9P,,(cos9),andsoon. 4.4.Integral Representations oftheLegendre Polynomials The Legendre polynomials have simple representations interms of deﬁnite integrals withthevariable xasparameter. Toobtain theﬁrstofthese representations, weassume thatxisarealorcomplex number, andchoose thepath ofintegration C’informula (4.2.6) tobeacircle ofradius \/Ix” ——1| with center atthepoint u=x.”Then u=x+\/x2—1e“", ——'r:$ <1-c, and(4.2.6) becomes 1==x2+2x\/x2 -16"» +(X2-l)e2“°-11» P""‘)=2“ 2\/ﬁe” l"‘°’ which reduces to-6 P,,(x) = [x+\/x2 —-1cos<p]"d<p. (4.4.l) Formula (4.4.l) iscalled Laplace’s integral. Here thechoice ofthevalue of thesquare root \/x2 —1does notmatter, since after raising theexpression inbrackets tothenthpower andintegrating theresult term byterm, odd powers ofthesquare root vanish. From (4.4.l) wecanderive animportant inequality satisﬁed byLegendre polynomials. Letxbearealnumber such that —l<x<1.Then |x+ \/x2—1cos<pl= \/x2+(l —X2)COS2<p <1, andhence lP,,(x)| <1, -1<s1. (4.4.2) Another important integral representation oftheLegendre polynomialsX 9According toCauchy’s integral theorem, replacing thecontour C’byanyother closed Jordan curve surrounding thepoint u=xdoes notchange thevalue ofthe integral. sEc.4.4 ORTHOGONAL POLYNOMIALS 49 canbededuced from (4.4.l) byassuming thatxisarealnumber such that —l<x<1.Inthiscase, setting x=cos6, 0<6<1:, wecanwrite (4.4.l) intheform P,,(cos6)=%In(cos9+isin0cos<p)"d<p.0 Ifweintroduce anewcomplex variable ofintegration t=cos9+isin6coscp thisformula becomes 18'“ z"dz P"1°“°>=l... ‘M’where theintegral isevaluated along thelinesegment ABjoining thepoints t=e*‘°(seeFigure 11),andthechoice ofthesquare rootisdetermined by A r 19 /1 I06l=1 FIGURE ll thecondition that itsvalue atthepoint t=cos6besin6.According to Cauchy’s integral theorem, theintegration along ABcanbereplaced by integration along thearcACB oftheunitcircle, since theintegrand isanalytic intheregion between thearcandthechord. Making thischange, andwriting t=e“",weﬁndthat P(Cos 6)_lfe ei(n+1/2)\l' dkp " T=-.V ’which becomes 9 P,,(cos6)=%f d¢, 0<6<TE,7l=O,1,2,...,O COS _ COS . (4.4.4) after taking therealpart. This integral representation isknown asthe Mehler-Dirichlet formula. , 50 oRTI-IoGoNAL POLYNOMIALS CHAP. 4 4.5. Orthogonality oftheLegendre Polynomials Oneofthemost important properties oftheLegendre polynomials istheir orthogonality ontheinterval [—1,1],which follows from thedifferential equation (4.3.7). Toprove thisproperty, wesubtract thedifferential equation forthenthpolynomial multiplied byP,,,(x) from thediﬁerential equation for themthpolynomial multiplied byP,,(x). This gives [(1—X2)P.§.(X)l’P»(X) —[(1-X2)Pl.(X)l'P...(X) +[m(m +1)—n(n+l)]P,,,(x)P,,(x) =0, or {(1-X2)lPh(X)P..(X) -P£(X)Pm(X)l}' +("1—"Xi" +'1+1)Pm(X)P»(X) =0- Integrating thelastequation over theinterval [—1,1]andnoting thatthe integral oftheﬁrstterm vanishes, weﬁndthat (m-n)(m+n-1)llP,,,(x)P,,(x) dx=0, i.e., llP,,,(x)P,,(x) dx=0ifm.4n. (45.1) Formula (4.5.l) shows thattheLegendre polynomials areorthogonal with weight p(x)=1ontheinterval [—1, 1]. Theorthogonality property (4.5.l) plays animportant roleinthetheory ofexpansions offunctions inseries ofLegendre polynomials (seeSec.4.7). Inthistheory, wewillalsoneed toknow thevalue oftheintegral (4.5.l) for m=n,which canbefound bythefollowing device (brought toourattention byV.L.Kan): Wereplace nbyn—1intherecurrence relation (4.3.l) and multiply theresult by(2n+1)P,,(x). Then from thisequation wesubtract (4.3.l) multiplied by(2n—1)P,,_1(x), obtaining n(2n +1)I’§(x) +(n—1)(2n +l)P,,_2(x)P,,(x) -('1+1)(2" T1)P..-1(X)P..+1(X) "-"(Zn—UP?-1(X) =0, n=2,3,... Finally, integrating thisrelation overtheinterval [—1,1],andtaking account of(4.5.l), weﬁndthat l1P2(x)dx=3”--1l11>2(x).1x n=23... -111 2n+1_1n—1 s as Repeated application ofthisformula gives flP2(x)dx—-3— flP2(x)dx=_2—-_1" 2n+l_11 2n+1 SEC.4.6 ORTHOGONAL POLYNOMIALS 5| Direct calculation shows thatthisresult isalsovalid forn=0,1,andhence 1 2 2 _i, :f_1P,,(x) dx-2"+1 n0,1,2, (4.5.2) Itfollows from (4.5.l—2) thatthefunctions <p,,(x)=\/n+4P,,(x), n=0,1,2,... form anorthonormal system ontheinterval [—1,1].1° 4.6.Asymptotic Representation oftheLegendre Polynomials for Large n TheLegendre polynomials P,,(x) (—1<x<1)have asimple asymptotic representation which describes their behavior forlarge values ofthedegree n. Toobtain thisrepresentation, weuseageneral method duetoSteklov.“ Ourstarting point isthedifferential equation (4.3.10)satisﬁed bythefunction u(6)=\/sin 0P,,(cos0). Writing thisequation intheform u”+(n+%)2u =—%, (4.6.l) taking account oftheinitial conditions TC ,1':__, d§=mm u@- mm andregarding theright-hand sideof(4.6.l) asaknown function, weﬁnd that” u((i)=P,,(0) cosKn+ —6)]+ sinKn+ —0)] +@{;f3@mm+ne4m§; (4.6.2) Equation (4.6.2) canberegarded asanintegral equation forthefunction u(6). 1°Asystem offunctions cp,,(x) (n=0,1,2,...) issaid tobeorthonormal onthe interval [a,b]if b 09 r L<1>..(x)<1>..(x) dx={L ZZ 1‘V.A.Steklov, Surlesexpressions asymptotiques decertaines fonctions, déﬁnies par lesequations dtﬂérentielles linéaires dusecond order, etleurs applications auprobléme du développement d’une fonction arbitraire enséries procédant suiuant les-dites fonctions, Communications delaSociété Mathématique deKharkow, (2),10,97(1907). 12SeeE.A.Coddington, AnIntroduction toOrdinary Diﬂerential Equations, Prentice- Hall, Inc.,Englewood Cliffs, N.J.(1961), Theorem 11,p.123. 52 ORTHOGONAL POLYNOMIALS CHAP. 4 Next, using formulas (4.2.6), (4.3.6), andtherelations (1.2.1, 4,6)involv- ingthegamma function, weobtain 1>....(0>=<-1)"iV;(,f’('m+,%)1)- P.....<0> =0. Pt..<0>=0.Pt....(<>> =<-1)“ Itfollows thatequation (4.6.2) canbewritten intheform u(0)=u,,{sin [(n+4)e+g]+r,,(e)}. (46.3) where andenotes theﬁrstorthesecond oftheexpressions PG+ 2P(g+1) ml?+1)“*1”+%lP(%’+$1depending onwhether niseven orodd, and 1 "/2 . dM9) =‘W L"(<P)$111l("+%)(9-191$" (4-6-4) Now suppose thatthevariable 0isconﬁned totheinterval 8<0<TC—8, where 8isaﬁxed positive number, andletM,,denote themaximum modulus ofu(0)inthisinterval. Then itfollows from (4.6.3) and(4.6.4) thatforevery 0in[8,11-8], 1tM,, 2<U."+ CSC 8, andhence 1':M,, 2\,,-__- s.M"<°‘ +4(2n+ 1)°S° Solving thislastinequality forM,,,weobtain _ T‘ 2*1, 1‘2 Mn<ot,,[1 4(2n +1)csc8] 2n+1>4csc8, which implies theestimate 1-:csc28 1': 2'1 1-:2lVn(e)l € " — C , 271-l-1>ZCSC Thus r,,(0) =O(n“1) uniformly intheinterval [8,1-: —8].Therefore (4.6.3) leads totheasymptotic formula u(0)zat,sin[(11+4)e+ n—-><50 (46.5) (D forall8< <n—8. sEc.4.7 oRTI-IoG0NAL POLYNOMIALS 53 Making some simple calculations based onStirling’s formula (1.4.25),1" weﬁndthat anzAlia fl——> GD, rm andtherefore (4.6.5) canbewritten inthesimpler form u(9)z sin[(1.+4)e+ n->66. (4.6.6) Recalling thedeﬁnition ofu(6), weﬁnally have thefollowing asymptotic representation fortheLegendre polynomials: P,,(cos0)z /%sin[(n+~]-)0+§]> n——>00, 8< <T=—3- (4.6.7)CD Formore exact asymptotic representations, werefer thereader toHobson’s treatise.“ 4.7.Expansion ofFunctions inSeries ofLegendre Polynomials Intheapplications itisoften necessary toexpand agiven realfunction f(x), deﬁned intheinterval (—-1,1),inaseries ofLegendre polynomials: f(x) =£0c,,P,,(x), —1<x<1. (4.7.1) Thecoeﬂicients c,,canbedetermined formally byusing theorthogonality property oftheLegendre polynomials (seeSec.4.5). Infact, multiplying the series (4.7.1) byP,,,(x), integrating term byterm over theinterval [—-1,1]and using (4.5.1—2), weﬁndthat l_11f<x>P..<x> dx=lc.P.<x>P..<x> dx ="Z cnJ11 Pm(x)Pn(-x) dx=mi Cm, O which implies 1 6,,=(n+4)If(x)P,,(x) dx, n=0,1,2,... (47.2)-1 However, itisnotknown inadvance whether f(x)canbeexpanded inaseries 1°Thefactthat lim(1+E)"=e"isalsoused. 1‘E.W.Hobson, TheTheory ofSpherical andEllipsoidal Harmonics, Cambridge University Press, London (1931). 54 ORTHOGONAL POLYNOMIALS CHAP. 4 oftheform (4.7.l), orwhether theterm-by-term integration used todeter- mine thecoefﬁcients c,,islegitimate. Therefore, itcannot beasserted without further study thattheseries (4.7.l) with thecoefﬁcients (4.7.2) actually con- verges andhasthesumf(x). Inorder toestablish simple sufficient conditions forsuch convergence (seeTheorem 1below), weﬁrstprove thefollowing LEMMA. Iftherealfunction cp(x)ispiecewise continuous“ in(—1,1) andiftheintegral 1<p2(X) dx (4.7.3)-1 isﬁnite,“ then lim\/n+1llcp(x)P,,(x) dx=0. (4.7.4)1|-woo _1 Proof. First wewrite (4.7.4) asasumofthree integrals ./,7:-gig =(/,ﬁT3U_'1‘*°...+f_1:6 ...+f1‘_G =1,+,2’.+;,. (4.7.5) Then, using Schwarz’s inequality" andformula (4.5.2), weﬁndthat I14<~/@ lFax)dx]"2[tux)dx]1'2 <\/LT) U_11P,%(x)dx]1'2U:_6,>2(x) dx]1'2=U11_5q>2(x) dx]“Z. andsimilarly, —1+6 1/2 I/.|< q>*<x>dx] - Itfollows from these estimates andtheexistence of(4.7.3) thatgiven any e>0,there isa8=8(a)>0,independent ofn,such that |,.e,|< |y,|< (4.7.6) 15Forthedeﬁnition ofpiecewise continuous andpiecewise smooth functions, see G.P.Tolstov, op.cit.,p.18. 1”Ifq>(x) isdeﬁned only in(—1,1),then (4.7.3) means limJ“0q>2(x)dx.1:.b—~0+ -1+4 Ifq>(x) ispiecewise continuous intheclosed interval [—1,1],then theﬁniteness of(4.7.3) isobvious. Inother words, weallow q>(x)tobecome inﬁnite attheendpoints -1and 1,provided theintegral (4.7.3) remains ﬁnite. 1’According toSchwarz‘s inequality, b b [l°f<x>.-(X) dx]”<lfro)dxlI-=<x>dx.G ll G provided theintegrals ontheright exist. SeeG.P.Tolstov, op.cit.,p.50. sEc.4.7 oRTI-IoGoNAL POLYNOMIALS 55 +J':n:—61 1 51 n TCA/2 .1§ I 2Assuming that8hasbeen chosen inthisway, wenowuse(4.6.3) towrite i 7r—61 f2-\/n+%l q>(cos6)P,,(cos (-))sin0d6 Z 1"-61 .—.=\/n+iu,, <p(cos 0)\/sin 0s1n (n+§)0d0 “_ +X/LEI G1<p(cos('))\/sin 0cos(n+})0d0 51 cp(cos0)sin0r,,(0) d05 where 8,=arccos(1—8)Since, byhypothesis, <p(cos6)\/mil) is piecewise continuous andhence absolutely integrable on[81,1:—8,], theﬁrsttwointegrals ontheright approach zeroasn—>co.“ Moreover, thelastIntegral also approaches zero asn—>oo,since r,,(0) =O(n‘1) uniformly in[81,7:—8,],asshown inSec.4.6,where itwasalsoproved that \/n +3]at—> — asn—>ooTherefore jz->0asn—>oosothatforasuitable choice of N=N(e), wehave |f<— (4.7.7) forevery n>NCombining (477)and(4.7.6) weﬁndthat f1+f+f3l<5> n<N> andthelemma isproved. Wearenow ready toprove THEOREM 1.Iftherealfunction f(x) ispiecewise smooth in(—1,1) andiftheintegral f_l1f2(x) dx (4.7.s) isﬁnite, thentheseries (4.7.l), withcoeﬂicients c,,calculated from (4.7.2), converges tof(x) atevery continuity point off(x). Proof. First wenote thattheconditions imposed onf(x) imply the existence oftheintegrals intheright-hand sideof(4.7.2),19 sothatthe coeﬁicients c,,canactually becalculated. LetS,,,(x) denote thesum of 1°G.P.Tolstov, op.cit.,p.70. 19Apply Schwarz’s inequality tothefunctions f(x)andP,,(x). 56 ORTHOGONAL POLYNOMIALS CHAP. 4 theﬁrstm+lterms oftheseries (4.7.l). Then itfollows from (4.7.2) that m m 1 sax)=ZcnPn(x)=Z(n+%)Pn(x)ff(y)Pn(y) dyn=0 n=0 "'1 1 (4.7.9) =I_1/o>1<,,.(x, y)dy,where K,..(X, y)=Z(H+&)P»(X)P»(y)- (4»7-10)n=0 The“kernel” K,,,(x, y)canbecalculated bythefollowing device! We multiply therecurrence relation (4.3.1)byP,,(y)andthenfrom theresult- ingequation wesubtract thesame equation with xandyinterchanged. This gives (n+1)[Pn+1(x)Pn(y) _Pn+1(y)Pn(x)] —n[Pn(x)Pn—l(y) "‘Pn(y)Pn—1(x)] =(Zn+1)(X—y)P,.(X)P,.(y)- Summing over nfrom 1tom,andnoting thatPo(x) =1,P1(x) =x,we obtain (x—y)Zon+1)Pn(x)-Pn(y)n=1 =(m+1)[Pm+1(X)Pm(y) —P,..+1(y)P».(X)] —(X—J’), which implies Km(x, =m+1Pm+1(x)Pm(y) _Pm+1(y)Pm(-x)_ 2 x—y Integrating (4.7.l0) with respect toybetween thelimits -1and1,and using (4.5.1—2),2° weﬁndthat L11K,,,(x,y)dy=1. (41.12) Now suppose xisapoint of(-1, 1)atwhich f(x) iscontinuous. Multiplying (4.7.l2) byf(x), subtracting theresult from (4.7.9), and using (4.7.1l), weobtain smo)-f(x)=fKm(-xvy>[/(y)—f(x)]dy =”’T'”Pm)fP,.+1<y>¢<x, y)dy (4.1-13> -’%‘Pm+1(x)fPm(y)<P(xs y)dy, 2°Since Po(y) =1,wehave I1=,.<y>dy =fj1P0(y)Pn(y) dy= ZZ31 SEC. 4.7 ORTHOGONAL POLYNOMIALS where _f(y)—f(X)<P(x,y) y_x Regarded asafunction ofy,<p(x,y)ispiecewise continuous in(—1,1), andmoreover fl<p2(x,y)dy (41.14) isﬁnite. Infact, ifyaéx,thepiecewise continuity ofq>(x,y)in(-1, 1) follows from thatoff(y),while <p(x,y)ispiecewise continuous aty=x since <P(X,X —0)=f'(X —0), <P(X,X+0)=f’(X +0) both exist ifxisacontinuity point off(x).“ Thefactthat (4.7.l4) is ﬁnite follows from (4.7.8) andthefactthat<p(x,y)isbounded inaneigh- borhood ofy=x,where both q>(x,x—0)and<p(x,x+0)exist. There- fore, according tothelemma, "}i__n§O~/m+%P.....<y><<><x,y> dy =limvm+eflP.<y><»<x.y> dy=0-1 m—>oo _ Moreover, using (4.6.7), weseethateach oftheexpressions m+l m+li Pm , L Pm 2\/m+% (x) 2\/m++} H(x) remains bounded asm—>eo.Itfollows that theright-hand side of (4.7.13) goes tozero asm->oo,i.e., lim S(x)=f(x) andtheproof ofTheorem liscomplete. Remark I.Thecase where xisadiscontinuity point off(x) isalso of interest. Itcanbeshown thatinthiscase, under thesame conditions asin Theorem 1,theseries (4.7.l) converges tothelimit22 limS(x)=%[f(x +0)+f(x—0)]. (4.7.15) "I m->00 Remark 2.Theorem lgives sufﬁcient conditions forexpanding f(x) ina 2‘Cf.G.P.Tolstov, op.cit.,p.73. 2“Thisshould becompared with thesimilar situation encountered inthetheory of Fourier series (ibid., p.75ﬁ'.). 58 ORTHOGONAL POLYNOMIALS CHAP. 4 series oftheform (4.7.l). These conditions canbeconsiderably weakened. Atheorem which isvalid foralarger class offunctions canbefound in Hobson’s book.“ 4.8.Examples ofExpansions inSeries ofLegendre Polynomials Wenowgivesome simple examples illustrating thetechnique ofexpanding functions inseries ofLegendre polynomials: Example 1.Letf(x) beapolynomial ofdegree m: f(x) =Zaux". n=0 Then (4.7.l) takes theform f(x)=2c,,P,,(x). (4.s.1)n=0 Inthiscase, there isnoneed tocalculate theintegrals (4.7.2), since the coeiﬁcients c,,caneasily befound bysolving thesystem oflinear equations obtained when theexplicit expressions fortheLegendre polynomials aresub- stituted into(4.8.l) andcoeﬁicients ofidentical powers ofxinboth sides of theequation areequated. Thus, forexample, X2=¢'oPo(x) 'l'¢'1P1(X) 'l‘¢'2P2(x) :Co+C1?‘'l'%¢'2(3x2 _‘1), sothat cL= %, cl=0, ca=§. Therefore X2=%Po(X) +%P2(X), anexpansion which isvalid forallx. Example 2.Suppose f(x)isthefunction _ 0, —-l <X<11, f(")‘l1, a<x<l. According toTheorem 1,f(x) canbeexpanded inaseries oftheform (4.7.l), with coeﬁicients C"=(n+-9I11>,.(><)dx. Using (4.3.5) andnoting thatP,,(l) =1,weﬁndthat Cu:_%lPn+1(a) '_Pn—1(a')]v CC: —a): 23E.W.Hobson, op.cit.,p.329. sac.4.8 ORTHOGONAL POLYNOMIALS 59 which leads totherequired expansion r(x)=an—co-ai1P....<~1> -P._.<<»)1P.<x). -1<x<1-<4-8-2)»- Next, weverify thattherelation (4.7.l5) holds atthediscontinuity point x=oz.Letting S,,,(x) denote thesum oftheﬁrstm+lterms oftheseries (4.8.2), wehave s..<<»>-so-1»)-e1P...<4>P.<4> -P.<<»>P._.<4>1 =%—%Pm+1(¢)P».(<1)- Since, according to(4.6.7), P,,(1x) ->0asn—>oo, lim5(<1)=~%=%[f(¢ +0)+f(<1 -0)]. m m-ow inkeeping with thegeneral theory. Example 3.Finally, let 1—xf(x) = This function satisﬁes theconditions ofTheorem l,andhence canbeex- panded inaseries oftheform (4.7.l). Thecoefficients c,,canbecalculated bythefollowing method, which isoften useful: Wemultiply theexpansion (4.2.3) byf(x)andintegrate overtheinterval [—l,l].After some elementary calculations, weobtain 1 _(1-¢)= 1+\/i:°°,,1A/l—x zl‘+’ T;‘°g—_._v;l .Z.‘l.. "-2”»"‘>""’ "'<11(4.8.3) where theterm-by-term integration isjustiﬁed bytheuniform convergence of theseries (4.2.3) intheinterval [—l, 1],which follows from theestimate (4.4.2). Expanding the‘left-hand sideof(4.8.3) inpowers oft,weﬁndthat 4 °° 1" °°J1Fe--4ii = " iP,.d,3"Z,(41.2-l)(2n+3)Z.’ _, 2(X)" which implies 11- 4L~/T"P,(x)dx =3. 11T" 4 l_,~/' 2xP"(")d" =T(4112-l)(2n+3)‘ 60 ORTHOGONAL POLYNOMIALS CHAP. 4 Wenow use(4.7.2) towrite therequired expansion intheform A/1_;_" =%P0(x) -2; . -1<x<1.(4.s.4) 4.9. Deﬁnition andGenerating Function ofthe Hermite Polynomials Another important class oforthogonal polynomials encountered inthe applications, especially inmathematical physics,“ consists oftheHermite polynomials H,,(x),25 which canbedeﬁned bytheformula 2dae-xi-’ H,,(x) =(—l)"e" W, n=0,1, 2,... (4.9.l) According to(4.9.l), theﬁrstfewHermite polynomials are H0(x) =l, H1(x) =2x, H2(x) =4x2—2, H3(x) =8x3—l2x,..., andingeneral, [1|I2] k _ (-1)'1! 4-21. H,,(x)-ZW”_2k),(2x) , (49.2)lc0 where [v]denotes thelargest integer <1».Itwillbeshown later (seeSec.4.13) that theHermite polynomials areorthogonal with weight p(x)=e"‘“ on theinterval (—oo,oo). The Hermite polynomials (ormore exactly, theHermite polynomials multiplied bytheconstant factor l/n!) arethecoeﬂicients intheexpansion U) H" w(x,t)=e2""‘° =ZT(,)Qt", |t|<oo, (4.9.3) n=0 - andhence w(x,t)iscalled thegenerating function oftheHermite polynomials. Toprove (4.9.3), weneed only note that w(x,t),regarded asafunction of thecomplex variable t,isanentire function, andtherefore hastheTaylor series 2 Q 3"w(x,t)=em” =Z% 0t", It]<oo, On: - = 2*Inproblems involving theintegration ofLaplace’s equation andHelmholtz’ equation inparabolic coordinates, inquantum mechanics, etc.(seeSecs. 10.7—8). 25Actually introduced in1859 byChebyshev, some years before thepublication of Hermite’s work. sac.4.10 ORTHOGONAL POLYNOMIALS 6| which immediately implies (4.9.3), since anw __ x2[an —(x—t)2] _ nx2[dne_u2] __ (at")t=0 _e atne 2=0_( 1)e dun u=x _Hn(x)' Formula (4.9.3) canbeused toderive various properties oftheHermite poly- nomials. Forexample, setting x=0in(4.9.3), expanding e"2 inpower series, andcomparing coefﬁcients ofpowers oftinboth sides oftheresulting equation, weﬁndthat 2l H..<0>=<-1)" H....(0) =0. <4-9-4) There isanother expansion closely related to(4.9.3), which wewillprove inSec.4.11, i.e., W(x,y, 1)=(1_I2)-1/2e[2zyt—(a:2+112)t2]/(1-:2) =ago tn, It]<1, (4.9.5) where theleft-hand side canberegarded asthegenerating function of products ofHermite polynomials. Setting y=xin(4.9.5), weobtain w 2 W(x,x,1)=(1-t2)'1’2e2"“'/(1+‘>= |1|<1.(49.6) Formulas (4.9.3, 4,6)playanimportant roleinthetheory ofHermite poly- nomials. 4.l0. Recurrence Relations andDifferential Equation forthe Hermite Polynomials Substituting (4.9.3) intotheidentity 8wFt—(2x—2t)w-0 (apower series canalways bedifferentiated term byterm), weﬁndthat °°H,,+1(x) n_ °°H,,(x) n °°H,,(x) n1_"Z0 n,r2xnZ0T! r+Znzoin! 1*_0, which gives H,,+1(x) —2xH,,(x) +2nH,,_,(x) =0, n=l,2,... (4.lO.1) when thecoeﬁicient oft"isequated tozero. Therecurrence relation (4.l0.1), connecting three Hermite polynomials with consecutive indices, canbeused tocalculate theHermite polynomials stepbystep, starting from H0(x) =1, H1(x) =2x. 62 ORTHOGONAL POLYNOMIALS CHAP. 4 Wecanderive another recurrence relation satisﬁed bytheHermite poly- nomials bysubstituting (4.9.3) intotheidentity“ 8w5 —21W —- This gives °°H;(x) ,. °°H..(X) ...,_ E0 n! I 21:20 n! t _0, or H,’,(x) =2nH,,_,(x), n=1,2,... (4.l0.2) Formula (4.l0.2) allows ustoexpress thederivative ofaHermite polynomial interms ofanother Hermite polynomial, andisvery useful. Using therecur- rence relations (4.l0.l—2), wecaneasily derive adifferential equation satisﬁed bytheHermite polynomials. Infact, eliminating H,,_1(x) from these two relations, weobtain H,,+1(x) —2xH,,(x) +H,’,(x) =0. Then, differentiating thisformula andusing (4.l0.2) again, weﬁndthat H,§’(x) -2xH,§(x) +2nH,,(x) =0, n=0,1,2,...,(4.l0.3) where thevalidity of(4.l0.3) forn=0canbeveriﬁed directly. Itfollows from (4.l0.3) that thefunction u=H,,(x) isaparticular integral ofthe second-order linear differential equation u”—2xu’ +2nu=0. (4.l0.4) Bymaking changes ofvariables, wecaneasily derive other differential equations whose integrals canbeexpressed interms ofHermite polynomials. Forexample, itiseasy toseethat u=e"‘2/2H,,(x) isaparticular solution oftheequation u”+(2n+1—x2)u =0. (4.l0.5) 2°Thejustiﬁcation fordifferentiating (4.9.3) term byterm withrespect toxfollows from theuniform convergence of(4.9.3) inthedomain |x|<aforarbitrary ﬁnite a>0. According to(4.9.2), |H.<><>1< 114<4. sothat(4.9.3) ismajorized bytheconvergent series EL"("a)ll‘ =eza|¢|+|»1¢"=0 i" n! andhence converges uniformly forIxl<a(cf.footnote 7,p.47). SEC.4.11 ORTHOGONAL POLYNOMIALS 63 4.lI.Integral Representations oftheHermite Polynomials TheHermite polynomials have simple anduseful representations interms ofdeﬁnite integrals containing thevariable xasparameter. Toderive these representations, westart from thefamiliar integral e"‘”=%fm e“2cos2xt.11, (4.11.1)TC 0 where xisanarbitrary realorcomplex number. Differentiating (4.l1.1) 2n times withrespect tox,”andcomparing theresult with (4.9.l), weﬁndthat 22n+1 _1n xi w H2,,(x) = ~j‘ e"2t2" cos2xtdt, n=0,1,2,...(4.11.2)TC O Similarly, foroddindices wehave 2n 2_ nx2 00 H2,,.,1(x) =£21] e"'2t2"*1 sin2xtdt, n=0,1,2,...,‘/7: 0 (4.11.3) which canbecombined with (4.11.2) intoasingle formula n_'nx2 eo H,,(x)= e“”*2”"t"dt, n=0,1,2,... (4.11.4)Tr —@ Toillustrate theutility ofthese representations, wenow derive formula (4.9.5). According to(4.11.4), for|t|<1wehave 0° 1; exz +112 Ln (_ 1; 2 2"”! ’ZTC2n!(2') n=0 n=0 XJun J-co e—u2—v2+2iux+2ivy(uv)ndudv =ex2+y2 Joe foo e—u2-112 +2iux+2ivy dudv E(_l)n(2uvt)" no n0 Ill Tr — -00 _ . x2+y2 eo ee =eT.[ I e-u2—v2+2iux+2ivy—2uvldu dU_ (4116) After twoapplications ofthefamiliar formula 00 _ \/Ie‘“2‘2'2’“ ds=le"2"‘2, Rea2>0, (4.1l.6)_..., a 2’Tojustify differentiating behind theintegral sign, seeE.C.Titchmarsh, op.cit. pp.99-100, noting that theintegral in(4.ll.1) isuniformly convergent inthedisk |x|<aforarbitrary ﬁnite a>O,since itismajorized bytheabsolutely convergent integral if“) -c2+2atdV; 0e I. 64 onrnooouxr. POLYNOMIALS CHAP. 4 theright-hand sideof(4.11.5)reduces to W(x, y’1)=(1_t2)—1/2e[21yt—(a:2 +1/2)t2]/(1—t2)_ Thelegitimacy ofthevarious formal calculations follows from theconver- gence oftheexpression lxl2+l l2 °° °° °° e yI f e—u2—v2+2|u||xl+2|vIl1/Idud” Z(Zlul lv'lltl)" 71: _,, _,,, "=0 n. forall|t|<1. 4.12. Integral Equations Satisﬁed bytheHermite Polynomials TheHermite polynomials satisfy simple integral equations withsymmetric kernels. Toderive these equations, wereplace xbyyintheexpansion (4.9.3) ofthegenerating function, multiply theresult byem“/2”” (—oo <x<oo) andintegrate over (-oo,oo).This gives -[co e21/t—t2+tx1/-1/21/2 dy=J-0° e'lxU"1/21/2 dyEH"('y) tn _m -..,, "=0 n. Q (4.12.1) =Z5Iewe‘/e’H.<y>dy. n=0 '-'°° Interchanging theorder ofintegration andsummation ispermissible, since co °° co °° n f dy2 14"<f dyZ5('-§l)H.<1|y1>"'°° n=0 ' -°° n=0‘ =J“) e-‘/21/2+2lulltl+l¢lz dy<oo, where wehave used theinequality 1H.<x>1<,%.H.<11x1>. implied by(4.9.2). Evaluating theintegral intheleft-hand sideof(4.12.1), weﬁndthat J00 ezyt-:2-1/,1/¢+ixy dy=1/2net2+2ixt—1/2x3 _°° ,0_ (4.12.2) =t/Q-W 20$H..<x>.n= ' Comparing coeﬁicients ofidentical powers oftin(4.12.1—2), weobtain the desired integral equation satisﬁed bytheHermite polynomials e"‘”’”H..(x) = e"‘”e'1‘“'2H,,(y) dy, n=0,1,2, (4.123)TE —uo SEC. 4.13 ORTHOGONAL POLYNOMIALS Ifweconsider separately thecases ofeven andoddn,bearing inmind that H,,,,(x) isaneven function andH2,“ 1(x)anoddfunction (ofthevariable x), then (4.12.3) implies thefollowing twointegral equations with realkernels: e-x“'2H...<x> =<—1>“/Ii lm"””’2H....o1 cosxydy,O [*2/2H2m+1(x) =(_1)"',/gt Jon€‘”2'2H2,,,+1(y) sinxydy, m=0,1, 2,... (4.12.4) 4.13. Orthogonality oftheHermite Polynomials Itiseasy toshow that theHermite polynomials areorthogonal with weight e"‘2 ontheinterval (—oo,oo),i.e., .__-fa’e"‘2H,,,(x)H,,(x)dx 1rm.411. (4.13.1) Infact, setting u,,=e""’2H,,(x) andusing equation (4.l0.5), wehave u§f+(2n+l—x2)u,,=O, u{{,+(2m+1—x2)u,,,=O. Multiplying theﬁrstofthese equations byumandthesecond byu,,,weseethat %(u{,u,,, —u,’,,u,,) +2(n-m)u,,,u,, =O. (4.l3.2) Then, integrating (4.13.2) over (—oo,oo),weﬁndthat (n—m)JW u,,,u,, dx=0, which implies (4.l3.l). The value oftheintegral (4.13.1) form=ncanbefound asfollows: Wereplace theindex nbyn-—1intherecurrence relation (4.l0.l) andmulti- plytheresult byH,,(x). Then from thisequation wesubtract (4.l0.l) multi- plied byH,,_1(x). This gives H?.(X)+2(n~1)H..(X)H.._2(X) —H..+1(X)H..-1(X) —ZHHZ-f(x) =0, n=2,3,... (4.13.3) Multiplying (4.l3.3) bye"‘2, integrating over (—oo, oo)andusing theortho- gonality property (4.l3.l), weobtain Ine"‘zH2(x)dx =Znfwe"‘2Hﬁ_1(x)dx, n=2,3,... 1| 66 ORTHOGONAL POLYNOMIALS CHAP. 4 Repeated application ofthisformula gives 28 fooe""H,f(x) dx=2"'1n! JEDe"‘2H§(x) dx=2"n!\/fr, n=2,3,... Direct calculation shows thatthisresult isalsovalid forn=0,1,andhence fa) e""2H§(x) dx=2"nl\/;, n=0,1, 2,... (4.13.4) Itfollows from (4.l3.1, 4)thatthefunctions <p,,(x)=(2"n1\/E)-We-*2/2H,(x), n=0,1,2,... form anorthonormal system ontheinterval (—oo,oo). 4.14. Asymptotic Representation oftheHermite Polynomials for Large n TheHermite polynomials have asimple asymptotic representation which describes their behavior forlarge values ofthedegree n.This representation wasﬁrstfound byAdamov,” andplays animportant roleintheproblem of expanding functions inseries ofHermite polynomials (seeSec.4.15). We again apply thegeneral method used inSec.4.6tosolve theanalogous prob- lemfortheLegendre polynomials. Our starting point isthedifferential equation (4.l0.5) forthefunction u=e"‘2’2H,,(x). Writing thisequation in theform u”+(2n+1)u=xzu, (4.l4.1) taking account oftheinitial conditions 14(0)=114(0). 1/(0)=1111(0). andregarding theright-hand sideof(4.l4.1) asaknown function, weﬁnd that u(x)=H,,(0)cos\/2n+lx+H,’,(0) ‘22%L11"n (4.14.2) + Lxy2u(y) sin[\/W1 (x—y)]dy. 2"Note that Jm e"‘2Hf(x) dx=4-[00 e‘*2x’ dx=2\/E. 29A.A.Adamov, Ontheasymptotic expansion ofthepolynomials e“"2"" d"(e“ ’“‘“'2)/dx" forlarge values ofn(inRussian), Annals ofthePolytechnic Insitute ofSt.Petersburg, 5, 127(1906). SEC.4.14 ORTHOGONAL POLYNOMIALS 67 Next, using formulas (4.9.4), (4.l0.2) and(1.2.l, 4),weobtain H...<0>=<-1)”M). H.....(0> -0. H.:..<<>>=11.H1....(0> -2<—1>'"F,H)- Itfollows thatequation (4.14.2) canbewritten intheform u(x)=ot,,[cos (x/2n+lx— +r,,(x)], (4.14.3) where andenotes theﬁrstorthesecond oftheexpressions F(n+1), 2P(n+1)1), (4_l4_4) F(g+1) \/2n+l1"(g+§ depending onwhether niseven orodd, and 1fr2~¢.— ,,=i—;—-_ +1 — d. 4.14.5 r(x) an‘/2n+1oyu(y)S1n[ H(Xy)]y( ) Toestimate theremainder r,,(x) forarbitrary realx,weuseSchwarz’s inequality (seefootnote 17,p.54).Taking account of(4.l3.4), wehave 4.1.11< if.-.1.][i,"*' .21...4]1 Ix! 1/2 no 1/2 o Mlll."iwyl (24111/E)1/2 |x|5/2 = %=i""""~Itfollows from Stirling’s formula (1.4.25) that =11,z2‘"*”'2n"l2e‘"'2, 2"n1\/E z2"“/1e"‘n"”‘/21: (414.6) asn—>oo,andhence theproduct 1-3,,n1’4 isbounded forarbitrary n>0. Therefore |r,,(x)| <C|x]5'2n‘1"*, (4.l4.7) where Cissome constant. This lastinequality shows thatforanyﬁnite xwe have theasymptotic formula 4 71717 8u(x)zancos(X/2n +lx—Y)» n—>oo (4.14. ) or H,,(x) z2‘"*1>’2n"'2e‘"’2e"2'2 cos(\/2n +1x—%), n—>oo. (4.14.9) 68 ORTHOGONAL POLYNOMIALS CHAP. 4 For more exact asymptotic representations oftheHermite polynomials H,,(x) forlarge n,werefer thereader tothemonographs bySzego andSan- sone, cited intheBibliography (seep.300). 4.15. Expansion ofFunctions inSeries ofHermite Polynomials Wenow show that arealfunction f(x) deﬁned intheinﬁnite interval (—oo,oo)canbeexpanded inaseries ofHermite polynomials f(x)=Zc,,H,,(x), -00<X<oo, (4.1s.1)n=0 provided f(x) satisﬁes certain general conditions. Thecoefficients c,,canbe determined formally byusing theorthogonality property oftheHermite polynomials (see Sec. 4.13). Infact, multiplying theseries (4.l5.1) by e"‘2H,,,(x), integrating term byterm over theinterval (—oo, oo),andusing (4.13.1, 4),weﬁndthat looe""2f(x)H,,,(x) dx=iC"Jme"‘2H,,,(x)H,,(x)dx =2'~m1\/¥=e,,,O which implies 0,,= J:e"°”f(x)H,,(x) dx, n=0,1,2,...(4.15.2) Inthecourse ofestablishing simple sufficient conditions fortheseries (4.l5.l) with these coefﬁcients toactually converge andtohave thesumf(x), wewill need thefollowing LEMMA. Iftherealfunction <p(x) deﬁned intheinﬁnite interval (—oo,oo)ispiecewise continuous inevery ﬁnite subinterval [—a, a]andif theintegral Kw(1+x2)e-~’<p2(x) dx (4.1s.3) isﬁnite, then _ nl/4 an 2 "1111;(Y/an Loe"‘H,,(x)<p(x) dx=0. (415.4) Proof. First wewrite theintegral (4.15.4) asasumofthree integrals n1/4 co n1/4 —a a w =y,+/,+ya. (4.1s.5) SEC. 4.15 ORTHOGONAL POLYNOMIALS Then, using Schwarz’s inequality, weﬁndthat nl/4 -11 2 l/1l< @ _l_w6”‘lHn(X)l l<P(X)l dx 1/4 —a _x2 1/2 -a. _x2 1/2 < ""lll...<‘+*2)’~’<P2<*>""l- < ll dx]1'2U (1+x2)e"‘2<p2(x) dx]1/21 (4.15.6) andsimilarly, \/_ 66 2 1/2 no 2 1/2 '43‘5 l-..1—+% ""lll.<1*"2)""W""l'(4.l5.7) Ournext stepistoshow thattheintegral \/' 1"H5J= _ €—x2 dx satisﬁes thecondition J=0(1), (4.15.9) i.e.,J’isbounded foralln.Toshow this,weusetheidentity Ii no 1_x2 ne—x2 J’— ma‘, <4-154°) proved inProblem 8,p.95.Writing (4.5.10) intheform .¢=2,/gU01...+f: andmaking thechange ofvariable x—>x'1inthesecond integral, we obtain n 11_x2 ne—x2 +(_1)ne-x“2 J’_2,/ggfo (1+X2) 1+x, dx.(4.1s.11) Since e-x“+(—1)"e-F” <2, 0<x<1, n1l—x2" dx K“' WTheintegral ontheright canbeevaluated bymaking thesubstitutionitfollows from (4.l5.l1) that A/T‘lol l—x2 —— 1-+—x§-“lrt 70 ORTHOGONAL POLYNOMIALS CHAP. 4 Then dx _ dt 1+xi4\/z(1 -1) andaccording to(l.5.2), 11—x2" dx _1_2 ,,_1),2 _ ln+l ‘ll, Eel, ’1’(1*’>‘ d’-Eli’ where B(x,y)isthebeta function. Using (1.5.6) and(1.2.5), weﬁnd that -n+1 1n+1‘/"Fl 2l BET) =in?“1"(5+1) andhence 1/Z1"(i’-%1) J’<4- P(%+1)The estimate (4.15.9) isnow animmediate consequence ofStirling’s formula (1.4.25). Since Jisbounded, itfollows from theexistence of(4.l5.3) that given anys>0,there isana=a(e)>0,independent ofn,such that |/.1<§»1/.1<§~ <4-15-12> Assuming thatahasbeen chosen inthisway, wenowuse(4.14.3) towrite “uni/4 “ 2 i— m-cfa= U“ e"‘l2<p(X)C0S (\/2n +1x —T)dx +fade"‘2/2<p(x)r,,(x) dx]- Since <p(x)e"‘2'2 ispiecewise continuous andhence absolutely integrable in[-—a, a],theﬁrst oftheintegrals ontheright approaches zero as n—>oo. The second integral also approaches zero asn—>oo, since, according to(4.l4.7), theintegrand isO(n“"‘) uniformly in[—a, a], while thefactor infront ofthebrackets isbounded, asfollows from (4.14.6). Therefore f2—>0asn—>oo,sothatforasuitable choice of N=N(e:), wehave |y,|<§ (4.1s.13) SEC. 4.15 ORTHOGONAL POLYNOMIALS forevery n>N.Combining (4.15.l3) and(4.l5.l2), weandthat If.+f2+f6l<4.11>N. andthelemma isproved. Wearenowready toprove THEOREM 2.Iftherealfunction f(x) deﬁned intheinﬁnite interval (—oo, oo)ispiecewise smooth inevery ﬁnite interval [—a, a],andifthe integral ﬂowe"‘2f2(x) dx (4.15.14) isﬁnite, then theseries (4.15.l), with coeﬂicients c,,calculated from (4.l.52), converges tof(x) atevery continuity point off(x). Proof. First wenote thattheconditions imposed onf(x) imply the existence oftheintegrals intheright-hand sideof(4.l5.2), sothatthe coefficients c,,canactually becalculated.” LetS,,,(x) denote thesum oftheﬁrstm+1terms oftheseries (4.l5.l). Then itfollows from (4.1.52) thatL1,) mM 1 All sxx)=goc..H.<x>="Z0H..<><>W3f_we-1"/<y>H.(y> dy -fa”e-1"/<y>K..(><. y)dy. (415.15) where L’"H..(X)H..(y)K,,,(x,y)_V; (4.15.16) The“kernel” K,,,(x, y)canbecalculated bythefollowing device: We multiply therecurrence relation (4.l0.l) byH,,(y) andthen from the resulting equation wesubtract thesame equation with xandyinter- changed. This gives [H..+1(X)H..(y) —H..+1(y)H.(X)] —2"[H..(X)H..-1(y) —H..(y)Hr.-1(X)] =2(x-y)H,,(x)H,,(y), n=1,2,...(415.17) Dividing (4.l5.l7) by2"nl, summing overnfrom 1tom,andnoting that H0(x) =1,H1(x) =2x,weobtain 2(x Z _Hm+1(x)Hm(y) _Hm+1(y)Hm(x) _2(x _y), "=1 2n. 2’"ml which implies K...(x.y)=H“‘(’2(H:(32;+I1”;l"'*\*/(;f)H'"(")- (415.18) 3°Apply Schwarz’s inequality tothefunctions e"‘2'2f(x) ande"‘2'2H,,(x). 72 ORTHOGONAL POLYNOMIALS C1-IAP, Wenote thatK,,,(x, y)satisﬁes theimportant identity fooe-1/”K,,(x, y)dy=1, (415.19) which isanimmediate consequence of(4.15.l6) and(4.13.1, 4).“ Now suppose xisacontinuity point off(x), andconsider thedif- ference S,,,(x) —f(x), which, according to(4.15.15) and(4.15.18, 19), canbewritten intheform sax)-f(x)=fe-y*1<m<x, y)/to)—f(x)]dy _ Hm(x) 0° _2 _2"'+1m!\/E I-is8yH'"+1(y)¢(x’y) dy(41520) Hm+1( ) 0° -3,2 - LeHm(y)<P(x1y)dy> where wey)= Regarded asafunction ofy,cp(x,y)ispiecewise continuous in(—oo,oo), forexactly thesame reasons asgiven intheproof ofTheorem 1,p.55. Moreover, theintegral (1+y2)@‘”2<P(x,y) dy isﬁnite, since <p(x,y)isbounded inanyneighborhood ofy=x(see p.57),andforsuﬂiciently large b>x, if<1+y2>e-w(x, y)dy=ff<1+me-1" dy =0(1)we"”’[f’(y) +f*<x>1dy, where thelastintegral isﬁnite, because of(4.l5.l4). Asimilar estimate canbegiven fortheinterval (—oo, —b). Therefore, according tothe lemma, . (m+l)1'4 J” _2 11mii? @"Hm+1(y)<P(X,y) dy ,,,_,,, m+1 !\/ 1/2 _w [2(m+1)”]M an (415.21). m _2 =$120 Jiw 91'Hm(y)<P(X>J’) d)’=0- 31Since Ho(y) =1,wehave J-Qe'”"H»(y) dy=F@'””Ho(y)H,-(y) dy={2} n762’-==> —== 1r, n=.4 sec.4.16 ORTHOGONAL POLYNOMIALS 73 Ontheother hand, according to(4.14.l9) andStirling’s formula, each oftheexpressions [22"'*1(m +1)n/E11/2 H,,,(x) (2mm!\G)1/2 H,,,+1(x) ("1+1)“ 2"'*1m!\/¥=’ ml" 2"‘+1m!\/E remains bounded asm—> oo.Itfollows that theright-hand side of (4.15.20) goes tozero asm—>oo,i.e., lim-5'm(X) =f(X), andtheproof ofTheorem 2iscomplete. Remark I.Thecasewhere xisadiscontinuity off(x)isalsoofinterest. Itcanbeshown thatinthiscase, under thesame conditions asinTheorem 2, theseries (4.l5.1) converges tothelimit %lf(X +0)+f(X —0)]- Remark 2.Other suﬁicient conditions forexpanding afunction f(x) ina series ofHermite polynomials canbefound inthebooks mentioned atthe endofSec.4.1.32 4.l6. Examples ofExpansions inSeries ofHermite Polynomials Inapplying Theorem 2toagiven function f(x), wehave toevaluate the integral in(4.l5.2). Inmost cases thisisdone byreplacing H,,(x) byits explicit expression (4.9.l) orbyoneoftheintegral representations given in Sec.4.11. Thefollowing examples serve toillustrate thetechnique ofexpand- ingfunctions inseries ofHermite polynomials: Example 1.Thefunction f(x)=x2p: P=011,2>--- satisﬁes theconditions ofTheorem 2.Inthiscase, P X2” =2c2nH2n(x)a n=O where l °° 2C2" = Jim €_x x2"H2,,(x) dx. 3”SeealsoJ.Korous, Onexpansion offunctions ofonerealvariable inaseries of Hermite polynomials (inCzech), Rozpravy Ceské Akademie, (2),37,no.11(1928). 74 oarr-rocomu. POLYNOMIALS CHAP. 4 Substituting from (4.9.1) andintegrating byparts ntimes, weﬁndthat 1 on d2n 2 C2" = m.X2pﬁ(€_x) dx 2(2n)!\/ - X __ 1 In —x2 2P—2nd z»»(2n)1~/;<2p -2”)!_we"" 2 1 (2p)l _ 22"(2n)!\/Z <21»-2”)!K””+ii" According totheduplication formula (1.2.3) forthegamma function, 22"'2"F(p -n+—§)(p-")1=\/5(2)» -2n)!, andtherefore theexpression for02,,simpliﬁes to C=__Q-L_.2" 22"(2n)!(p —n)! Thus thedesired expansion is (2P)l P H»(X)x2p=F'nZo 9 —0O<X<0O, p=0,l,2,... (4.l6.1) Inthesame way, weﬁndthat 2+1! P H,x2P*1=(€2p+1) 2(2)? +21)*!(1S‘)_ n)!, —oo<x<oo, p=0,l,2,... 1|.O (416.2). Example 2.Letf(x) =e‘“‘,where aisanarbitrary realorcomplex num- ber. Then thesame method asused inExample 1shows that @ B“=Z@,.H,.(X),n=O where _ 1 0° —x2 a __ ‘[00 axdn —x2 *"“’"<">"""T,1v; dxde"”‘ an co ax—2 _ an a sothat ea"=e“”"‘Z H,,(x), -00<x<oo. (4.163)n=0 ' Wegetthesame result bysetting t=a/2intheexpansion (4.9.3) ofthe generating function. sec.4.16 ORTHOGONAL POLYNOMIALS 75 Example 3.Consider thefunction f(x) =e“‘2"’, Rea2>—l. Inthiscase, ewzxz =E:c2nH2n(x)a where "-0 cm= ‘lime“‘“2 *1”‘2H2,,(x) dx. Toevaluate theintegral, wereplace H2,,(x) byitsintegral representation (4.ll.2). Making anappropriate change ofvariable and again using the duplication formula (l.2.3), weobtain (cf.footnote 12,p.6) 2 11 Tl w @ C211.= Le"’t2" dtJ:soe““2"2 cos2xtdx =All J“)e"2(1+“_2>t2"dt=¥:1)n l<Tlm e‘ss"“/¢ds0 0 \/1-c(2n)la \/1r(2n)l(l +41”)"‘/= (_1)n a2n P (__1)na2n = -ti 4-— + 2 - \/n(2n)! (1+a2)"+‘/2 (ni)22"nl(l +a2)"*1/2 With thisvalue ofc2,,,wehave °° _ n2n €_a2x2 =n;o H2"(X), —@ <X<@, RC G2 > (4.16.4) Example 4.If 1! 0, then Sgnx =2¢'2n+1H2n+1(-X), n=0 where 1 G) c2,.+1 = J_w e"" H2,,+1(x) sgnxdx 1 no = J-0@"‘””="~<*> dx-Using theidentity e~=*H..(x> =-g1e-*”H._1(x)1. <4-16-5) 76 ORTHOGONAL POLYNOMIALS CHAP. 4 which follows from (4.l0.l) and(4.l0.2), weﬁndthat C : H2n(0) = , 2"“ 22"(2n +1)1\/E 2=="(2n +1)n1\/E andhence sgnx=-1:3 H2,,+1(x), -66<x<06.(4.166)\/TCF02 (2n+1)n! Example 5.Byintegrating (ordifferentiating) these formulas with respect tothevariable xortheparameter a,wecanderive further expansions ofthe same type. For example, integrating (4.l6.4) with respect toxover the interval [0,x]andusing (4.l0.2), weobtain 1 °° (_1)na2n+1 H (x) (Max)=\/¥=..;,22~n1(1 +a*)"*‘/= 2;”:1’“°°<"<°°’(4'16"7) where <D(x) istheprobability integral. Another interesting expansion is obtained ifwemultiply theseries (4.l6.4) by(l+a2)‘1 andintegrate with respect toafrom 0tooo.This gives 2“°(—1)" H.1(X)e"2[1-(x)]=;nZ0FnT27*-W;-1» 0<x<66,(4.16.s) where wehave used theidentity (2.l.7). Other examples ofexpansion offunctions inHermite polynomials are given intheproblems attheendofthechapter (seep.93). 4.l7. Deﬁnition andGenerating Function ofthe Laguerre Polynomials Stillanother important class oforthogonal polynomials encountered in theapplications, especially inmathematical physics,” consists ofthe Laguerre polynomials L$,‘(x),3“ deﬁned bytheformula LZ(x)=ex %(e"‘x""°‘), n=0,1,2,... (417.1) 33Inproblems involving theintegration ofHelmholtz’s equation inparabolic coor- dinates, inthetheory ofthehydrogen atom, inthetheory ofpropagation ofelectro- magnetic waves along transmission lines, etc. 3‘Thepolynomials L$,‘(x) diﬁ"er byonly aconstant factor from thepolynomials T;‘,‘(x) investigated byN.Y.Sonine, Recherches surlesfonctions cylindriques etledéoeloppe- ment desfonctions continues enséries, Math. Ann. 16,1(1880). Laguerre studied only thespecial caseoz=0.Intheliterature, thepolynomials L:(x) aresometimes called the generalized Laguerre polynomials. SEC. 4.17 ORTHOGONAL POLYNOMIALS forarbitrary real on>—1.According to(4.l7.l), theﬁrst fewLaguerre polynomials are L$(x) =l, L‘{(x) =l+at—x, L§(x) =%[(1+ot)(2+ac)—2(2+<x)x+x2],..., andingeneral, using Leibniz’s formula, wehave a__"F(n+<x+ l) (—x)" where forallk<ntheratio ofgamma functions canbereplaced bythe product (n+ot)(n+o1— 1)---(n+oc—(n—k— 1)). Itwillbeshown below (seeSec.4.21) thattheLaguerre polynomials L%(x) areorthogonal with weight p(x)=x°‘e"‘ ontheinterval 0<x<oo.The polynomials L,‘{(x) =L,,(x) form thesimplest class ofLaguerre polynomials. Another important class consists ofthepolynomials L,$1'2(x) which are simply related totheHermite polynomials (seeSec.4.19). Asthestarting point forthetheory ofLaguerre polynomials, webegin with thefollowing expansion no w(x,7)=(1-t)‘°‘"‘e"‘”‘1'” =2Lﬁ(x)t", |r|<1(417.3)n=0 ofthegenerating function w(x,t).Toprove (4.17.3), wenote that theleft- hand side, regarded asafunction ofthecomplex variable t,isanalytic inthe disk lt|<1,andhence must have anexpansion oftheform w(x1)=(1-7)-~=-18-*1/<1-*> =2cﬁ(x)t", |z|<1. 9 1l=O According toafamiliar theorem from complex variable theory, theco- efﬁcients c$,‘(x) canbewritten ascontour integrals c$,‘(x)= (1_t)“°“‘e"“/“">t“"“1dt, (4.17.4) evaluated along anyclosed contour Csurrounding thepoint t=Oandlying inside thedisk lt|<l.Choosing acontour ofsuﬂiciently small sizeand introducing thenewvariable ofintegration u=x/(1 —~t),weﬁndthat X —0t *1‘ n+0! c°‘(x)= faea) du (4.17.5) Tl T: l(u__xn+1 ’ where C’isasmall closed contour surrounding thepoint u=x.Evaluating thisintegral byresidue theory, weobtain exx_a dn —u 1|, oz Ci(x) ="";!- [E17,611+°‘L=x ELn(-x): thereby verifying (4.l7.3). 78 ORTHOGONAL POLYNOMIALS CHAP. 4 There isanother expansion closely related to(4.l7.3), i.e., 2 W(x’y, 1)=(1_t)-1e—<x+y>2/(1-t>(xyt)—a/2Ia[ ] ”mmmmm “HQ =7;:o t", |t]<l, ot>—l, where I,,,(z) isthemodiﬁed Bessel function oftheﬁrst kind (deﬁned in Sec. 5.7).“ Here thefunction W(x, y,t)canberegarded asagenerating function ofproducts ofLaguerre polynomials. Thefollowing special caseof (4.l7.6), obtained bysetting y=x,isimportant intheapplications: 1/2 W(x,x,t)=(1-t)"1e"“”<1-"x-°‘t"“'2I,,<%) w'[Lu(x)]2 (4.17.7)_ n.,, n __nZ?,(n+a+l)z, |t|<l, ot> 1. 4.I8. Recurrence Relations andDifferential Equation forthe Laguerre Polynomials Substituting (4.l7.3) intotheeasily veriﬁed identity (1—F)?’ +[x—(1—t)(l+ ot)]w =0, weﬁndthato0 00 (1-12)2nL$‘,(x)t"‘1 +[x-(1-t)(l+“)1ZLﬁ(x)t" =0, which gives (H+1)L%+1(X) +(X—<4—2"—1)L%(X) +(H+4<)L%-r(x) =0, n=1,2,... (4.18.1) when thecoefﬁcient oft"issetequal tozero. Similarly, substituting (4.l7.3) intotheidentity“ 6w(l—i)Er_-l"lW=0, 35SeeE.Hille, OnLaguerre’s series, I,Proc. Nat. Acad. Sci.,12,261(1926); PartII, ibid.,12,265 (1926); Part111,ibid.,12,348(1926). 3°Thejustiﬁcation fordifferentiating (4.17.3) term byterm with respect toxfollows from theuniform convergence of(4.l7.3) inthedomain |xl<aforarbitrary ﬁnite a>0. According to(4.17.2), iL§(x>| <L:(—a), lxl<4.<1>—1. sothat (4.l7.3) ismajorized bytheconvergent series Zr:<-@111" =<1~|»|>-~-1e“""<1-~'>.n=0 andhence converges uniformly for|x]<a. SEC. 4.18 ORTHOGONAL POLYNOMIALS weobtain (1-1)Z1"““di)E") +2LZ(x)t"*1 =0,n=0 n=0 which implies dL;'{ dL%_ a-21%) —#00 +L,,_1(x) =0, n=1,2,... (4.18.2) Elimination ofLZ_1(x) from (4.l8.l—2) leads totheequation 37 l"‘”_1)dL%Ex) +("+1)w%(x) (4.1s.3) +(2n+2+or—x)L§(x) —(n+l)LZ+1(x) =0,n=0,1,2,. .. Finally, replacing nbyn—1in(4.l8.3) and using (4.18.2) toeliminate (d/dx)Lﬁ_1(x), weobtain x =nL§(x) —(n+ot)L§_1(x), n=l,2,... (4.l8.4) Formula (4.l8.4) allows ustoexpand thederivative ofaLaguerre polynomial interms ofanother Laguerre polynomial. Recurrence relations ofanother type, involving Laguerre polynomials with different superscripts canbeobtained byregarding thegenerating func- tionasafunction oftheparameter 0t,andthen writing equations connecting w(x,t,oz)andw(x,t,at+1).Thus, substituting (4.l7.3) intotheidentity (1—t)w(x, t,on+1)=w(x, t,or), andcomparing coeﬁicients ofidentical powers oftinboth sides oftheresult- ingequation, weobtain L2‘,*1(x) —L$,‘I}(x) =LZ(x), n=1,2,... (4.l8.5) Similarly, substituting (4.l7.3) intotheidentity 8w(x, t,<1)_Y _—tw(x, 1,ot+l), weobtain another formula ofthistype: a'LZ7&1) =—LZi}(x), n=1,2,... (4.186) Using therecurrence relations (4.l8.2, 4),wecanderive adifferential equation satisﬁed bytheLaguerre polynomials. Infact, differentiating 3"Insome cases, thevalidity ofarecurrence relation forsmall ndoes notfollow from thegeneral argument, butthen onecanalways verify therelation bydirect sub- stitution ofLg(x) =1,Li‘(x) =l+at—x,... 80 ORTHOGONAL POLYNOMIALS CHAP. 4 (4.l8.4) with respect toxandthen using (4.l8.2, 4)toeliminate (d/dx)LZ_1(x) andLﬁ_1(x), weﬁndthat 20: or x +(a+ 1—x)‘%Q+nLZ(x)=0, "=0, 1,2,... (4.1s.7) Itfollows from (4.18.7) thatu=LZ(x) isaparticular solution ofthesecond- order linear differential equation xu"+(at+1—x)u’+nu=0. (4.18.8) Equation (4.l8.8) isencountered inmathematical physics andplays anim- portant roleinthetheory ofLaguerre polynomials. Bymaking changes of variables, wecaneasily derive other differential equations whose integrals canbeexpressed interms ofLaguerre polynomials. Forexample, itiseasy toseethatthedifferential equations rr / _ xu+(ot+l—2v)u +[n+%~§+$]u=O (418.9) and 1/ 2 %— a2u+4n+2a+2—x +xi2]u=O (4.l8.10) have theparticular solutions u=e""2x”LZ(x) and u=e—x2/2x6i+ %L%(x2) respectively. 4.19. AnIntegral Representation oftheLaguerre Polynomials. Relation between theLaguerre andHermite Polynomials TheLaguerre polynomials have asimple representation interms ofde- ﬁnite integrals containing thevariable xasparameter. Toobtain thisrepre- sentation, weassume thatxisapositive realnumber. Then e"‘x"*°‘ =loo(\/§)"*°‘J,,+,,(2\/§)e"dt, (4.191)O where J,,(x) istheBessel function oforder v.38Differentiating (4.l9.l) with 3°Here weanticipate some results onBessel functions, proved inChap. 5.Formula (4.19.1) isaspecial caseofformula (5.15.2), obtained bysetting a=1, 1>=2~/}, x=\/7, v=n+oc. sec.4.19 ORTHOGONAL POLYNOMIALS 8| respect toxandtaking account oftheidentity if-‘u“'2Jv(2\/E) =u“’“1>/2Jv_1(2\/u), obtained bysetting z=2\/u intheﬁrstoftheformulas (5.3.6), weﬁndthat £(e"‘x'""°‘) =la”(v§)"-m+===J,,_,,,,.,,(2\/E)e-'1'" dt,m=0,1,2...,0 (419.2) where itiseasy tojustify thedifferentiation behind theintegral sign. Setting m=nin(4.l9.2) and taking account of(4.l7.l), weobtain thedesired integral representation oftheLaguerre polynomials: -0:/2 to _ Lﬁ(x)=‘Z5-I t"*‘/2°‘J,,,(2\/xz)e"dt, at>1,n=o,1,2,...0 i (4.l9.3) Although thisformula hasbeen derived under theassumption that xisa positive realnumber, itcaneasily beextended toarbitrary complex values of xbyusing theprinciple ofanalytic continuation. Wenow setoz=i4in(4.19.3) andusethefamiliar formulas (5.8.l—2) from thetheory ofBessel functions. Then wehave L,j1/2(x) =-'—e:/-= Ine“t""/2 cos(2\/Y) dt n.1:o =it-AJime‘“2u2" cos(2\/E) du.71'\/7': 0 (4.19.4) L;/2(x) ="fig lowwt"sin(24/E) dt =-—ex—_ 2:Juge“‘zu2""1 sin(2\/E) du. TE nl\/x\/ o which, taken together with (4.11.2-3), imply L.:1'2<x> = H...(v¥>._ (419.5) L;/2(x) = These formulas establish aconnection between twoclasses oforthogonal polynomials, andallow ustoregard thetheory ofHermite polynomials asa special branch ofthetheory ofLaguerre polynomials.” 39Onecanalsoprove theformulas (4.19.5) directly from theexpansions (4.17.2) and (4.9.2). 82 ORTHOGONAL POLYNOMIALS CHAP. 4 4.20. AnIntegral Equation Satisﬁed bytheLaguerre Polynomials TheLaguerre polynomials satisfy asimple integral equation with asym- metric kernel. Toobtain thisequation, wereplace xbyyintheexpansion 11> (1-1)-=1-12-xi/<1-¢>= ZL2,‘(x)t”, |t|<1,6.>-1, (4.20.1)n=0 multiply theresult by e-1’2y“'2J.<\/5), where J,,(z) istheBessel function oforder et,and then integrate from 0tooo.This gives (1_t)—a-1J_m e-y(1+1)/2(1-¢)ya/2Ja(\/5,) dy ° (4.20.2) =Zr"ine-~/*x'2J.(~/»Ty>L:(y> dy.O provided that theprocess ofterm-by-term integration ispermissible. To prove thelegitimacy ofthis process, suppose |t|<4.Then, using the inequalities 4° |Lﬁ(x)| <L$,‘(—x), |.I,,(x)| <I,,,(x), x>O,ot>—l, where I,,,(x) isthemodiﬁed Bessel function oftheﬁrstkind (seeSec.5.7),we have lo|J.(\/5)|@'*"2y°"* Zlt|"|L%(y)|dyn=0 <fw1.(v1?i>e-W/2 §l1l"L%(—r) to0 n0 =(1—|t|)'°"1J:o [a(\/)5)ya/2e—1/(1—3lt|)/2(1-|t|>dy, where, inevaluating thesum, (4.20.1) hasbeen used again. For ltl<4, at>—1thelastintegral ontheright converges, ascanbeveriﬁed bycon- sidering theasymptotic behavior ofthefunction I,,,(x) forlarge andsmall x (seeChap. 5).Therefore theright-hand side of(4.20.2) isabsolutely con- vergent, which guarantees thevalidity ofreversing theorder ofsummation andintegration.“ 4°Theﬁrst inequality follows from (4.17.2), thesecond from thepower series expan- sions oftheappropriate Bessel functions (seeChap. 5). ‘*1E.C.Titchmarsh, op.cit.,p.45. sac.4.21 ORTHOGONAL POLYNOMIALS 83 Wenow set4/;=uintheleft-hand side of(4.20.2) anduseformula (5.l5.2). This gives (1_I)-<1-1 fooe-u<1+t>/2(1-0ya/2Ja(\/5,) dy O :2(1 +t)—oc—lxot/2e—x(1—t)/2(1+t) =2x01/2e—x/2 2L%(x)(_t)n’ n=0 forltl<1.42Thus, forall|t|<4,wehave theidentity 2e-*'2x°=/2 Z<-1)"L:<x)r" =Z1"e-4/2y~'2J.(v5>L:(y> dy.n=0 n= m andthen, comparing coeﬁicients ofidentical powers oft,weobtain thede- sired integral equation .-~/2x~/2L:(x> = fw1.<~/?y>e-W/2L:<y> dy.° (420.3) ot>—l, n=O,l,2,... Forat=iilthisequation reduces tothecorresponding integral equations (4.11.4-5) fortheHermite polynomials. 4.21. Orthogonality ofthe Laguerre Polynomials Wenow prove oneofthemost important properties oftheLaguerre polynomials, i.e.,their orthogonality with weight e"‘x°‘ ontheinterval 0<x<oo.Setting u"(x) :e—x/2x6:/2L%(x) andrecalling (4.18.9),. weseethat u,,(x) and u,,,(x) satisfy thedifferential equations 2 (xu;,)’ +(n+0%-‘-1 —E—%)u,, =0, x 612 Subtracting thesecond ofthese equations multiplied byu,,from theﬁrst multiplied byum,andintegrating from 0tooo,weobtain x(u{,u,,, —u{,,u,,) +(n—m) u,,,u,, dx=O. ‘*2Forsuch 1, ReLt>0,1—t andhence theconvergence condition issatisﬁed. 84 ORTHOGONAL POLYNOMIALS CHAP. 4 Foron>—1theﬁrstterm vanishes atboth limits,“ andhence J‘:u,,,(x)u,,(x) dx=0 ifmaén or fooe""x°‘L°,§,(x)LZ(x) dx=0 ifmaén,ot>~l. (4.2l.1) O Thevalue oftheintegral (4.21.l) form=ncanbefound asfollows: We replace theindex nbyn-1intherecurrence relation (4.18.l) andmultiply theresult byLZ(x). Then from thisequation wesubtract: (4.l8.l) multiplied byLZ_1(x), obtaining nlL;'€(x)l2 -('1+4<)[L;‘.‘-1(X)l” -('1+1)Lli+1(x)Lii-r(x) +2L2‘,(x)Lﬁ_1(x) +(n+or-1)L‘3,‘(x)Lﬁ_2(x) =0, n=2,3,... Multiplying thisequation bye"‘x°‘, integrating from 0tooo,andusing the orthogonality property (4.2l.1), weﬁndthat nla”e"‘x°‘[Lﬁ(x)]2 dx=(n+6.)lo”e"‘x°‘[L2,‘_1(x)]2 dx, n=2,3, Repeated application ofthisformula gives ‘*4 lowe“"x°‘[L%(x)]"‘ dx=(”"L“)§,'Z,,+_°‘1§. .1.)51J“+2)m@"‘X°‘[L‘i‘(x)]’ dx =Fii(n+“+1), n=2,3,...nl Itfollows bydirect substitution thatthisformula isalsovalid forn=0,1, andhence Jage"‘x°‘[Lﬁ(x)]2 dx= , at>-1, n=0,1,2,...O (4.21.2) Obviously, thefunctions n! 1'2<p,,(x) = e""2x°"2L§(x), n=O,1,2,... form anorthonormal system ontheinterval 0<x<oo. Formulas (4.2l.l—2) play animportant roleintheproblem ofexpanding functions inseries ofLaguerre polynomials (seeSec.4.23). *3Substituting foru,,,andu,,,weeasily verify thatthisterm is0(x‘*°‘) asx—>0. 4‘Direct calculation shows that low¢"‘X“[L'i‘(X)]’ dx=J’:e"‘x°‘(oc +1-x)’dx=(06+1)r(6+1). sac.4.22 ORTHOGONAL POLYNOMIALS 85 4.22. Asymptotic Representation ofthe Laguerre Polynomials for Large n Like theother orthogonal polynomials, theLaguerre polynomials have a simple asymptotic representation which describes their behavior forlarge values ofthedegree n.Toobtain thisrepresentation, wewrite u=e“"’2Lﬁ(x), (4.22.1) andnotethatuisthesolution ofthedifferential equation xu”+(<1+1)u'+(n+°‘—'2l—1)u =% (4.22.2) which isanalytic inaneighborhood ofthepoint x=0andsatisﬁes the initial condition F(n+on+1)0=Li =4- 4.2.”() (0) nlP(ot+1) (23) Therestoftheargument issomewhat dependent onwhether atispositive or negative, butsince thisdifference isnotofafundamental nature, wewillonly consider thecaseat>0. Regarding theright-hand sideof(4.22.2) asaknown function, weﬁnd that u(x)=A.u.(x)+A.u.<><>+5(Ny)°‘*‘u(y)[u1(y)u2(x) -u.<x>u.o>i dy. (4.22.4) where uxx)=<~/W)-"J.(2~/W). uxx)=WW)-4 Y.<2v%€). N=n+L?’ andJ,,(x), Y,,(x) aretheBessel functions oftheﬁrstandsecond kinds, re- spectively (seeChap. 5).“ Taking account oftheasymptotic behavior ofthe Bessel functions, described byformulas (5.l6.1, 2),weﬁndthatasx—>0, u1(x) —>fﬁ, u2(x) ->oo, *5Here u1and 112areapair oflinearly independent solutions ofthehomogenous equation u”+ﬂu’+L[u=0,x x with Wronskian W141.-1.1=§<Nx>-""4 Seeequations (5.4.11—12) and(5.9.2) 86 ORTHOGONAL POLYNOMIALS CHAP. 4 while theintegral is0(x2).‘*° Therefore thevalues oftheconstants ofintegra- tionare r++1.4,= A,=0, (4.22.5) and(4.22.4) canbewritten intheform u(x)=A1[u1(X) +u(x)]. (4-22-6) where no)-ivlx(Ny)°‘“u(y)[u1(y)u2(><) -u.<x>u.o>1dy. <4-22-7) Itwillnow beshown thatforﬁxed x20thesizeoftheremainder in (4.22.6) issmall compared totheﬁrstterm. Inproving this, wedistinguish twocases: (a)0Qx<N'1and(b)x>N‘1.First weﬁndanupper bound (denoted byM,,)fortheabsolute value of|u(x)] intheinterval 0<x<N'1. According toSec.5.16, for0<x<N” wehave O(N‘°‘x"°‘), 6.>0, u1(x) =0(1), u2(x) ={oﬁog L), at:0 (4.22.8) Nx ' Therefore, ifon>0,itfollows from (4.22.4—5) that |u(x)| <A10(1) +M,,N'1_l0 (Ny)°"'1[0(N'°‘x'°‘) +0(N"°‘y"°‘)]dy =A1O(1) +M,,x2O(l) =A10(l) +M,,O(N‘2), which implies that M,,=A,0(1) (4.22.9) forlarge n,aresult which remains valid foron=0.Using (4.22.9), weﬁnd that |r,,(x)] <x20(1) =0(N'2) (4.22.10) forO <<N“, ot>0,whereas |r,,(x)| <x2log(N"x‘1)0(l) =0(N‘2) (4.22.ll) forO<x< N‘1,a=0. Toestimate r,,(x) forx>N-1,wewrite (4.22.7) asasumofintegrals:>< r,,(x)=4%,U01/N +EN =1.+yr, (4.22.12) According toSec.5.16, intheinterval N'1<x<oowehave u1(x) =O(N"/*°“‘/4x"/1°‘"‘/*), u2(x) =0(N‘ ‘/2”"/4x“/*°"‘/*). (4.22.l3) ‘*6Except inthecaseat=0,where theintegral is0(x’ log SEC.4.22 ORTHOGONAL POLYNOMIALS 87 Therefore, ifat>0,weﬁndasbefore that |/.|<N-11/”<Ny>“+1<~»<>"'/2~5Y*10(1)+ o<N-ﬂy-~>1dy =N'20(N‘ ‘/*°‘"‘/4x“/2°" ‘/4), (4.22.14) aresult which remains valid forat=0,andmoreover 11.1<<A.~>-10<~-4“-ax-W-4) x(Ny)“*‘|u(y)|(Ny)-‘/=““/4dr1/N <A;1N‘/2“-‘/*0(N-‘/="-‘/wt-‘/="-‘/4) lxy‘4"*%|u<y)|dy-O Using Schwarz’s inequality andformula (4.21.2), wehave loxy‘/*°‘*“’*lu(y)ldr <U:y°‘*%dyl1/Ell: u’(y)dyl1/2 =A}’2x‘/*°“"/*(=x +%)'1’2. andhence M2]<A1-1/2Nn6-xxa@+%0(N-n=1-%x-as-n), which becomes lfgl <N'1"‘x‘/’°‘*%0(N"%°“%x'%°‘"%) (4.22.l5) since A1=0(N°‘), according to(4.22.5). Itfollows from (4.22.l2, 14,15)that [r,,(x)| <O(N“/W"/*x'%°"%)[N'1/‘x%°‘*% +N"20(l)]. (4.22.16) Acomparison of(4.22.8) with (4.22.l0-ll), andof(4.22.l3) with (4.22.l6), shows thatthesizeoftheremainder term in(4.22.6) issmall compared to u1(x) forall0<x<aandarbitrary ﬁnite a>0,provided thatnislarge. Therefore, ﬁnally, wehave theasymptotic formula u(x)zA1u1(x), n—>oo (4.22.l7) or Lax) zP_('Ll'n;'i) ex/2(Nx)-6./2_]m(2\/M), n_>oo, N:n+%1. (4.22.18) Intheinterval 0<8<x<awecanreplace theBessel function byits asymptotic representation (5.16.l). This reduces (4.22.l8) tothesimpler form LZ(x) z-rc‘1’2e"2n‘/=°“‘/*x'%°"‘/1 cos<2\/E —%—Z), n—>oo. (4.22.19) 88 ORTHOGONAL POLYNOMIALS CHAP. 4 4.23. Expansion ofFunctions inSeries ofLaguerre Polynomials Oneofthemost important properties oftheLaguerre polynomials isthe factthatarealfunction f(x) deﬁned intheinﬁnite interval (0,00)canbeex- panded inaseries oftheform G) f(x)=Zc,,LZ(x), 0<x<66, (423.1) provided f(x) satisﬁes certain general conditions. Thecoefﬁcients c,,canbe determined formally byusing theorthogonality property oftheLaguerre polynomials (seeSec.4.21). Infact, multiplying (4.23.l) bye"‘x°‘Lﬁ(x) and integrating term byterm over theinterval (0,oo),weﬁndthat ! OD c,,= Le"‘x°f(x)L§(x) dx. (4.23.2) This expansion isvalid iff(x) ispiecewise smooth inevery ﬁnite interval [x1,xz]and suitably well-behaved near thepoints x=0andx=oo.In particular, wehave THEOREM 3.Iftherealfunction f(x), deﬁned intheinﬁnite interval (0,oo),ispiecewise smooth inevery ﬁnite subinterval [x1,x2], where O<x1<x2< oo,andifthe integral Lne"‘x°f2(x) dx isﬁnite, thentheseries (4.23.1), withcoeﬂicients calculated from (4.23.2), converges tof(x)atevery continuity point off(x). Atadiscontinuity point, theseries converges to %[f(X +0)+f(X —0)]- Theorem 3canbeproved byamethod similar tothatused inproving the corresponding theorem forHermite polynomials (Theorem 2,p.71).“ 4.24. Examples ofExpansions inSeries ofLaguerre Polynomials Inapplying Theorem 3toagiven function f(x), wehave toevaluate the integrals in(4.23.2). Inmost cases thiscanbedone byreplacing LZ(a) byits explicit expression (4.l7.1) orbytheintegral representation (4.l9.3). Itis ‘*7SeeJ.V.Uspensky, Onthedevelopment ofarbitrary functions inseries ofHermite’s and Laguerre’s polynomials, Annals ofMath., (2), 28,593 (1927). For thecase on>—4,Uspensky imposes alessrestrictive condition onthebehavior off(x)nearx=0. Forexpansion theorems valid under other conditions onf(x), seeG.Szego, op.cit.,and J.Korous, Onseries ofLaguerre polynomials (inCzech), Rozpravy Ceské Akademie, (2), 37,no.40(1928). sec.4.24 ORTHOGONAL POLYNOMIALS 89 sometimes helpful tomake useofthegenerating function (4.l7.3). Thefol- lowing examples serve toillustrate thetechnique ofexpanding functions in series ofLaguerre polynomials: ‘*8 Example 1.Thefunction f(x)=X“ satisﬁes theconditions ofTheorem 3ifv>—%(ot +1),andwehave 00 xv=72;)c,,LZ(x), where nl 00 —xv ozocC"— )L 8 X+Ln(X)dX. Substituting from (4.17.1) andintegrating byparts ntimes, weﬁndthat 1 °oVdn—1lot “"mTﬁmlxw““““ _(—l)"v(v—1)---(v— +1) °°_xHa-ii———iF(n+m+1)n Le x dx _(__1)n F(v+ot+l)F(v-l-1) _ F(n+a+1)+1‘(v-n+1)’ andhence x"=F(v+6+l)F(v+1)2) (4.24.1) 0<x<oo, ot>—l. Inparticular, ifvisapositive integer p,theseries (4.24.l) terminates after aﬁnite number ofterms, andwehave xp:F(p+°‘+l)p! (4242) O<x<oo, ot>—1,p=0,1,2,... Example 2.Thefunction f(X)=8”" satisﬁes theconditions ofTheorem 3ifa>—%.Inthiscase, ® e"“‘=Zc,,LZ(x),n=0 “Itshould benoted thattheconditions imposed ontheparameters inExamples 1-4 aresufﬁcient, buttheexpansions maycontinue tohold inlarger regions. 90 ORTHOGONAL POLYNOMIALS CHAP. 4 where | Q c,,=1 ‘loe“‘“*1>"x°‘LZ(x) dx 1 OD —axdn —xnoc =1~(..J.?...t1)l, 6w<@"*>"" = an m —(o.+1)x n+ot r(n+a+1)l,,e Xdx nQ — ! l'l—0,l,2,... With these values ofc,,wehave 8-...=(a+1)_._. 3(%)"Lg(x), 0<X<66.(4.243) O Wegetthesame result bysetting t=a/(a+1)intheexpansion (4.l7.3) of thegenerating function. Example 3.Consider thefunction f(x)=(ax)-4/2J,(2\/E<), x>0,a>0,6.>-1. Inthiscase, thedesired expansion is (ax)-8/*1.<2v$¢) =Zc.L:(><).n=0 where n! oo _x Xor/2 __ O‘ C": J1) € Ja(2\/l1.X)Ln(X) dx. Toevaluate theintegral, wemultiply theidentity (4.l7.3) by _x xon/2 _ e(Z) J,,(2\/ ax) andintegrate with respect to,x from 0to00.Then, assuming that [t]is sufficiently small, weobtain EX(1-t)-“"1 lowe""‘1‘”(—)°‘/2J,,(2\/ﬁ)dx =e““1‘” NQ="7120?; in="Z0inLoo@—x(§)u/2J,(2\/ax)LZ(x) dx, where wehave used formula (5.l5.2). Comparing coefﬁcients ofidentical powers oft,weﬁndthat e—l1aﬂ F(n+on+1)’c,,= sec.4.25 ORTHOGONAL POLYNOMIALS 9| andhence °° n -W1.2v="a—% L:,(ax) (ax)8.2,F(n+<1+1)(x) (4.24.4) x>O, a>0, 01> -1, Example 4.Ifwemultiply (4.24.3) by(a+l)°“1 and integrate with respect toafrom 0tooo,weobtain go —ax 01-1 _ w oz 00 a n da _ no lee (a+l) da_nZ0L”(x)L la+l) (a+l)2—nZon+1 Theintegral intheleft-hand sidecanbeexpressed interms ofthecomple- mentary incomplete gamma function (seeProblem 10,p.15).This gives °°L$.‘(X)e"x‘°‘F(<z,x) =2i, 0<x<oo, at>-1, (4.24.5),,=0n +1 which foron=0reduces to . °°L(x)—I— E"F = —"—, . .. eEz( x) e(0,x) 12:0” +1 0<x<oo (4246) Some other expansions inseries ofLaguerre polynomials aregiven in Problems 19-20, p.96. 4.25. Application totheTheory ofPropagation ofElectromag- netic Waves. Reflection from theEndofaLong Transmis- sion Line Terminated byaLumped Inductance Asacurious example oftheapplication ofLaguerre polynomials, we consider theproblem ofpropagation ofelectromagnetic waves along atrans- mission lineoflength l.Suppose the lineterminates atoneendinacoilof /Y X=1 inductance Lo,while attheother enda source ofconstant d-cvoltage V0is Z suddenly switched onattime t=016? 0 (seeFigure l2).Lettheinstantaneous values ofthevoltage and current be denoted byV=V(x,r)andI=I(x,1), FIGIJRE 12 andlettheinductance andcapacitance perunitlength ofthelinebedenoted byLandC.Then theproblem reduces totheintegration ofthefollowing system oflinear diﬁ"erentia1 equations,“ aV a1 a1 av-5_L5t, -5- cw (4.254) ‘*9SeeS.Ramo and J.R.Whinnery, Fields and Waves inModern Radio, second edition, John Wiley andSons, New York (1953), p.24. 92 ORTHOGONAL POLYNOMIALS CHAP. 4 subject totheinitial conditions V|t=o Z I|t=0 I 0 andboundary conditions 8]V|x=0 =V0, V|,,=, =L0-5 _l- (4.25.3) Tosolve these equations, weusethemethod oftheLaplace transform (seeSecs. 2.6,8),which converts (4.25.l) intoapairofordinary differential equations. Asusual, let denote theLaplace transform ofthefunction f: f=fooe"”fdt. (4.2s.4)0 Then (4.25.l) goes into dI7 - di - andeliminating I,weobtain asecond-order differential equation d2I7 _W --LCp2V =0, subject totheboundary conditions __V0 417L__1/|,,=,,_7 5;+toV|,,=,_0. (425.6) Itfollows from (4.25.5) and(4.25.6) that coshg(l —x)+lsinhZ(l —x) V=V° " L21’ ” , (4.257/) ‘D coshEl+ —sinhelv Lop v where v=1/\/LCisthevelocity ofwave propagation along theline, and Z=\/L? isthecharacteristic impedance.“ Wenow return totheoriginal function Vbyusing theFourier-Mellin inversion theorem (cf.p.25) __ 1 t_V—2T1, J;€p where theintegral isalong alineAparallel totheimaginary axisandtothe right oftheorigin. Being primarily interested inthevoltage attheendofthe line, wesetx=lin(4.25.7—8). Then 1 l ep‘ 7,"let"7- "P, (4-25"’) 5°S.Ramo andJ.R.Whinnery, op.cit.,p.27. PROBLEMS ORTHOGONAL POLYNOMIALS where at=Z/Lo, andT=I/visthetime ittakes thewave togofrom one endofthelinetotheother. Toobtain theanswer inaform which hasa simple physical interpretation, weexpand l7|,,=, inpowers ofe‘2"T and integrate term byterm. This gives 1 °° n1 p__atnep[t—(2n+1)T] —V..-=-1—.—— ——— ,2V0l‘ )21=1lllp+<»l p+<» "P or,ifweintroduce thenewvariable ofintegration q=(p+ot)/2oz,H _1_V|x =E(_l)ne—ot[t—(2n1-1)T]iiJ‘ (1_§)ne2qa[t—(2n+1)T]d?q, 2V0 =’ "=0 21: A, (4.25.l0) where A’isalineparallel toandtotheright ofA. Theevaluation oftheintegral in(4.25.10) 1~"(¢)=-1-I (1-1)n€:dq (4.2s.11)Zrri A. q q isaccomplished byusing residue theory applied totheclosed contour con- sisting ofA’andthearcofthecircle |q|=R(where Risarbitrarily large) lying totheleftofA’if-r>0ortotheright ofA’if-r<0.Intheﬁrstcase, wehave 1d" e‘d"Fe)=,5[Wm—1>e“}]q:0 =H<y"e~i>]y=T =Lm.<4-25.12) where L,,(-r) isthenthLaguerre polynomial (seeSec. 4.17), while inthe second case F(*r) =0.Substituting (4.25.l2) into (4.5.10), weﬁnd that V|,,=, =0for0<t<T,and N—1 iV|,,=,=Z(—l)"e‘°‘”“(2""1>T1L,,{2ot[t -(2n+1)T]} (425.13)n=0 for (2N—1)T<t<(2n+l)T, N=l,2,... Formula (4.25.l3) represents thesolution inclosed form, andtheappearance ofnewterms atintervals of2Tseconds corresponds tothearrival ofaddi- tional reﬂected waves atthepoint x=I. This method isapplicable totransmission lines terminated byloads of other kinds, andinmany other cases theanswer canalsobeexpressed in terms ofLaguerre polynomials. PROBLEMS 1.Show thatalltheroots oftheequation P“(x)=0arerealandlieinthe interval (—1,1). Hint. UseRolle’s theorem. 2.Show thatalltheroots oftheequation H,,(x) =0arereal. 94 ORTHOGONAL POLYNOMIALS CHAP. 4 3.Prove theinequality“ 112 (1-—x"’)1’4|P,,(x)| < 1 -1<x<1, n=1, 2,... 4.Using theexpansions (4.9.2) and(4.l7.2), prove Uspensky’s formula _ n 1 _ Lao)=‘—¥‘l’ii'i+—‘) I(1-12>“-‘/2H..<~/mdr. '1>—l,'\/TCF(Ot +-})(2n)! -1 which expresses theLaguerre polynomials interms oftheHermite poly- nomials. 525.Prove Koshlyakov’s formula Ls,=+ﬂ(x) = i5iL1¢~(1_t)°‘1Lﬁ(xt)dt, a>-1,ta>0. Hint. Replace theLaguerre polynomial L‘3,‘(xt) byitsexpansion (4.l7.2), andintegrate term byterm. Comment. For at=_%, (3=at+%,Koshlyakov’s formula reduces to Uspensky’s formula. 6.Inmany cases, theevaluation ofintegrals oftheform Inoe"‘2f(x)Hﬁ(x) dx canbeaccomplished bythefollowing device: Multiply equation (4.9.6) by f(x), integrate from —ootooo,andevaluate theintegral intheleft-hand side, calling theresult cp(t). Then expand q>(t)inpowers oftandequate coefficients ofidentical powers oftinboth sides oftheequation soobtained. Applying thismethod, show that Fe"‘2H,2,(x)dx =2"nWE, lae""2HZ(x)x2 dx=2~n!\/Em +9. Jae e'2"“H§(x) dx=2"“/=1"(n +'5')- (2n)'\/_ 1-2" no -ax ' a -..,‘”H2"(")""=Rea2 >O,n=O,1,2,..., w _x2 2 21!-+1: 2!D2‘/T J14” 3 Hv(X)H2n(X) dx = v7.Prove that p=O, 1,2,..., n=0,1,2, ...,p. Hint. Toderive thesecond formula, usethemethod ofProblem 6. 51Forasimple proof, seeG.Szegti, Orthogonal Polynomials, revised edition, Ameri- canMathematical Society, New York (1959), Theorem 7.3.3, p.163. 5’N.S.Koshlyakov, OnSonine’s polynomials, Messenger ofMathematics, 55,152 (1926). PROBLEMS ORTHOGONAL POLYNOMIALS 95 8.Provethat 1 °°H"(x) _ °°1—x2 "e"2 "2""=l-...lIT?l Trade "=°"-2~-- Hint. Usethemethod ofProblem 6. Comment. This formula wasused intheproof ofTheorem 2,p.71. 9.Derive theintegral representation 1 e -e""2L,,(x) = J; 8_’2Hﬁ(l) COS (\/2X!) dt. Hint. Tocalculate theintegral ontheright, usethemethod ofProblem 6. 10.Derive theformula 2"n' °° 2e"‘2H3.(x) = Ie"”"L,,(%) cossxds.71: 0 Hint. Usetheresult ofProblem 9andtheFourier integral theorem.“ 11.Derive thefollowing integral equation forthesquare oftheHermite poly- nomial ofoddindex: —:2 2 _ no Z - 2 .\/_ ) =IJ1(2\/xy) . dy_vx 0 \/y Hint. Tocalculate theintegral ontheright, usethemethod ofProblem 6. 12.Derive thefollowing integral equation forthesquare oftheLaguerre polynomial: e-~x~tL:<x>12 =1..(2~/5)e-”y~[L:<y)12dy. <»>~%- Comment. The result ofthepreceding problem isaspecial case ofthis formula. 13.Prove theexpansions Q(-1)"H (X) ,.e‘2cos2xt=Z0 l2 , ltl<°°, . °°(—l)"H2 u(x)e'2S1l'12.Xf =Zo t2"*1, [t|<oo. Comment. Theexpressions ontheleftinthese formulas canberegarded asgenerating functions fortheeven and odd Hermite polynomials, respec- tively.Z3 3 14.Verify thefollowing expansions inHermite polynomials (cf.Secs. 2.1.3): ._ _5‘”<—1)"H..<x> 8*[1 (I)?-’(X)il _Tr720 23n+ 1/2"! 2”T1» F(x)=~/E H,,,.,(x). 53G.P.Tolstov, op.cit.,p.190. 96 ORTHOGONAL POLYNOMIALS CHAP. 4 15.Derive thefollowing expansion ofthesquare ofaHermite polynomial in aseries ofHermite polynomials: H§(x) =2p(pl)2 5: v p=0,1, 2,... "=0 . . Hint. Usetheresult ofProblem 7. 16.Derive thefollowing expansion ofaproduct ofHermite polynomials with different indices inaseries ofHermite polynomials: _ P, VP H2n+r(x) _ Hp(X)Hp+1(X) ~217-(I1 +")~2 ’ P.P‘—0,1. 2.---,,=o . . . Hint. For r=1therequired result isobtained bydifferentiating the formula found inthepreceding problem. Thegeneral casecanbeobtained by using mathematical induction. Comment. Thisexpansion canbewritten inthesymmetric form rn1n(P.q> 2nH _ Hp(x)H.(x) =olq!"Z0 ,* 11.11=0,1.2.--- 17.Using thegenerating function (4.9.3), prove thefollowing addition theorem fortheHermite polynomials: - "H(x)H-(y) -_H,,(x cosat+ysinat)=pl"Z0 cos" onsin” "ct. 18.Prove theformula __(_1)p PH2n(x)H2p~2n(y) _ “("2J’Y2)'2“ n!(P—n)!’1’“°’1’2"" Hint. Usetheexpansion (4.17.3). 19.Derive thefollowing expansion oftheincomplete gamma function (see Problem 10,p.15)inaseries ofLaguerre polynomials: X_°lY(Ot,X) =T20 v 0<X<O0, 0t> 20.Derive theexpansions P I-;'§*”“(X +y)=ZLZ(x)1-‘.3-t(y). P=0.1.2,---."=0 Lax)= )L:<x). 11=0.1.2.... Hint. Usethegenerating function (4.17.3). 21.TheJacobi polynomials PS,“-">(x) aredeﬁned bytheformula Pt“-ﬂ><x> = <1-x)-“<1+x)-6% [<1-x>"+~<1+x)"*°]. ot> -1, 13> -1, n=0,l,2,... PROBLEMS ORTHOGONAL POLYNOMIALS 97 Using themethods ofthischapter, show thattheJacobi polynomials have the following properties: (a)Thefunction u=P$.°"‘”(x) satisﬁes thedifferential equation (1—X’)1/’+[ﬁ—<1—(<1+B+2)xlu’+n(n+<><+B+1)u=0; (b)Thepolynomials P§,°"'”(x) areorthogonal with weight P(X)=(1—x)°‘(l+X)“ ontheinterval [—-1,1]; (c)Thepolynomials P§,°""’(x) aretheexpansion coefﬁcients ofthegenerat- ingfunction w(x,t)=2“*°R-1(1 —t+R)-“(1 +t+R)"° =2P§.°"“’(x)t", |2|<r,7'-=0 where R=(1—2xt+t2)1'2, andrisgiven byformula (4.2.4). 22.TheChebyshev polynomials“ aredeﬁned bytheformula T,,(x) =cos(narccosx), n=0,1,2,... Show thattheChebyshev polynomials have thefollowing properties: (a)Thefunction u=T,,(x) satisﬁes thedifferential equation (1—x2)u” -—xu’+nzu=0; (b)Thepolynomials T,.(x) areorthogonal with weight t>(X)=(1—x”)'1” ontheinterval [—1,1]; (c)Thepolynomials T,,(x) aretheexpansion coefﬁcients ofthegenerating function 1—:2 "°w(x,F)=1—f§§J—_|_*? =To(x) +2"; Tt(X)l"', ltl<F, where risagain given by(4.2.4). Comment. TheChebyshev polynomials play animportant roleinthe theory ofapproximation. 5’Sometimes transliterated asthe“Tchebichef polynomials”, asinG.Szego, op. cit.,and intheBateman Manuscript Project, Higher Transcendental Functions, Vol.2, Chap. 10.Werefer thereader tothese sources forfurther information ontheJacobi and Chebyshev polynomials. CYLINDER FUNCTIONS: THEORY 5.I.Introductory Remarks Byacylinder function wemean asolution ofthesecond-order linear differential equation 1/+1u'+(1-f)u=0 (511)z zz ’ " where zisacomplex variable andvisaparameter which cantake arbitrary realorcomplex values. Equation (5.1.1), called Bessel’s equation oforder v, isencountered instudying theboundary value problems ofpotential theory forcylindrical domains (seeSec.6.3), which explains theorigin oftheterm cylinder function. Certain special kinds ofcylinder functions areknown inthe literature asBessel functions, andthisterm issometimes applied tothewhole class ofcylinder functions. The cylinder functions, with their manifold applications, have been studied ingreat detail, andextensive tables ofsuch functions areavailable. These functions areamong themost important special functions, with very diverse applications tophysics, engineering andmathematical analysis itself, ranging from abstract number theory andtheoretical astronomy toconcrete problems ofphysics andengineering. Some ofthese applications, mainly from theﬁeld ofmathematical physics, willbeconsidered inChapter 6.The present chapter isdevoted toabrief exposition oftheelementary theory of cylinder functions. Thereader whowishes togofurther inhisstudy ofthese functions should consult thespecial literature devoted tothesubject (seethe Bibliography onp.300), notably theclassic treatise byWatson} towhich wewillmake frequent reference. 1G.N.Watson, ATreatise ontheTheory ofBessel Functions, second edition, Cambridge University Press, London (1962). 98 SEC. 5.2 CYLINDER FUNCTIONSZ THEORY 5.2.Bessel Functions ofNonnegative integral Order Inmany applied problems, oneneed only consider aspecial class of cylinder functions, corresponding tothecasewhere theparameter vinequa- tion(5.1.1) isanonnegative integer n.Thiscaseismuch simpler than thecase ofarbitrary v,andwillserve tointroduce thegeneral theory. Webegin byshowing thatoneofthesolutions ofBessel’s equation ...1. rt”u+;u+(1-?p=0, n=aLaU. wan isthefunction u,=.I,,(z), known astheBessel function oftheﬁrst kind of order n,anddeﬁned forarbitrary zbytheseries Ma=§§%%§%i |n<w. (mmO Using theratio test,weeasily verify thatthisseries converges inthewhole complex plane, andhence represents anentire function ofz.Suppose we denote theleft-hand sideof(5.2.l) byl(u), andintroduce theabbreviated notation _(—1)*”‘r”m@+m forthecoefﬁcients oftheseries (5.2.2). Then wehave l(u,)=Z[(n+2k)(n +2k—1)+(n+2k)—n2]ot,,z"+2""2 +2ot,,z""2" lc=O k=O =24akk(n +k)zn+2k—2 +2akzmztt t¢=1 lc=0 =2l4°1;¢+1(k 'l'1)(n ‘l‘k+1)'l‘°‘t¢:lZ"+2kt l¢=0 andtherefore l(u,) E0,since theexpression inbrackets vanishes. Thus J,,(z) satisﬁes Bessel’s equation (5.2.1), i.e.,J,,(z) isacylinder function. The simplest functions ofthiskindaretheBessel functions oforders zeroandone: Z(Z/2>2(Z/2>* (Z/2)“ (523)-’1<Z>=§l‘ _T1i'+T3.1"3W+"'l' Wenowshow thattheBessel functions ofhigher order canbeexpressed interms ofthetwofunctions JO(z) andJ,(z). Assuming thatnisapositive I00 CYLINDER FUNCTIONSZ THEORY CHAP. 5 integer, wemultiply theseries (5.2.2) byz"andthen differentiate with respect toz.Thisgives £2[Z1tJ"(Z)] :_ Z2n+2k—1 R‘ OM8[Q/'\3 _nW (__1)k Zn—1+2k n "Z (2 =Z1"-1(2)’OI" %[z"J,,(z)] =z"J,,_1(z), n=1,2,... (5.2.4) Similarly, multiplying (5.2.2) byz'",weﬁndthat %[z‘"J,,(z)] =—z"‘J,,+1(z), n=0,1,2,... (5.2.5) Performing thedifferentiation in(5.2.4—5) anddividing bythefactors 2*", wearrive attheformulas 1:.<z)+§1.(z)= 1._.(z). 14(2)-§1.<z)=-J...<z). (52.6) which immediately imply thefollowing recurrence relations satisﬁed bythe Bessel functions: 1._.<z)+J...<z)=2;”J.(z). n=1,2,... <5-2.1) J,,_,(z) —J,,.,,(z) =2J,’,(z), n=1,2,... (5.2.8) Repeated application of(5.2.7) allows ustoexpress aBessel function of arbitrary order v=n(n=O,1,2...) interms ofJo(z) andJ1(z), thereby greatly simplifying theeffort needed tocalculate tables ofBessel functions. Formula (5.2.8) allows ustoexpress derivatives ofBessel functions interms ofother Bessel functions. Forn=0,(5.2.8) should bereplaced by J(,(z) =—J,(z) (5.2.9) [inkeeping with (5.2.5)], which isanimmediate consequence oftheformulas (5.2.3). TheBessel functions oftheﬁrstkind J,,(z) aresimply related tothecoefﬁ- cients oftheLaurent expansion ofthefunctionz w(z,1)=e/z=<*-*"*> =Zc,,(z)t", 0<1t|<OO.(5.2.10)1l=—tD 2Regarded asafunction oft,w(z,t)isanalytic intheannulus 0<8$t<A<oo, andtherefore thisexpansion exists. SEC.5.2 CYLINDER FUNCTIONS2 THEORY l0l Tocalculate thecoeﬂicients c,,(z), wemultiply thepower series 2 e=*/2=1+ ——(z1’,2)t +——(Zg) 12+---, _2 _ (z/2) 1 (z/2)2 1 6”’-1-T7+T?2'+'" andthen combine terms containing identical powers oft.Asaresult, we obtain c,,(z) =J,,(z), n=O,1,2,..., C,,(Z)=(—1)nJ_,,(Z), n=-1,-2,...., (5211) which implies w(z,t)=e‘/==""'"‘> =J,,(z) +2J,,(z)[t" +(—1)"t'"], 0<ltl<oo. n=1 (5.2.12) Thefunction w(z,t)iscalled thegenerating function oftheBessel functions ofintegral order, andformula (5.2.l2) plays animportant roleinthetheory ofthese functions. Toﬁndageneral solution ofBessel’s equation (5.2.1), thereby obtaining anarbitrary cylinder function ofintegral order v=n(n=0,1,2,...),we must construct asecond solution of(5.2.l) which islinearly independent of J,,(z). Forsuch asolution wechoose uz=Y,,(z), called theBessel function ofthesecond kind, which willbedeﬁned inSec.5.4.Itwillbeshown inSec. 5.5thatthisdeﬁnition leads totheseries expansion Yn(z) =%Jn(z) log; _ ! (3)2).-1. (s.2.1s)1°°(__1)k( /2)n+2k -;kZo_7d(7’I7),_ [toe+1)+=l)(k+n+1)], where 1 1l.l)("’l‘l‘I)=—Y"l‘l'l"2'l'...'l";s tl)(1)=—y, YisEuler’s constant (seeSec. 1.3), andinthecase n=0,theﬁrstsum in (5.2.l3) should besetequal tozero. Thefunction Y,,(z) isanalytic inthe complex plane cutalong thesegment [—oo,O],andbecomes inﬁnite asz—>0. Thus, thegeneral expression forthecylinder function oforder v=nisa linear combination ofBessel functions oftheﬁrstandsecond kinds, i.e., u=Z,,(z) =AJ,,(z) +BY,,(z), n=0,1,2,..., (5.2.14) where AandBareconstants. CYLINDER FUNCTIONSZ THEORY CHAP. 5 5.3.Bessel Functions ofArbitrary Order The Bessel functions considered inthepreceding section areaspecial caseofthemore general Bessel functions oftheﬁrstkind ofarbitrary order v. Todeﬁne these functions, consider theseries i<-1)'@(z/2)v+" ,,,=0F(k +1)F(k +v+1) where zisacomplex variable belonging totheplane cutalong thesegment [—oo, 0],andvisaparameter which cantake arbitrary realorcomplex values.“ Itiseasily seen that(5.3.l) converges forallzandv,andthatthe convergence isuniform ineach variable intheregion |z[<R,|v|<N(where RandNarearbitrarily large). This follows from thefactthatstarting from some sufﬁciently large k,theratio oftheabsolute value ofthe(k+l)thterm tothatofthekthtenn equals lz|2 < R2 4(1<+1)|k+ 1+v|‘4(k+ 1)(/<+1-1v)’ where theright-hand sideispositive, independent ofzandv,andapproaches zero ask—>co.‘Since theterms of(5.3.l) areanalytic functions ofzinthe plane cutalong [—oo,0],thesumoftheseries isananalytic function ofzin thesame region. Wecallthisfunction theBessel function oftheﬁrst kind of order v,anddenote itbyJ,,(z), i.e., °° R.’ v+2lc J,(z)=go |2|<oo,|arg2|<TE.(53.2) Toshow thatthefunction (5.3.2) satisﬁes Bessel’s equation with para- meter v,wewrite(5.3.l) 2 l(u)Eu”+éu’+(1-—%)u =0, ul=J,(z), andrepeat thederivation given inSec.5.2,5 obtaining (D la.)=Z14=»...(k +1)(k+»+1)+<><.1zv+2k.k=o 3Ingeneral, thecondition imposed onzisnecessary forthefunction z"tobesingle- valued, butcanbeomitted ifvisaninteger. 4Aseries offunctions 3u).(z)k=0 converges uniformly inadomain Dif 14tc+1(Z) u|.(z) forallzinDandk>M,where qisindependent ofz.SeeE.C.Titchmarsh, op.cit.,p.4. 5Recall thatauniformly convergent series ofanalytic functions canbedifferentiated term byterm.<q<1 sac.5.3 CYLINDER FUNCTIONS! THEORY l03 where (—Wof-k: '2 F(k+l)F(k +v+l) Using (1.2.1), weseeatonce thatl(u,) E0. Since forﬁxed zintheplane cutalong thesegment [—oo,0],theterms of theseries (5.3.2) areanalytic functions ofthevariable v(seeSec.1.1),thefact that(5.3.2) isuniformly convergent implies thattheBessel function ofthe ﬁrstkind isanentire function ofitsorder v.Forintegral v=n(n=0,1, 2,...),l"(k+v+l)=(n+k)!and(5.3.2) reduces to(5.2.2). Therefore thefunctions deﬁned inthissection arethenatural generalizations ofthose studied inthepreceding section. For negative integral v=—n(n=1, 2,....),theﬁrstnterms oftheseries (5.3.2) vanish (seeSec. 1.2), andthe series becomes °° __1k 2-—n+2k °° ___1n+s 2n+2s 3 andhence J_,,(z) =(—-l)".I,,(z), n=1,2,... (5.3.3) Thus, theBessel functions ofnegative integral order differ only bysignfrom thecorresponding functions ofpositive integral order. Itfollows thattheex- pansion (5.2.12) canbewritten intheform w(z,1)=e’/1"““_1)= 2J,,(z)t". (53.4) Many oftheformulas derived earlier forBessel functions ofnonnegative integral order remain thesame forBessel functions ofarbitrary order. For example, %[ZvJv(z)] :Zv-Iv—1(Z)a %lZ_vJv(Z):l :—Z_vJv+1(Z)s 1._.<z)+1...<z)=J.<z). 1._.<z)-1...(z)=2Jc(z).(5.3-6) generalize formulas (5.2.4-5, 7-8), andareproved inexactly thesame way. Wealsohave 1zv.(z)1=Z"-'"1.-..<z).(5.3.?) [Z-v.(z)1 =<—1)“z-V-'"J....<z). which areproved byrepeated application of(5.3.6). I04 CYLINDER FUNCTIONS: THEORY CHAP. 5 5.4.General Cylinder Functions. Bessel Functions ofthe Second Kind Bydeﬁnition, acylinder function isanarbitrary solution ofthesecond- order linear differential equation ,, 1, v21(1))=ll+Eu+(1-z—2)u=0, (5.4.1) andhence hasthegeneral form u=Z,,(z) =C1u,(z) +C2u2(z), (5.4.2) where u,anduzarearbitrary linearly independent solutions of(5.4.l), and C1,C2areconstants which, ingeneral, arearbitrary functions ofthepara- meter v.Itiseasytoobtain anexpression forthegeneral cylinder function in thecase where visnotaninteger. Infact, choosing u,=J,,(z), where J,,(z) istheBessel function deﬁned inSec.5.3,wetake thesecond function tobe uz=J_,,(z), which isalsoasolution of(5.4.l), since (5.4.l) does notchange ifvisreplaced by—v.Fornonintegral v,theasymptotic behavior ofthese solutions asz—>0isgiven by u,z » u2z 1 (5.4.3) andtherefore these solutions arelinearly independent.“ Thus, thedesired expression forthegeneral cylinder function canbewritten as u=Z,,(z) =C1J,(z) +C2J_\,(z), v#0,i1,1-2,... (5.4.4) If)/isaninteger, then, because of(5.3.3), theparticular solutions u,andu2 arelinearly dependent, and(5.4.4) isnolonger ageneral solution ofBessel’s equation (5.4.l). Toobtain anexpression forthegeneral cylinder function which issuitable forarbitrary v,weintroduce theBessel functions ofthe second kind, denoted byYv(z) anddeﬁned bytheformula I _J,,(z) cosvn—J_,(z)Y,(z) -‘___? W (5.4.5) forarbitrary zbelonging totheplane cutalong thesegment [—oo,0].”For integral v,theright-hand sideof(5.4.5) becomes indeterminate [cf.(5.3.3)], andinthiscasewedeﬁne Y,,(z) asthelimit Y,,(z) =limY,(z). (5.4.6) 6This argument breaks down ifvisaninteger (including zero). 7Thefunction wedenote byY,(z) issometimes denoted byN,,(z) intheliterature on Bessel functions. sac.5.4 CYLINDER ruucrrousz THEORY I05 Since both thenumerator anddenominator areentire functions ofv,and since d. .a—S1nv1r=rrCOsvrc;é0 Ifv=n,v thislimit exists andcanbecalculated byL’Hospital’s rule, application of which gives _13J,,(z) __n8J_,,(z)Y,,(z)_;[_5V-V H(1)-_av V (5.4.?) Itfollows from itsdeﬁnition thatY,(z) isananalytic function ofzintheplane cutalong [—oo,0],andanentire function oftheparameter vforﬁxed z. Inview of(5.4.4), thefactthat Y,(z) isacylinder function, i.e.,satisﬁes Bessel’s equation (5.4.l), isobvious fornonintegral v.Toshow that Y,(z) isacylinder function forintegral v,weusetheprinciple ofanalytic continua- tion, noting thatsince l(Yv)isanentire function ofv,l(Yv)E0forvaén implies l(Y,,)forallv.Thefactthatthesolutions ul=J,,(z) andU2=Y,,(z) arelinearly independent follows from thelinear independence ofthesolutions J,(z) andJ_,(z)fornonintegral v,andfrom acomparison ofthebehavior of u,and112asz—>0[cf.(5.4.3) and(5.5.4), proved below] forintegral v.Thus, ﬁnally, theexpression u=Zv(Z) =C1Jv(Z) "I"C2Yv(Z) forthegeneral cylinder function Z,(z) issuitable forarbitrary v. TheBessel functions ofthesecond kind satisfy thesame recurrence rela- tions asthefunctions oftheﬁrstkind, e.g., 5212"Y.<z)1=z~Y.-.(z). £1:-~Y.(z)1 =-Z-"Y...(z).(5.4.9) Y.-.<z)+Y...<z)=Y.<z). Y.-.(z)-Y...<z)=2Y¢<z)- Fornonintegral v,thevalidity ofthese formulas follows from thedeﬁnition (5.4.5) andthecorresponding formulas forJ,(z). Toobtain thesame formulas forintegral v,weneed only pass tothelimit v—>n,observing that allthe functions involved arecontinuous with respect totheindex v.Wealsonote that(5.4.7) implies therelation Y_,,(z) =(—1)"Y,,(z), n=0,1,2,..., (5.4.10) which allows ustoreduce thecalculation offunctions ofnegative integral order tothatoffunctions ofpositive integral order. Bymaking changes ofvariables inBessel’s equation (5.4.l), wecaneasily obtain anumber ofother differential equations whose general solutions can I06 CYLINDER FUNCTIONSZ THEORY CHAP. 5 beexpressed interms ofcylinder functions. Ofthevarious equations ob- tained inthisway, those ofgreatest practical interest are 2 22 u”+-il-220‘ u’+[(l*3yz"‘1)2 +-———-—iIIl-2;Y]u=0, (5.4.l1) u”+azlu =0, with solutions u=z°‘Z((3z") u=Z1’2Z.E z1"“"2))) (5.4.12) V 1 1/((+2) Y+ 2 where Z,(z) denotes anarbitrary cylinder function. 5.5.Series Expansion ofthe Function Y,,(z) Toderive aseries expansion ofthefunction Y,,(z), weusetheexpansion (5.3.2) tocalculate thederivatives with respect totheindex vwhich appear in (5.4.7). Because of(5.4.10), weneed only consider thecase v=n(n=0, 1,2,...).Since, asalready shown, theseries (5.3.l) converges uniformly inv, wecandifferentiate itterm byterm, obtaining“ 3V 0°_lc n+2k: % n=Zo [1eg§-¢(k+n+1)].P? where _F12I‘I"(z) _ isthelogarithmic derivative ofthegamma function (seeSec.1.3). Similarly, wehave =~.1.);*1”25;*:;[~+»V+1»)-Fork=0,1,2,...,n— 1, I F(k—v+l)—>00, tIJ(k-v+l)—>0OPi‘ ,l\/18E/\ asv—>n,sothattheﬁrstnterms ofthelastseries become indeterminate. However, using familiar formulas from thetheory ofthegamma function [see(1.2.2, 4)and(l.3.4)], weﬁndthat .tI.i(k—v+l) . . kI)(V—k)-I'TCCOtTC(V—k)___.i_ = F__ ______._i___._ £15171‘I,(k_V+1) (v k)S1nTc(v k) T: =(—l)""‘(n-—k—l)l, k=0,1,...,n—1, BThepassage tothelimit v—>nbehind thesummation signislegitimate, since a series obtained byterm-by-term differentiation ofauniformly convergent series of analytic functions isitself uniformly convergent. sec.5.6 CYLINDER FUNCTIONSZ THEORY I07 andtherefore @J_.() __."-‘<-k-1)1_2"-"dvz,=,_( 1),2,n/<1 °° (_1)o I: Z ](z)2P+n +-1"—————— —1—+ +1- 1 where wehave introduced thenewsummation index p=k—n. Itnow follows from (5.4.7) that thedesired expansion ofthefunction Y,,(z) is l"'1(n —k—l)lz2"'" Y4”)=“.2,2,T' I2) +71%} [21<>g§-¢(k+ 1)—tI»(k+n+ 1)]O [argz|<1:,n=0,1,2,..., (5.5.1) where theﬁrstsumshould besetequal tozeroifn=0[cf.(5.2.l3)]. Accord- ingto(l.3.6—7), thevalues ofthelogarithmic derivative ofthegamma func- tionaregiven by ~)<1)=-Y.¢<m+1)= —Y+1+%+"'+%* m=1.2..... (5.52) where Y=0.57721566. ..isEuler’s constant. Using (5.2.2), wecanwrite theexpansion (5.5.l) inasomewhat different form: 1/"(2.) =%_]"(z)10g§ _ <%)2lc—1I. = I (5.53) -%k§o [¢(k +1)+tl)(k+n+1)]. Finally, wenote that(5.5.1) implies theasymptotic representations Y0(z) z5loggt z—>0 (n_1)‘Z_" (5.5.4) Y,(z)z—-——T'(§) , z—>0, n=l,2,..., which show that Y,,(z) becomes inﬁnite asz->0. 5.6. Bessel Functions ofthe Third Kind Next wediscuss stillanother class ofcylinder functions, i.e.,theBessel functions ofthethird kindorHankelfunctions, denoted byH$1)(z) andH§2’(z). I08 CYLINDER FUNCTIONSZ THEORY CHAP. 5 These functions aredeﬁned interms oftheBessel functions oftheﬁrstand second kinds bytheformulas H$"(Z) =Jv(Z)+iYv(z)> H$”’(Z) =Jv(z)—1'Yv(Z)s (5-6-1) where visarbitrary andzisanypoint oftheplane cutalong thesegment [—oo, 0].Themotivation forintroducing thefunctions (5.6.l) isthatthese linear combinations ofJ,,(z) and Y,,(z) have very simple asymptotic expres- sions forlarge |2|(seeSec. 5.11) andarefrequently encountered inthe applications. Itfollows from (5.6.1) thattheHankel functions areentire functions ofv, andanalytic functions ofzintheplane cutalong [—oo, 0].Clearly, the functions H§”(z) andH§2’(z) arelinearly independent ofeach other, andeach islinearly independent of.I,,(z). Therefore wecanwrite thegeneral solution ofBessel’s equation (5.4.l) inanyoftheforms u=Z.(Z)=A1J.(Z) +A2H$1’(Z) (562) =B.1.(z)+B.Hs2><z) =1>.Hs1>(z) +1>.Hi*>(z). '' where A1,...,D,arearbitrary constants, aswellasintheform (5.4.8). Since theHankel functions arelinear combinations ofthefunctions J,,(z) andY,,(z), they satisfy thesame recurrence relations asthese functions, e.g., %[z“H6"’(Z)l =z"H6’l’.(Z). £12-~HsP><z)1 =-Z-"Hit>.<z). () Haw)+Hem)=11%).Hsexz)—H5'i31(z)=2 (5.6.3) where p=1,2.Using (5.4.5) toeliminate Y,,(z) from (5.6.1), weobtain H51)( )_ , H52)(z) = , (5_6_4)ZT zsinwe zsinvr: which imply theimportant formulas H92,(z)=e""'H,§"(z), H91(2) =e“”“H,§2’(z). (5.6.5) 5.7.Bessel Functions ofImaginary Argument Intheapplications, onefrequently encounters twofunctions I,,(z) and K,,(z), which areclosely related totheBessel functions. LetDbethecomplex plane cutalong thenegative realaxis. Then, forallzinD,I,,(z) andK,(z) aredeﬁned bytheformulas w (Z/2)v+2Ic Iv(Z) =go )9 IZI <(X3, |arg ZI<TC, K,(z) = |argz|<TC, vaé0,i1,i2,...(5.7.2) SEC.5.7 CYLINDER FUNCTIONSI THEORY I09 where, forintegral v=n, K,,(z) =limK,,(z), n=0,i1,i2,... (5.7.3) Repeating theconsiderations ofSecs. 5.3-4, weﬁndthatI,,(z) andK,,(z) are analytic functions ofzforallzinD,andentire functions ofv. Thefunctions I,,(z)andK,,(z) aresimply related totheBessel functions of argument ze*"”2. If -1:<argz <gti.e., —g<arg(ze"”2) <TI, then (5.3.2) implies °° v2 Jv(ze7!i/2) =evm/2 12:0 kT_Tn =em/2Iv(z), sothat Iv(z) =e‘"‘”2J,,(ze"”2), -1:<argz< (5.7.4) Similarly, according to(5.6.4), forthesame values ofzwehave J_,,(ze"”2) —e‘‘”"J,,(ze"”2) . lsinvrc -var!/2] ()_e—v1ti/2Iv(Z) 26 _ Z 2 i-—i Z i.e_‘/7“/2Kv(z), lSln VTC 7171H‘(1)(zenf/2) : andhence 1<,(z)=Qe""”"’H,§1>(ze"”2), --TE<argZ< (5.75) Ontheother hand, if —g<argz <1:, -1:<arg(ze"‘”2) <-5, then itiseasily veriﬁed that Iv(z) =evni/2JV(Ze—ni/2), Kv(Z) =_ge“Vﬂl/2H$2)(Ze"ﬂll2)- (5_7_6) Because of(5.7.4—6), I,,(z) andKv(z) areoften called Bessel functions of imaginary argument. However, thisterm isnottoofortunate, andinstead we willusually refer toI,,(z) asthemodiﬁed Bessel function oftheﬁrst kind and toK,(z) asMacd0nald’sfuncti0n.9 9K,(z) iscalled themodiﬁed Bessel function ofthethird kindintheBateman Manu- script Project, Higher Transcendental Functions, Vol.2,p.5. I CYLINDER FUNCTIONS! THEORY CHAP. 5 Itisanimmediate consequence oftheformulas justderived thatI,(z)and K,(z) arelinearly independent solutions ofthedifferential equation u”+lu’—(l+v—2)u—0 (577z zz _’ ") which differs from Bessel’s equation only bythesignofoneterm, andgoes intoBessel’s equation ifwemake thesubstitution z=iit.Equation (5.7.7) isoften encountered inmathematical physics, anditsgeneral solution, for arbitrary v,canbewritten intheform u=C,I,,(z) +C2K,(z). (5.7.8) Thefunctions I,,(z) andK,(z) satisfy simple recurrence relations, e.g. ,%[z“I.(z)i =z"1.-.(z). iz—v.<z)i =z—v...<z). %1z"K.(z)1= -z"K.-.<z). £12-"I<.(z)1 --Z-"K...(z)(5.7.9) 1._.<z)+1...(z)-zlctz). 1._.<z)-1...<z)=1.<z). K.-.(z)+K...<z)--21<¢<z). 1<._.<z)—1<...<z)-—3,1K.(z). The recurrence relations involving I,,(z) areproved bysubstituting from (5.7.1). Then, using these formulas and(5.7.2), wederive thecorresponding formulas involving K,(z) fornonintegral v.Finally, weextend theresults to thecase ofintegral vbyusing thecontinuity ofK,(z) with respect tothe index v. Two other useful formulas are I-1(1) =[n(z)s n=0.11.12.---, K—V(z) = KV(z)9 where theﬁrstfollows from (5.7.l) ifwenote thattheﬁrstnterms ofthe expansion vanish ifv=——n,while thesecond isanimmediate consequence ofthedeﬁnition (5.7.2). Using (5.7.3) andthemethod ofSec.5.5,wecanderive aseries expansion ofthefunction K,,(z). Theresult ofthecalculations is(51.10) Kn(z) = l (5)2):-,. +%(-1)"-1k§° [2log§ -4»(k+1)-\I)(k+n+1)]. |argz| <rt,n=0,1,2,..., (5.7.1l) SEC.5.8 CYLINDER FUNCTIONS2 THEORY III where 1l)(z)isthelogarithmic derivative ofthegamma function [whose values canbefound from (5.5.2)], andtheﬁrstsum should besetequal tozero if n=0.Wenote that(5.7.l1) implies theasymptotic representations K0(z)zlog Z-—>0, 1 _n (51.12) K,,(z)z2(n—l)l(g) ,z—>0, n=l,2,..., which show thatK,,(z) becomes inﬁnite asz—>0. 5.8.Cylinder Functions ofHalf-Integral Order Wenow consider thespecial class ofcylinder functions oforder n+% (n=0,i1,12,...).Inthiscase, thecylinder functions canbeexpressed interms ofelementary functions. Toseethis, weﬁrstﬁndthevalues ofthe functions .Ii1,2(z). Setting v=i-§in(5.3.l) and using theduplication formula (1.2.3) forthegamma function, weobtain wtowawwW)=Z.22 1/2 ‘*7 (_1)I€z2I€ 2 1/2 I =fa).Z.,n2'tT1) =fa) 2 1/2 J_,,,(z) = cosz. (5.s.2)(5.8.1) andsimilarly, Thefactthat anyBessel function oftheﬁrst kind ofhalf-integral order canbeexpressed interms ofelementary functions now follows from the recurrence relation Lﬂ@+LH®=€L® [see(5.3.6)], repeated application ofwhich gives 1 21/2- -Ia/2(2) =2-71/2(2) "-/-1/2(1) = —C052]’ 21'2.J_3,2(z) =— [S111z+$1, andsoon.Using (5.3.7), wecanwrite thegeneral expression forJ,,.1/,(z) in terms ofelementary functions. Forexample, setting v=Qinthesecond of theformulas (5.3.7) andtaking account of(5.8.l), weﬁndthat 21/2 1 d 1i' JM1/,(z)=(—1)"(;-E) z"*/4272) "=0, 1,2,... (5.s.3) H2 CYLINDER FUNCTIONSI THEORY CHAP. 5 Toderive thecorresponding formulas forBessel functions ofthesecond andthird kinds, westart from theexpressions (5.4.5) and(5.6.4) ofthese functions interms ofBessel functions oftheﬁrstkind, anduse(5.8.l—2). Forexample, 21/2 Y1,2<z>= -1-1/42) =- cow, (1) ‘21/2 12 (2) '21/2 —izH1,2(z) =-1E) e, H1,2(z) =4; e, andsoon. Finally, wenote that 21/2_ 21/2 Tc1/2 I1/2(2) = $111112, I-1/2(2) = 995112, K1/2(2) = 9”, (5.8.5) where theformulas forgeneral index n+=}areobtained from (5.8.5) and therecurrence relations (5.7.9). Ithasbeen shown byLiouville thatthecase ofhalf-integral order istheonly casewhere thecylinder functions reduce to elementary functions. 5.9. Wronskians ofPairs ofSolutions ofBessel’s Equation BytheWronskian ofapair u1(z), u2(z) ofsolutions ofalinear homo- geneous second-order differential equation ismeant thedeterminant _141(1) 112(2)W{u1(z)> 142(2)} _141(2) 115(2) where theprime denotes differentiation with respect totheindependent variable z.Thesolutions ulandu2arelinearly independent ifandonly if theWronskian does notvanish identically.” Wenow calculate theWron- skians ofvarious pairs ofsolutions ofBessel’s equation 2 u"+lu'+(l—L2)u=0,z 2 thereby obtaining anumber offormulas which areuseful intheapplications. Inparticular, these formulas show thatthesolutions inquestion arelinearly independent, afactproved earlier byother means. Tocalculate theWronskian, wewrite theequations forulanduzinthe form d, 2 d, 2;;(Zu1)+(Z-“;)u1=0,;Z<zu2>+(Z—“;)u2=0, 1°E.A.Coddington, op.cit.,Theorem 6,p.111. sac.5.10 CYLINDER ruucrrous: 'r1-nsoav ll3 andthensubtract theﬁrstequation multiplied byuzfrom thesecond equation multiplied byul.Theresult is 5‘;[zW{u1(Z), um]=0, which implies Wale),112(2)}= where Cisaconstant, independent ofz,whose value canbedetermined, for example, from therelation C=limzW{u1(z), u2(z)}.Z-'0 Inparticular, choosing ul=.I\,(z), u2=J_v(z),where visnotaninteger, andusing theexpansion (5.3.2) andformulas (l.2.l-2) from thetheory ofthe gamma function, weﬁndthat _. --2v 2____2sinvnC-l1_n31.,(l +V)P(1__ V)[1+0(2)] - —T , which implies2. W{Jv(z),J-42>}=—ii:-‘,—”1‘ (5-9-1) The validity of(5.9.l) forintegral vfollows bycontinuity, andwehave WE0,asmust beexpected. TheWronskians ofother pairs ofsolutions of Bessel’s equation canbefound inthesame way, orelsetheycanbededuced from (5.9.1)andtherelations (5.4.5), (5.6.4). Wealways begin byconsidering thecase ofnonintegral v,andthen usecontinuity toextend theresult to arbitrary values ofv.Inthisway, weﬁndthat W{.I,,(z), Yv(z)}= (59.2) W{Jv(z),Haw}=— (5.9-3) W{H:1><z>, Hm}=- (19.4) andsoon.FortheBessel functions ofimaginary argument wehave W{Iv(z),Kv(z)}=— <5-9-5) 5.l0. Integral Representations oftheCylinder Functions Thecylinder functions have simple integral representations interms of deﬁnite integrals andcontour integrals containing zasaparameter. The II4 CYLINDER FUNCTIONSI rrnsonv CHAP. 5 representations bycontour integrals have greater generality, andareusually valid inlarger regions ofvalues oftheargument zandparameter vthan the representations bydeﬁnite integrals, butthelatter aremore frequently en- countered intheapplications. Therefore wewillbeprimarily concerned with representations bydeﬁnite integrals.“ Oneofthesimplest integral representations oftheBessel functions isdue toPoisson. Consider theidentity 1 1 1 2 2v—12 F(k+v+ 1):I‘(k+—§)I‘(v+—§).i_1tk(1_t) /d" R“>_i’ (5.10.1) implied by(l.5.6). Substituting (5.l0.l) into theexpansion (5.3.2) and reversing theorder ofintegration andsummation,” weobtain Jv(Z) =kg‘) 21¢F(k +%)1F(v + Jill t2k(1 _I2)»/-1/2 dt </2>v 12 "°<—1>"<1>2*=P<vz+i‘)“‘’)/"’,Z.22"P<k +1>ZI"<k +s) __ (/2)” 1 2v_.,_1%) L(1-z)/cosZ!dt, (s.1o.2) where wehave used theduplication formula (l.2.3) forthegamma function: 22’“l‘(k +1)I‘(k+-5)=F(§)I‘(2k +1)=l“(~})(2k)!. Thus JV“)=1% l_.<1“‘W’”"’* (5.103)Rev >—%, |arg2|<1:, orequivalently, (z/2)“ " .Jv(z) =I,—I;T——— cos(zcos6)s1n2" 6d6, Rev>——},|arg2|<TC,(%(+%L Qmg where wehave made thesubstitution t=cos6. 1‘The reader with aspecial interest inintegral representations ofcylinder functions should consult G.N.Watson, op.cit.,Chap. 6. 1’Tojustify reversing theorder ofintegration andsummation, wenotethat °°IZ/2Iv+ 21¢ 1 1 _V2 To P<k+are+l)‘W1‘‘if"’_ °° [Z/2!v+2k = _g;Fw+1ww+v+1y‘M”D<”' ifRev >—§. 55¢,5,10 CYLINDER FUNCTIONS2 THEORY ll5 Toobtain another important integral representation ofJ,,(z), westart from theformula 1 1 _.1:(k_HT_T) =ﬁfeas"‘*"*1)ds, (5.105) proved inProblem 9,p.15,where Cisthecontour shown inFig. 13.Sub- stituting (5.l0.5) into(5.3.2), weﬁndthat O =(§)v%:;_Le‘s“"1dsi:o (;F1(3;i4f/-iii‘-)’: (s.10.6) where reversing theorder ofintegration andsummation isagain easily justi- ﬁedbyanabsolute convergence argument. Assuming temporarily thatzisapositive real number andsetting s=zt/2, wecanwrite (5.l0.6) intheform C Jv(Z)=Le‘/=2"-"*>¢-V-1 dz,(5.10.?) 0 where C’isacontour resembling C.Bythe principle ofanalytic continuation, thisresult isvalid inthewhole region [argz|<1:/2. Writing t=pe‘°and choosing theradius ofthecircular partofC’tobe1,wehaveFIGURE 13 Jv(z) =grJ:cos(zsin6—v0)d6—$J‘1m e“/=*<P‘F"‘>p“"1 dp, which, after thesubstitution p=e°‘,becomes J,,(z) =éfncos (zsin6—v6)d6— e'“‘““°“"°‘ doc, Rez >0,0 O (5.l0.8) where visarbitrary. Inthecasev=n(n=O,i1,i2,...),thesecond term ontheright vanishes, and(5.l0.8) takes asimpler form. Inmany cases, onecanderive integral representations ofBessel functions ofthesecond andthird kinds from thecorresponding integral representations ofBessel functions oftheﬁrstkind, byusing formulas (5.4.5) and(5.6.4). II6 CYLINDER FUNCTIONS2 THEORY CHAP. 5 Forexample, ifRez>0andvisnonintegral, itfollows from (5.4.5) and (5.l0.8) that Y(z) =SEE “cos (zsin6—-v0)d6—L813-t coe'”‘““°""°‘ dotV ‘"7 0 TY 0 ._ cos (Zsin 6+V9) _ e-—zs1nhoz+voI da_ T" 0 T‘0 Replacing 0by1:—6inthethird integral ontheright, weﬁndafter some simple calculations that Yv(z) =;1c-J0" sin(zsin9—v9)d9—éjjj e_“‘“h °‘(€'°‘ +€"’°‘ COSvrc)dot. (5.109) Inproving (5.l0.9), itwasassumed thatvisnonintegral, buttheformula holds forarbitrary vbytheprinciple ofanalytic continuation, since both sides areentire functions ofv. Integral representations oftheHankel functions canbeobtained byusing (5.l0.8—9) andthedeﬁnitions (5.6.l). Forexample, ifRez>0, H‘51)(z) =_]v(Z) _|_iYv(Z) =éfn et(zSill(-)—v8) dg 0 + llfm e—zs1nha[evo: +e-V(Ol+7!l)] dd 11'0 1 0 1 11: __: ezslnh ot—voz dm + ezslnh 19-—v19 rcl _,0 1:19:0 1 no + ezs1nh(o1+1:1)—v(ot+11:t) d(a +Tu"), Tu oc=O which, after thesubstitution t=oz+ii),reduces to H,§1>(z) =l.Ie""““'"dt, ReZ>0, (5.10.10)Til Q1 where C1isthecontour shown inFig.l4(a). Similarly, H,§2’(z) =— ezsi““'""‘dt, Rez >O (5.l0.1l)TC! C2 where C2isthecontour shown inFig. l4(b). Thus (5.l0.10) and(5.l0.1l) arethesame, except forthechoice ofthecontour ofintegration. Substituting t=ui%1ciinto(5.lO.10—1l), weﬁndthat -—wit/2 H51>(Z) ="_-,- e‘“°S“““"‘du, ReZ>0,(5.10.12)TF1 D1 vni/2 H§2’(z) =_ e‘”du, Rez>0,(5.10.13)D2 where thepaths ofintegration D1andD2areshown inFigure 15. Tofurther transform these integrals, weassume temporarily thatzisa sEc.5.10 CYLINDER FUNCTIONSZ THEORY II7 positive realnumber and that theparameter visconﬁned tothestrip -1<Rev<l.Then, according toCauchy’s integral theorem, theintegral 1r/' 6‘, 0I > ; >- 0 ‘YT/‘ C2 (0) (bl FIGURE 14 along theleft-hand partofthebroken lineD1(orD2),uptothepoint u=0, canbereplaced byanintegral along thenegative realaxis, andtheintegral 15/ vr/L 2 01 2 > 0 —> 0 _11 _L/2 2 02 (0) (bl FIGURE 15 along theright-hand part ofthebroken linecanbereplaced byanintegral along thepositive realaxis.” Thus formulas (5.l0.l2—l3) become —vni/2 on H51)Z)=5-.— e“’°°’“‘"-1"du, (5.10.14)Till _w evni/2 co _ H,§2’(z) =---W-1,-I e'”du, (5.10.15) 13Itiseasily veriﬁed that theintegral along thevertical segment needed tocomplete each contour towhich weapply Cauchy’s integral theorem approaches zero astheseg- ment ismoved indeﬁnitely fartotheleft(ortotheright) oftheimaginary axis. Toshow that thecondition —1<Rev <1guarantees theconvergence of(5.10, 14-15), con- sider thesubstitution y=e“. II8 CYLINDER FUNCTIONSZ THEORY CHAP. 5 where z>0,—l<Rez<1.Using theprinciple ofanalytic continuation, weeasily seethat(5.10.l4) remains valid for0<argz<TC,while (5.lO.l5) remains valid for-71:<argz<0,since ineach caseboth sides of(5.l0.l4—l5) areanalytic functions ofzintheindicated region. Moreover, thecondition —l<Rev <lcanbedropped ifImz >0in(5.10.14), orifImz <0in (5.lO.l5). Finally, therefore, wehave theintegral representations e—v11:i/2 no H51>(z) =TI2" du, ImZ>0,(510.16) evni/2 w _ 1152(2) =-Tl.Ie“Zdu, ImZ<0,(510.17) where visarbitrary. Formulas (5.10.l6—l7) arethebasic integral representations oftheHankel functions. Other integral representations oftheHankel functions, useful in theapplications, canbederived bymaking suitable transformations ofthe integrals in(5.lO.l6—l7). For example, consider formula (5.l0.l6), let Rev>-4},andforthetime being assume thatargz=-rc/2,sothat —izis positive. According to(l.5.l), 10° ,y""‘/= =Hviﬁﬁfo 2-xvxvwdx, Rev>-1,(510.12) andhence, setting y=e“in(5.l0.l6), wehave H(1>(z) _6-‘/M/2 fage‘/ziz(u+y'1) -—v—1dV __ Tci 0 y y e—v1riI2 Jae 1/i 1 12 00 / =-—.?i e”‘”+”_ )1"'dyf e"‘” x"“‘1dx1=11"(\'+5)0 0_ e-—v1:l/2 no ~12 w iz iz -12 if """”‘l.°"Pl"y(" ﬁlial’ ’dy’where thereversal oftheorder ofintegration iseasily justiﬁed byproving the absolute convergence ofthedouble integral. Tocalculate theinner integral, weusetheformula“ "° \/E - _av2_ U2) _ _2~/ab J0e ‘bldv-K/Z e , a>0,b>0. (5.l0.l9) This gives Hé1)(Z) = f—v1ri/2 jwoe-2¢Tm~/1~(iz/2) xv_% dx, ix/T=1‘(v +1.)<1\/x-(iz/2) 14After making thetransformation 1=02,theintegral (5.l0.l9) becomes theLaplace transform ofthefunction 1;-I-‘"2 e'”", evaluated atp=a. sEc.5.10 CYLINDER FUNCTIONS! THEORY lI9 or 2e—v1Ii Zv co _ H61) =_____g(_)-J‘ 12t2_1v—%d’ R _ ,(z) ix/nF(V+%) 2 1e(t ) t ev> Q» (5.10.20) where weintroduce thenewvariable ofintegration t=\/X-(iz/2)_ \/ zz/2 Bytheprinciple ofanalytic continuation, thisformula, proved under the assumption that —iz>0,remains valid forarbitrary complex zbelonging tothesector 0<argz<11:.Injustthesame way, wehave theformula _2evm Zv eo _ Hf,” =_i (-)I-R112-1)-‘/221,(Z)i\/1=I‘(v +5)21e( )t(510.21) Rev >-5, -1:<argz <0 forthesecond Hankel function. The integral representations (5.l0.20—2l) playanimportant roleinthederivation ofasymptotic representations ofthe cylinder functions as|z|—>oo. Integral representations fortheBessel functions ofimaginary argument caneither beobtained directly byaslight modiﬁcation oftheconsiderations ofthissection, orelsededuced from (5.7.4—6) andthecorresponding integral representations oftheBessel functions and Hankel functions. Thus, it follows from (5.l0.3) that I,,(z) =—_i2); fl(1—t’)“"/1 cosh ztdt, \/1=P(» +5)-1 (5.10.22) [argz] <1:,Rev >-5, andfrom (5.l0.l6, 20)that Kv(Z) =gym} e—zc0shu—vu du=Jun e—zco5huCOSh vudu, 2-00 0 (5.l0.23) Rez>0,varbitrary, Kz =—-\/i- Evwe-"12-1)“-‘/=dr“U 1‘(v+5)(2)ii ( ’ (5.10.24) Rez >0,Rev >—-5. Wealsocallattention toanother integral representation 1ZVno-1-1 2/41>---1 T‘K,,(z) =—— e ZtVdt, [argz|<-, (5.lO.25)22 O 4 I20 CYLINDER FUNCTIONS2 THEORY CHAP. 5 which isuseful intheapplications, andisobtained from (5.10.23) bychanging thevariable ofintegration. Some other useful integral representations ofthecylinder functions and their products aregiven inProblems 1-9,p.139. 5.lI.Asymptotic Representations oftheCylinder Functions for Large lz| There aresimple asymptotic formulas which allow ustoapproximate the cylinder functions forlarge |z|and ﬁxed v.The leading terms ofthese asymptotic expansions canbederived starting from thedifferential equations satisﬁed bythecylinder functions, buttoobtain more exact expressions, itis preferable tousetheintegral representations found inthepreceding section. Asymptotic representations ofthecylinder functions forlarge |v|and ﬁxed zcanbeobtained rather simply from formulas (5.3.2), (5.4.5), (5.6.4) and(5.7.l.—2) byusing Stirling’s formula (l.4.22). Theproblem ofapproxi- mating thecylinder functions when both |z|and|v|arelarge isoneofthe most diﬁicult problems ofthetheory. Some basic results along these lines can befound inChapter 8ofWatson’s treatise, andnewformulas ofthistype have been obtained inrecent years byLanger“ andCherry.“ Ofallthecylinder functions, theHankel functions have thesimplest asymptotic representations. Wenow derive anasymptotic representation of thefunction H,$1>(z), starting from formula (5.10.20). Making thesubstitution t=1+2s,weﬁndthat 2v+1 i(z—vn) v w H61) Z _e ZI 2zis v—1/21 v—% d’ (Z)ix/-=1"(»+g) 0eS(+s) S (5.11-1) Rev >-5, 0>argz <1:. Replacing (1+s)”'‘/2byitsbinomial expansion <1.-___<-1>*<;,—)1.)+('_1)n+1€I%! —'_V)n+1 sn+1 fol __t)n(1 +st)v—n—@/2 dt (5.11.2) 15R.E.Langer, Ontheasymptotic solutions ofordinary differential equations, with an application totheBessel functions oflarge order, Trans. Amer. Math. Soc., 33,23(1931); Ontheasymptotic solutions ofdifferential equations, with anapplication totheBessel functions oflarge complex order, ibid., 34,447(1942). 16T.M.Cherry, Uniform asymptotic expansions, J.Lond. Math. Soc., 24,121(1949). Onexpansion ineigenfunctions, particularly inBessel functions, Proc. Lond. Math. Soc., 51,14(1949); Uniform asymptotic formulae forfunctions with transition points, Trans. Amer. Math. Soc., 68,224(1950). sEc.5.11 CYLINDER FUNCTIONS! THEORY |2| with remainder," andintegrating term byterm, weobtain Pf ol\4=® H‘(1)(z) :(%)1/2 ei(z—1/gvn—‘/41\')[ (2Zi)—k +rn(z)]_ Here _<—1)"+1e -v)...<—2z1>"*‘/2’"(Z)' n!F(v+5) ><Ine2z“s"”""‘/¢dsf1(l -t)"(1+st)"‘"'“/1 dt,0 O andwehave used theformula in ds=P0+5)(v+5).<—2zi)-<*+"%>. 0 Rev >-5, 0<argz <1c,k=O,1,2,..., implied by(l.5.l). Now suppose that 8<argz <1:-8,where 8isanarbitrarily small positive number, andforthetime being, assume that Rev —n-%s0. Then, estimating |r,,(z)|, weﬁndthat1*‘ Rev+%at|Im vl l(5-v)....|(2|z|) e"*2"‘ n11P(»+5)lco 1 XI e-2|z|ss1n5sR.ev+n+% dsf __t)n dt 0 0 _l(5—V)n+1i(2i l)R°"+1/’ "'I'“V'F(R@\' ++5)_ _,._1 " (n+1)!llf(v+5)|(;|z| sins)Rev+"+:h '“|2| ) forﬁxed v.Therefore H51)(z) =(7-T2z)1/2 e1<z-%vn—%u>[£o (2zi)-k +0(|zl-1»-1)], Rev >-5, 3—<argz <1:-8,n2Rev —%(5.ll.3) forlarge Actually, thecondition imposed onncanbedropped, since if Rev—n—%>0 1"Note that <1+0“=ZO(-1)*5%-“tr +<-1)"+1‘;‘j,),-*1 <1—r)"<1+:1)“-"-Id». where l=1rs(1+01<e.0).=1. 0).=W =)0+1)---<1 +k-1). 1°Forcomplex aandbwehave |a|>l =|a|H.ab e—1mb-area I22 CYLINDER FUNCTIONSI THEORY CHAP. 5 wecanalways ﬁndaninteger m>nsuch that Rev—m———§~<O. Then, representing H§‘>(z) by(5.11.3) with nreplaced bym,andnoting that ki0...+O(Iz|'"'“1)=k2)... +i +0(|z|""‘1)= = k=n+1 IL =Z +O(|z|'"‘1),k=0 weagain arrive at(5.11.3). Moreover, therelation H51>(z) =__e-vmH£13(z) [cf.(5.6.5)] allows ustoeliminate thecondition imposed ontheparameter v, andinfact, byusing anintegral representation ofasomewhat more general type than (5.10.20), itcanbeshown that theasymptotic formula (5.11.3) remains valid inthelarger sector |arg2|<1-:—8.19Finally, therefore, we have Helm= e*"-‘/2""-‘/*">[ i<—1>'<<».k><21"z)-* +0<|z|-"-1>]»k=0 |arg2|<1:—8(5.ll.4) forlarge |z|,where weintroduce thenotation <»,k)=(+6?e~we+V).=(‘"2‘‘W'32§;.Jk',(4”2 '<2"'“2), (v,0)=1. Anasymptotic representation ofthefunction H,§2’(z) canbeobtained in thesame way, starting from formula (5.l0.2l). Theresult is (2) 21/2 —i(—1/vn—V1t) n '—k —-1H.(Z)= e=24[k§0<v./<><21z> +0<|z|">], [arg2|s1:—8,(5.11.5) which differs from (5.ll.4) only bythesignofi. Asymptotic representations fortheBessel functions oftheﬁrst and second kinds canbededuced from formulas (5.ll.4-5) andtherelations (5.6.l). Thus weﬁndthat2° J,,(z) =(gym cos(2—%vrc-—%1c)[go(—1)"(v, 2k)(2z)‘2" +O(|z| '2"‘2)] 8' -(Q1/2 sin(Z-gm-in) ><[£0(—l)"(v, 21¢+l)(2z)‘2"'1 +0(|z;-2"~=*)], ___ |arg2]<1-:-—8,(5.1l.6) 1”G.N.Watson, op.cit.,p.196. 2°In(5.ll.6—8) theinteger nneed notbethesame inboth sums.R‘ SEC.5.11 CYLINDER FUNCTIONS! THEORY I23 and Y,(z) =<A%Z)1/2 cos(z--5w:—%,;1:) >< (—l)"(v, 2k+l)(2z)‘2"‘1 +O(]z|‘2"“3)] P? O 2 1/2 + sin(Z-gm-gn)[§o(-1)k(v,21<)(2z)-2" +0(]z|'2"'2)], |argz| s1:—8.(5.ll.7) Similarly, asymptotic formulas fortheBessel functions ofimaginary argu- ment canbederived from theintegral representations (5.l0.22, 24),orelseby using therelations given inSec.5.7,inconjunction with formulas (5.ll.4—5). Inthisway, weﬁndthat 1.<z>=e=<2wz>-1'2 <—1>'<(»./<><2z>-k +0<|zl'""1>]Pf +e‘”"“"*V*’(21:z)'1/2[kﬁ0 (v,k)(2z)"‘ +O(|z|'"‘1)], |argz]<1:—8, (5.ll.8) and K,,(z) =(5-:—z)1I2 e'Z['§:0 (v,k)(2z)"‘ +O(|zI)'"‘1], |argz|<1:—8, (5.ll.9) where in(5.ll.8) wechoose theplus signifImz>0andtheminus signif Imz<0.Thesecond term in(5.ll.8) willbesmall if|argz|<$1:—8,and then Iv(z) =eZ(21:z)‘1'2[ i0(—l)"(v, k)(2z)"‘ +0(|z|“""1)]» larg2|é5—8. (5.11.10)Pf The divergent series obtained byformally setting n=ooineach ofthe formulas (5.11.4~lO) istheasymptotic series (seeSec. 1.4)ofthefunction appearing intheleft-hand side. Themethod used here toderive asymptotic expansions gives only the order ofmagnitude oftheremainder term r,,(z), anddoes notfurnish more exact information about thesizeof|r,,(z)|. With suitable assumptions con- cerning zandv,theconsiderations given above canbemodiﬁed toyield much more exact results. Forexample, itcanbeshown“ thatifzandvare 21G.N.Watson, op.cit.,p.206. I24 CYLINDER FUNCTIONS2 THEORY CHAP. 5 positive realnumbers, andifnissolarge that2n2v—1},thentheremainder intheasymptotic expansion ofJ,,(z) orY,,(z) issmaller inabsolute value than theﬁrstneglected term, while thesame istrueoftheasymptotic expansion of Kv(z) ifn>v— 5.|2. Addition Theorems fortheCylinder Functions Given anarbitrary triangle withsides r1,r2andR,let6and:1»betheangles opposite thesides Randr1,respectively (seeFigure 16),sothat R==\/r§ +r§—2r1r2 cos6, sinii;=%sin6. Byanaddition theorem forcylinder functions wemean anidentity oftheform Zv(7\R) =fv(r1,'2,9)Z<1>$""(M)‘I’§/’"’(>\r2)@$'"’(9), (5-12-1)<m> where Aisanarbitrary complex number with |arg7\|<1:(forintegral v,this condition canbedropped), andmranges over some setofindices. Formula (5.l2.l) isanexpansion ofthegeneral I‘ R cylinder function Z,(7\R) inaseries whose terms areobtained bymultiply- 9 '4’ ingsome function fv(r1, r2,6),which is /2 independent ofthesummation index m,bythree factors, each ofwhich de- pends ononly oneofthevariables r1, r2,6. Formulas ofthiskind play animportant roleintheapplications, espe- cially inmathematical physics. Thesimplest such formula isthefollowing addition theorem fortheBessel function oftheﬁrstkind oforder zero:FIGURE 16 100R)=Z1.(m>1..<~2>e""""=-“° ,0 (s.1z.2) =J0(7\r1)Jo(7tr2) +22J,,,()\r1)J,,,(7\r2) cosme.m=1 Toprove (5.l2.2), weﬁrstnote that J(z)= e‘/=2“"'1’t‘"'1dt, n=0,+1,+2,... (s.12.3)" 21:6 _ _ where Cisanarbitrary closed contour surrounding thepoint t=0.22 Intro- ducing anewvariable ofintegration ubywriting t__r1e‘° —r2uR ’ 2”Formula (5.12.3) isaspecial caseof(5.10.7) andcanbeproved immediately by using residues, after recalling theexpansion (5.3.4). SEC.5.12 CYLINDER FUNCTIONS! THEORY I25 andusing thefactthat R2=(Hem —"2)(r1e_m _72), wehave l A l /\ 1d./o()\R) =ETTJ-C’ exp[% (item —@ —%(u—an-5, where theintegration isalong acontour C’resembling C.Moreover, accord- ingto(5.3.4), exp (uew — =mggw J,,,()\r1)e‘""’u”', (5.l2.4) where theconvergence isuniform inuonthecontour C’.Therefore, sub- stituting (5.l2.4) into(5.l2.3) andintegrating term byterm, weﬁndthat °° l lJ.,(iR) =mgwJ,,,(7\r1)e"“°2Tn,L’ eXp[— (-3)]um-1du [\/18w?-1’no‘Q =Z1.0101- ..(-~.>e""° = J.<~.)1..(~.)e*"'@.m=—so m=—-an which proves (5.l2.2). Wenowgivetwogeneralizations offormula (5.l2.2) tothecaseofBessel functions ofarbitrary order v,referring thereader elsewhere forproofs.” Theﬁrstgeneralization isoftheform“ cosval.»_°° cosm6J,(1R) sinW_mzwJ,,,,,(1r,).I,,,(1r,) Sinme. (5.12.5) where upisshown inFigure 16,andr2>r1ifvisnonintegral (forintegral v, thisrestriction canbedropped, i.e.,r1andr2canbeinterchanged). The second generalization of(5.12.2) isgiven bytheformula _ V 0° Jv+m()\r1)Jv+m(7\r2) V_-OR), _2r(»)M;(v+m)i——(lr1),(lr2), cm(cose), vaé0,-1,—2,..., (5.l2.6) where r1andr2arearbitrary. Here thefunctions C,‘{,(x), m=O,l,2,..., known astheGegenbauer polynomials, aredeﬁned asthecoefficients inthe expansion (1-2tx+12)-1=ZC,¥,(x)t"', (5.121)m=0 [sothatthefunction ontheleftisthegenerating function ofthepolynomials C,X,(x)], andhave thefollowing explicit expressions: V [m/2] m_ 1-‘(V +m_k) _ C,,,(x)1;)(-1)~2 2” x“ 2* (5.128) 2°G.N.Watson, op.cit.,Chap. 11. 2‘Formula (5.l2.5) isanabbreviated way ofwriting two formulas, oneinvolving cosines inboth sides, theother sines. I26 CYLINDER FUNCTIONSI THEORY CHAP. 5 [C5(x) =l].Forv=%theexpansion (5.l2.7) reduces toformula (4.2.3), and then theGegenbauer polynomials coincide with theLegendre poly- nomials: C,}/2(x) =P,,,(x). (5.12.9) Forv=0wehave C,‘,’,(x)EO, m=l,2,..., buttheproduct F(v)C,‘,’,(x) approaches aﬁnite limit asv->0: lim)F(v)(v +m)C,‘,’,(x) =2cos(marccosx), m=1,2,...(5.l2.l0) Therefore both formulas (5.l2.5-6) reduce to(5.l2.2) inthelimit v->O. Forcylinder functions ofother kinds, wehave similar addition theorems, among which wecitethefollowing: cosv:I.1 °° cosm6Z,,(7\R) .=2Zv+,,,()\r2)J,,,(7\r1) ..(5.12.11)s1nv<I.1 m=_m smmﬁ w =2~1*(v)§O(v +m) cy,(cos0),(5.12.12) 1* eI.,(kR) =2(—l)"‘Iv+,,,(>\r2)I,,,().r1) Z6, (512.13) Iv()‘R) _ v so m Iv+m()‘ 2)Iv+m()‘ 1) vT); _21*(v)mZ0(-1) (V+m) cm(cos0),(5.12.14) “Q eK,,(>\R) =2Kv+,,,(>\r2)I,,,()\r1) Z6. (5.12.15) 1%? =2VF(v)go(v+m) cy,(cose).(512.16) Informulas (5.12.l1—l3, 15-16), itisassumed that r2>r1unless visan integer orZ,+,,, =Jv,,,, in(5.12.12). Animportant special case ofthese addition theorems, encountered in mathematical physics, occurs when v=-1».Theformulas corresponding to thiscase areeasily obtained byusing (5.l2.9), together with theresults of Sec.5.8.25 5.I3. Zeros oftheCylinder Functions Insolving many applied problems, oneneeds information about theloca- tionofthezeros ofcylinder functions inthecomplex plane, andinparticular, 25G.N.Watson, op.cit.,p.368. sEc.5.13 CYLINDER FUNCTIONS! THEORY I27 onemust beable tomake approximate calculations ofthevalues ofthese zeros. Here wecitewithout proof some important results along these lines.” Webegin byconsidering thedistribution ofzeros oftheBessel functions of theﬁrstkind, i.e.,roots oftheequation .I,,(z) =0. (5.13.l) Theorem 1deals withthecaseofnonnegative integral v,andTheorem 2with thecaseofarbitrary realv: THEOREM l.Thefunction J,,(z), n=0,l,2,...hasnocomplex zeros, andhasaninﬁnite number ofrealzeros symmetrically located withrespect tothepoint z=O,which isitself azeroifn>0.Allthezeros ofJ,,(z) are simple, except thepoint z=0,which isazerooforder nifn>0. THEOREM 2.Letvbeanarbitrary real number, andsuppose that [argzl<1:.Then thefunction J(,(z) hasaninﬁnite number ofpositive real zeros, andaﬁnite number 2N(v) ofconjugate complex zeros, where l.N(v) =Oifv >—lorv =—l, —-2,...; 2.N(v) =mif—(m +1)<v<—m, m=1,2,... (Inthesecond case, if[—v] isodd,there isapairofpurely imaginary zeros among theconjugate complex zeros.) Moreover, allthezeros aresimple, except possibly thezeroatthepoint z=O. Thefollowing generalization ofequation (5.l3.1) isoften encountered in mathematical physics (AandBarereal): AJv(z) +BzJ§(z) =O, v>—l, |argz| <1:. (5.l3.2) Itcanbeshown thatthisequation hasinﬁnitely many positive realroots and nocomplex roots, unless 24,-+v<0, inwhich case(5.l3.2) alsohastwopurely imaginary roots.” The distribution ofzeros ofthefunction Iv(z) canbededuced from Theorem 2andtherelations ofSec.5.7.Inparticular, itshould benoted that allthezeros ofI,,(z)arepurely imaginary ifv>—1.Ifvisreal, Macdonald’s function K(,(z) hasnozeros intheregion |argz|<1:/2. Intherestofthe z-plane cutalong thesegment [—oo,0],K,,(z) hasaﬁnite number ofzeros.” 2“The problem ofthedistribution ofthezeros ofcylinder functions isalso ofcon- siderable theoretical interest, butliesoutside thescope ofthisbook. Weagain refer the reader interested indetails tothespecialized literature, e.g., Chap. 15ofWatson’s treatise. Itshould benoted thatsome oftheresults onzeros ofcylinder functions can bederived byarguments ofacompletely elementary character. 2"G.N.Watson, op.cit.,p.482. 2°Ibid., p.511. I28 CYLINDER FUNCTIONSI THEORY CHAP. 5 Tomake approximate calculations oftheroots ofequations involving cylinder functions, onecanusethemethod ofsuccessive approximations, where inmany cases agood ﬁrstapproximation isgiven bytheroots ofthe equations obtained when thecylinder functions arereplaced bytheir asymp- totic representations. 5.I4. Expansions inSeries andIntegrals Involving Cylinder Functions Inmathematical physics, itisoften necessary toexpand agiven function interms ofcylinder functions, where theform oftheexpansion depends on thespeciﬁc nature oftheproblem (seeSecs. 6.3-6.7). Wenow consider the most important ofthese expansions, whose roleinvarious problems involving cylinder functions resembles thatofFourier series andFourier integrals in problems involving trigonometric functions. Foremost among such expan- sions areseries oftheform fr=¢,,,J,x.,,5, O<r<a, v>- (5.14.1)H<1Z(Q) -1. where f(r)isagiven realfunction deﬁned intheinterval (0,a),J,,(x) isa Bessel function oftheﬁrstkind ofrealorder v2-inand O<x,,1< ---<xv,,,< arethepositive roots oftheequation .Iv(x) =0.Theexpansion coefﬁcients cmcanbedetermined byusing anorthogonality property ofthesystem of functions J.(x,,, m=1,2,..., (5.142) which isproved asfollows: LetozandBbedistinct nonzero realnumbers, and let 1/ 1/ 2 V2 II 1 I 2 V2 l.la+~;_lla+d"-;_—2'lla=O, HD+-I-_UB+B—72llB=0 betheequations satisﬁed bythefunctions um=Jv(otr) and u,,=J,,(Br). Subtracting thesecond equation multiplied byru,,from theﬁrstequation multiplied byrug,andintegrating theresult from 0toa,weﬁndthat ll I1 (ot2—(32)Lru,,u,, dr=r(u.,u,’, —u[,u,’, which implies La ',Jv(“r)Jv(Br) dr = : SEC.5.14 CYLINDER FUNCTIONSI THEORY I29 ifv>-1.Setting or=x.,,,,/a, [5=xv,/a in(5.l4.3), weobtain theformula larJ,(x,,,, I)J,(x,,1) dr=0ifm44n, (5.14.4)0 a a which shows that thesystem (5.l4.2) isorthogonal with weight ronthe interval [0,a](seeSec.4.1). Taking thelimit of(5.l4.3) as[5—>(Z,with theaidofL’Hospital’s rule, andusing Bessel’s equation toeliminate JZ,weﬁndthat29 a 2 2 I113(1)”) dr="5[J,j2(aa) +(1--§_,)J3(m)]. (5.14.5)0 U.£1 or,using therelations (5.3.5), a 2 2 fr1t(x...5)41=%J:-'*<x..>=a;1a..<x...>. <5-14-6)0 Then, assuming thatanexpansion oftheform (5.14.1) ispossible, multiplying byrJ,,(x.,,, r/a)andintegrating term byterm from 0toa,weobtain thefol- lowing formal values ofthecoefﬁcients cm: 2 “ rCm = J0 rf(r).I(,(xv,,, dl‘, m=1,2,... Theseries (5.l4.l), with coefﬁcients calculated from (5.14.7), iscalled the Fourier-Bessel series ofthefunction f(r). Wenowciteatheorem which gives conditions under which theFourier- Bessel series ofthefunction f(r)actually converges andhasthesumf(r): THEOREM 3.3°Suppose therealfunction f(r)ispiecewise continuous in (0,a)andofbounded variation inevery subinterval [r1,r2],3‘ where 0<r,<r2<a.Then, iftheintegral I“~/7lf(r)l41 isﬁnite, theFourier-Bessel series (5.l4.l) converges tof(r)atevery con- tinuity point off(r), andto %[f(r +0)+f(' —0)] atevery discontinuity point off(r). Next, weconsider animportant generalization oftheconcept ofa Fourier-Bessel series. Suppose thefunction f(r)isexpanded inaseries ofthe form (5.l4.l), where thistime thenumbers 0<x,,, <---<x,,,,, <--- ’9Thedetails aregiven inG.P.Tolstov, op.cit.,p.218. 3°Fortheproof, seeG.N.Watson, op.cit.,p.591. 31Concerning functions ofbounded variation, seeE.C.Titchmarsh, op.cit.,p.355. I30 CYLINDER FUNCTIONS; THEORY CHAP. 5 aretheroots oftheequation AJ.,(x) +BxJ§(x) =0, (5.148) instead oftheequation Jv(x) =0.Then itisanimmediate consequence of formulas (5.14.3, 5,8)that a r 0 ifm¢n, Jv( vm wt“) d = 2 2 - lb’"4"4'“5[Jc*(x..) +(1—):T)J3(xvn)] 1rm=11. (5.14.9) andtherefore thecoefﬁcients c,,,arenowgiven by 2 G ""=4*11e(x...)+11 -(v2/>e..)1Je<x..)1 ’f(’)’"l"""§l ""‘5"“"‘°) Theseries (5.l4.l), with coefﬁcients calculated from (5.l4.l0), iscalled the Diniseries” ofthefunction f(r). Iff(r)satisﬁes theconditions ofTheorem 3, andifAB“1+v>0,then theDini series off(r) actually converges tof(r)at every continuity point.” Both Fourier-Bessel series andDini series play an important roleinproblems ofmathematical physics, andexamples ofsuch expansions willbegiven inSecs. 6.3and6.7. Wenow turn toexpansions ofafunction f(r) deﬁned intheinﬁnite interval (O,00),interms ofintegrals involving Bessel functions. Among such expansions, theoneofgreatest practical importance istheFourier-Bessel integral, deﬁned by /to=wv.or>d1 wP-7v(7\P)f(P)dP, 0<1<co.»>-5. (5.14.11) Formula (5.14.l 1)issometimes called Hankel’s integral theorem, andisvalid atevery continuity point off(r) provided that 1.Thefunction f(r), deﬁned intheinﬁnite interval (0,oo),ispiecewise continuous and ofbounded variation inevery ﬁnite subinterval [r1,r2], where 0<r1<r,<oo; 2.Theintegral lw~/Firm! dr isﬁnite.“ 32Called aFourier-Bessel series ofthesecond type inG.P.Tolstov, op.cit.,p.237. 3“For theproof, seeG.N.Watson, op.cit.,p.596ﬁ'.,where onewillalso ﬁnd the modiﬁcations that must bemade intheDini series ifAB“ +v€0. 3*Fortheproof, seeG.N.Watson, op.cit., p.456ff.Atdiscontinuity points, the integral intheright-hand side of(5.4.11) equals %[f(r +0)+f(r—0)]- SEC.5.15 CYLINDER FUNCTIONSI THEORY l3l Asexamples ofFourier-Bessel integrals, consider theexpansions 1 0°_Z =J; €MlJ0()\I') dh, —k‘/22 rz co _i 1;: =Ie—|»=-'|~/H+k2 M d;\ (5_14_13) 0 \/z2 +r \/X2 +k2 (with realzandr),implied byformulas (5.15.1, 7)below. Theauthor hasstudied another integral expansion ofacompletely diﬁ"erent type, involving integration withrespect totheorder ofthecylinder function.“ Thisexpansion, which turns outtobeveryuseful insolving certain problems ofmathematical physics (seeSecs. 6.5—6) isoftheform 2°°- Kn) °° KnE)f(x)=?L1s1nhnrT(; d1Lf(a)-7%dE, x>0,(s.14.14) where Kv(x) isMacdonald’s function ofimaginary order v=i1.Formula (5.l4.l4) isvalid atevery continuity point off(x) provided that 1.Thefunction f(x), deﬁned intheinﬁnite interval (O,oo),ispiecewise continuous and ofbounded variation inevery ﬁnite subinterval [x1,x2], where 0<x1<x2<oo; 2.Theintegrals lo“lf(x)]x'1'2 log;dx, fjz|f(x)|dx (5.14.15) areﬁnite. Example. Anexpansion ofthistype is36 f(x)=We->= e=gJ: d1‘. (544.16) 5.l5. Deﬁnite Integrals Involving Cylinder Functions Intheapplications, itisoften necessary toevaluate integrals involving cylinder functions incombination withvarious elementary functions orspecial 35N.N.Lebedev, Suruneformule d’inversion, Dokl. Akad. Nauk SSSR, 52,655 (1946); Expansion ofanarbitrary function inanintegral with respect tocylinder functions ofimaginary order andargument (inRussian), Prikl. Mat. Mekh., 13,465(1949); Some Integral Transformations ofMathematical Physics (inRussian), Dissertation, Izd. Leningrad. Gos. Univ. (1951). Atdiscontinuity points, theintegral intheright-hand side of(5.l4.l4) equals *l[f(X +0)+f(x-0)]- “6Toderive (5.14.16), use(5.l4.l4) andtheBateman Manuscript Project, Tables of Integral Transforms, Vol.1,formula (24), p.197. I32 CYLINDER FUNCTIONS! THEORY CHAP. 5 functions ofother kinds. Such integrals areusually evaluated byreplacing thecylinder function byaseries orbyasuitable integral representation, and then reversing theorder inwhich theoperations arecarried out. Since an extremely detailed treatment ofthiswhole topic isavailable intheliterature,“ weconﬁne ourselves heretoafewexamples which illustrate themethod and leadtosome results needed later inthebook. Example 1.Evaluate theintegral fooe"“‘Jo(bx) dx, a>0,b>0.O Replacing J0(bx) byitsintegral representation (5.l0.8), weﬁndthat co uo 11:2 Le“"‘J0(bx)dx=L e—=~=dx%f0' cos(bxsin4»)as 2 1:/2 w _ =—J~ doIe'“" cos(bxsintp)dx7‘0 0 _ZJW:/2 ad‘? _1: 0a2+b2sin2<p’ where theabsolute convergence ofthedouble integral justiﬁes reversing the order ofintegration. Evaluating thelastintegral, wehave ~=> 1loe“”‘J(,(bx) dx=7‘ﬁ=+-F, a>0,b>0. (s.1s.1) Example 2.Evaluate Weber’s integral fooe“’“”"'.I.,(bx)x""‘1 dx, a>0,b>0,Rev >-1.0 Replacing J.,(bx) byitsseries expansion (5.3.2) andintegrating term byterm, weﬁndthat coH1212 V _ 00_u2x2 V °°(_1)k(bx/2)v+2k Le J.,(bx)x *1dx-J0e x*1dxkzoi———k!F(k +V+1) _0° (_1)k (§_)v+2kJm —a2x2 2v+2lc+1d ‘go/<!r(k+»+1) 2 06" X °°(-1>'< b 1e=,2,/<!r(k +V+1)li) 2a2v+2'<+2l., e’d’ bv E:(__b2/4a2)k :(2a2)v+1 kzo k! 3"G.N.Watson, op.cit.,Chaps. 12-13, theBateman Manuscript Project, Higher Transcendental Functions, Vol. 2,Chap. 7,and ibid., Tables ofIntegral Transforms, Vols. 1,2.Seealso F.Oberhettinger, Tabellen zurFourier Transformation, Springer- Verlag, Berlin (1957). SEC.5.15 CYLINDER FUNCTIONS2 THEORY I33 where reversing theorder ofintegration andsummation isagain justiﬁed by anabsolute convergence argument. Summing thelastseries, wehave co by L e—a.2x2Jv(bx)xv+1 dx = e—b2/40,2, a>0,b>O, Rev> —l. Example 3.Evaluate theintegral °°x“*1J(,(bx) i‘ﬂ 1dX, a>0, b>0, 1<RCV<2RelL+;, often encountered intheapplications. First wereplace thefunction (x2+a2)‘“‘1 byanintegral ofthetype(l.5.l), i.e., 1 _ 1 0°-—(x2+a2)t _(X,+a,),,, _W+1)lo e 1“dt,Re(J.>1,(515.3) assuming temporarily that —l<Rev<2Reit+%(this guarantees abso- luteconvergence oftherelevant double integral). Then, using (5.l5.2) and theintegral representation (5.lO.25) ofMacdonald’s function, weﬁndthat onxv+1Jv(bx) _ 1 O0-<12: J00 —x2t v1I0(X2+a2)u+1 dx-Wfo e t“dt0eJv(bx)x “dx __ by w -a2i—(b2/4t) dt _2v+1I'\(H+ 1)0e tv+1—u bva2v—2u. 00 _u_ a u du av—ubll -WT) Kt-"(“”)' Theextension ofthisresult tovalues oftheparameter (1.satisfying theweaker condition -1<Rev <2Rep.+%isaccomplished byusing theprinciple ofanalytic continuation. Thus wehave °°x”*1J(,(bx) _ a“‘“b“ lo(x2+a2)“*‘dx_2“l“(11+1)K"‘“(“b)’ (5.154) a>0,b>0, -1<Rev<2Rept+%. Inparticular, setting (1=——%,v=0andusing (5.8.5), weobtain theintegral 0°"J°(b") d6'“ 0b0 515 i =——--9 / , , ,,5 ll,\/x2+a2 "1»”> > (l Example 4.Evaluate theintegral °°K(M/X’ +1*’) 1 yo — Jv<I).X).X +1dx, a>0, b>0, y>0, Rev>—1, I34 CYLINDER FUNCTIONS! THEORY CHAP. 5 which alsohasnumerous applications tomathematical physics. Using the integral representation (5.lO.25) andformula (5.l5.2), weﬁndthat °°K(M/X” +1'2) Vloix,+y,),,, .I(,(bx)x +1dx _all 0°bv+1d 0°—t—[a2(x2+142)/4t] db _2uH 0J.,(x)x x0e [M1 an no-2-(112 2/4:) dt 0°-112-762/4! v+1=-W e 1’W; e J.,(bx)x dx2 o F 0 2v—u 11-2v-2 vno—t(1+b2/t12)—(a21/2/4t) dt= (1 b 6 "I1,0 I =Zvwbv (az+b2)u.—v—1 °°e—u—[y2(a2+b2)/4u]_iiLall uu-—v0 =(-L;"")“‘"“‘1<.-.-.<yvﬁ?>. Bychoosing various values oftheparameters intheidentity LooK———%ix;;;ug2) Jv(bx)x" ”1dx=g(______\/a2y+ bz)u—V_1K,_V_1(y\/11-23), a>0, b>0, y>0, Rev>—l, (5.l5.6) wecanderive anumber ofuseful formulas encountered intheapplications. Forexample, setting p.=i,v=0,wehave noe—a~/x2+112 e—y*/a2+b5 J0 J0(bX)X dx = 5.l6. Cylinder Functions ofNonnegative Argument andOrder Wenow collect some elementary andeasily veriﬁed results pertaining to theveryimportant caseofcylinder functions where both theargument xand theorder varenonnegative realnumbers: l.Bessel functions oftheﬁrst kind. Forx20andv2O,thefunction Jv(x) isrealandbounded, andhasanoscillatory character. Itsbe- havior forsmall andlarge values ofxisdescribed bytheasymptotic formulas xv Jv(X) ~ 2 X—>0, (5.l6.1) Jv(x) z cos(x—»}w:—;1;1c), x—> oo. sac.5.16 CYLINDER FUNCTIONS! THEORY I35 J,(x) hasinﬁnitely many zeros, including thepoint x=0ifv>O. Thegraphs ofJ,,(x) andJ1(x) areshown inFigure 17. l +1‘ 1/1lXl I|l 0 1 t/olll —1O 5 1O 15 20 25 FIGURE 17 2.Bessel functions ofthesecond kind. Forx>0andv>0,thefunction Yv(x) isanoscillatory realfunction, which isbounded atinﬁnity. Its behavior forsmall andlarge values ofxisdescribed bytheasymptotic formulas Y.,(x)z—% x—>0, v>O, Y,,(x)z,/%sin(x-gm-in), x—>66, (5.16.2) 22Yo(x) z—;log;, x—>0, which show, inparticular, that Y.,(x) —>-ooasx—>O. 3.Bessel functions ofthethird kind. Forx>0andv>0,theHankel functions H§1’(x) andHf,2’(x) areconjugate complex functions, which arebounded atinﬁnity. Their behavior forsmall andlarge values of xisdescribed bytheasymptotic formulas H§”’(x) z$i(2)V l)» x—>0, v>O,x 1: H§"’(x) z,/gce*“"‘1/M-r~=>, X->66, (5.163) H(§”’(x) ziiélog £5 x—>O, I36 CYLINDER FUNCTIONSI THEORY CHAP. 5 where theupper signcorresponds tothecasep=1,andthelower sign tothecasep=2.Obviously, H§">(x) ->ooasx->0. 4.Bessel functions ofimaginary argument. Forx>0andv>0,I(,(x) is apositive function which increases monotonically asx-> oo,while K(,(x) isapositive function which decreases monotonically asx—>co.“ Forsmall xwehave theasymptotic formulas xv I\,(X) ~ ! X9 0, v—1 1<,(x)z X->0, (5.16.4) K0(x) zlogit x—>O, andtherefore I,(0) =0 ifv>O, 10(0) =1,K(,(0) =oo. Theasymptotic behavior ofthese functions asx—>ooisgiven by ex Iv(X) z ’ X9 Q), (5.165) K(,(x) z e"‘, x—>00. Clearly, neither function hasanyzeros forx>0. 5.I7.Airy Functions Thesolutions ofthesecond-order linear differential equation u”—zu=0 (5.17.1) arecalled Airyfunctions. These functions areclosely related tothecylinder functions, andplayanimportant roleinthetheory ofasymptotic representa- tions ofvarious special functions arising assolutions oflinear differential equations.” Inparticular, theAiry functions turnouttobeuseful inderiving asymptotic representations ofthecylinder functions forlarge values of|z| and|v|,valid inanextended region ofvalues ofzandv.TheAiry functions alsohave avariety ofapplications tomathematical physics, e.g.,thetheory ofdiﬁraction ofradio waves around theearth’s surface.“ 3°This factabout Kv(x) follows from theintegral representation (5.l0.23). 39SeeR.E.Langer, op.cit.,T.M.Cherry, op.cit.,andV.A.Fock, Tables ofthe Airy Fuctions (inRussian), Izd. Inform. Otdel. Nauchno-Issled. Inst., Moscow (1946). 4°SeeV.A.Fock, Diﬂraction ofRadio Waves Around theEarth’s Surface (inRus- sian), Izd.Akad. Nauk SSSR, Moscow (1946). sEc.5.17 CYLINDER FUNCTIONS2 THEORY I37 Wenowpresent therudiments ofthetheory ofAiry functions. Choosing at=—1,Y=1inthesecond oftheequations (5.4.11—12), andusing there- sultsofSec.5.7,weﬁndthatthegeneral solution of(5.17.1) canbeexpressed interms ofBessel functions ofimaginary argument oforder v=i§.Inparti- cular, twolinearly independent solutions of(5.l7.1) are Z1/2 223/2 223/2 14="1=Ai(z) =Y[I-1/ah?) 7"I1/s(_3_')] 1 1/2 2a/2 2 E7-C K1,3(zT), [arg2|<g, _ 1/2 2:1/2 23/2 2 U=U2 =Bl(Z) = [I-1/3(Z?') +I1/3(—Z§—):|$ larg Zl<£9 (5.17.2) called theAiryfunctions oftheﬁrst andsecond kind, respectively. Replacing I*1/3bytheseries expansion (5.7.l), weobtain theexpansions °° 3k “O Z3k+1 A’ = 2% “ —i."_—'i’ 5‘lzl,2,32""/*k!l‘(k +4).2.,32""/=~k!I‘(k +%)IZI<°° . °° Zak 0° Zak-I-1 M2)=31/2lg.32”’/1k!I‘(k +%)+..Z..32"+%k!l‘(k +%)l’IZI<°°’ (517.3) which show thattheAiry functions areentire functions ofz. Wecanalsowrite (5.l7.3) inanother, somewhat more concise form. For example, theﬁrstexpansion isequivalent to -2 Ail’)=33/6,2, I35/8)’[z|<66.(5.17.4) Using the“triplication formula” forthegamma function [Problem 4,formula (i),p.14]wecantransform (5.17.4) into k 1.23_2,3 66 s1n%(k +1) Ai(z) =Tc20 k! (31’3z)", |zl<oo.(5.17.5) Itfollows from _(5.l7.3) that theAiry functions Ai(z) andBi(z) canbe deﬁned asthesolutions ofequation (5.17.1) satisfying theinitial conditions ‘ 3-2/3 I ‘I 3-4/3 141(0) =A1(0) =‘IE’ 111(0) =A1(0) =—T%)’ 112(0)=151(0)= 115(0)=Bi’(0)=(5.17.6) I38 CYLINDER FUNCTIONS2 'rHEoRY CHAP. 5 TheWronskian ofthispairofsolutions is W{Ai(z), Bi(z)} =W{Ai(z), Bi(z)}z=0 =it (5.17.7) where weagain usethetriplication formula forthegamma function.“ We canalsocalculate (5.17.7) directly from (5.17.2) and(5.9.5). Asymptotic representations oftheAiry functions forlarge |z|canbe deduced from thecorresponding results ofSec.5.11. Inparticular, wehave .rc—1/2 Ai(z)=TZ-1/*6-'/==='"[1 +O(|z|'3’2)], larg2|<3;-s,(517.8) 131(2)=Tr‘/22'1"-‘e’/@Z”’[l +0(lZ|_3I2)], [arg1|<g-s.(517.9) Itfollows atonce from (5.17.3), (5.7.1) and(5.3.2) thattheAiry functions of argument -zcanbeexpressed interms ofBessel functions oftheﬁrstkind oforder v=i§;: . 21/2 2717 A1(*Z) =TI-1-1/a(%Z3I2) +J1/a(%Z3/2)], largZl<?' (5.l7.lO) . z1/2 27': B1<-z)= (3)11_.,.ea/2) -1.,.ez")1. largzl<T Then, using theasymptotic representation (5.11.6),weﬁndthat Ai(—x) z7c“l2x‘1"‘ cos(:3-xalz —E), x—> oo, (5.17.11) Bi(—x) z-7:”/2x"‘/4 sin x3’2—Z), x—> oo, which shows thattheAiry functions have anoscillatory character forlarge negative values oftheargument. Finally, wenotethatthedeﬁnition ofAi(x) andtheintegral representation ofMacdonald’s function given inProblem 6,formula (ii),p.140,imply . 2x1l2 °° 2x3’2 . yAi(x) —Y J;cos(T s1nhy)cosh 5dy, x>0. After making thesubstitution sinhg =%x‘1'2t, thisgives thefollowing integral representation ofAi(x): .1"0Al(x)=Efa66$(ire+xt)dt, x>0. (5.17.12) *1Foraproof oftheﬁrstequality in(5.17.7), cf.E.A.Coddington, op.cit.,Theorem 8,p.113. PROBLEMS CYLINDER FUNCTIONS! THEORY Asomewhat more complicated argument gives thefollowing integral repre- sentation ofBi(x):*2 Bi(x)=iIn[e"/"‘“""‘ +sin(.313+xt)]dt, x>0.O Foranintegral representation of[Ai(x)]2, seeProblem 22,p.142. PROBLEMS 1.Derive theintegral representation“ 1! 7!/2 J§(z) = I2Jz11(2Z cos0)d6=(-1)" J6(2z cos6)cos2n0d6, 0 O n=0,1,2,... 2.Derive thefollowing formula involving products ofBessel functionsz“ 11/2 J,.(z)J.,(z) = J,.,..,(2z cos6)cos(pt—v)6d6, Re((1.+v)>-1. O 3.Prove that 11v Mi? J,.(z)J,,(z’)= gjl J0(\/22 +z’2—2zz’cos6)cosn6d6, n=0,1,2,...O Hint. Usetheaddition theorem (5.l2.2). 4.Derive theintegral representations Jv(x) =gfm sin(xcosht - coshvtdt, -1< Rev <1, x>O, O 2°° v11: Yv(x)= —-r cosxcosht—-5 coshvtdt, —1<Rev<1, x>0.0 Hint. Useformulas (5.10.14, 15). 21/2et(z— l/2vn—l/411:) co 1 Sv-V, <1) =_ M —v-/ ____Hy(z) To+2)Le‘s1121,2 ds, Rev> -2, ——<argz<71:,5.Derive theformulas (4) \_/< $102 .3“21/2 —l(z-1/2vn—l/,7!) co -5v_ — ”i”’<’>=l;2) "’y’l‘+§2 Rev >-1}, -1:<argz < ‘*2H.Jetfreys andB.S.Jeffreys, op.cit.,p.510. 4“G.N.Watson, op.cit.,p.32. *1Ibid.,p.150. I40 CYLINDER FUNCTIONSI THEORY CHAP. 5 6.Prove thefollowing integral representations ofMacdonald’s function?“ V; zv 0° -2 COSII l ' 2VKv(Z) = 06 Slllh ldl, RC2 >0, RCV >-2-, 2vI1 1 <10 ‘ Kv(x)= J; dt, x>0, Rev> -5., (1) K.,(x)=-177: loo.566(XS1I'1h 1)66611vtdz, x>0,|Rev|<1, (ii) cos— °2 7.:1/2 e-2 w _sv_V Sv-5/2 Kv(Z) — J‘0 6S 2 ‘l' dS, largzl <1:,Rev >--1;. 7.Prove thefollowing formulas involving products ofMacdonald functions :46 Kv(x)Kv(y) =éfw e‘!/z[t+(x2+y2)It]Kv(.'%/)%l 0 =fooK0(\/x2 +yz+2xycosh t)cosh vtdt, x>O,y>0, 0 K.,(x)K.,(y) = (I)J0(\/2xy cosht —xz—y2)sinh vtdt,OE yI x>0,y>0,[Rev|<1. (iii) 8.Derive theintegral representation 1=0 Mi? Iv(x)K.,(y) =-2]; (I)Jo(\/2xy cosh t—x2-yz)e““dt,Og 1/I x>0, y>0, Rev>-%. 9.Derive theintegral representation K,,(x)K.,(x) =faK,,_.,(2x cosh coshIJ%_—vtdt, x>0,y>0.O 10.Derive thefollowing asymptotic representations forlarge values ofthe order |v|: 1 JV(Z) z7? ev-1-vl0g(z/2)-(v+'/g)loE v’ lv|__)oo, |arg v|<N_8, 7t '\ 1/2 K,.(x)z it e""'2sin 5+ 'rlOg'c— -r--rlogf . -r:—>00.1 4 2 (Inthesecond formula, xisaﬁxed positive number.) ‘*5G.N.Watson, op.ci't., 172, 183. ‘*6Concerning Problems 7-9,seeibid., p.439. Themost detailed investigation of various integral representations ofproducts ofcylinder functions isduetoA.L.Dixon andW.L.Ferrar, Integrals fortheproduct oftwoBessel functions, Quart. J.Math. Oxford Ser., 4,193(1933); Part II,ibid., 4,297(1933). PROBLEMS CYLINDER FUNCTIONSZ THEORY I4I ll.Prove theformulas Jv(—x +i0)—J\,(-x —i0)=2isinvr:J.,(x), Yv(—x +i0)—Y.,(-x -i0)=2i[J.,(x) cosvrr+J_.,(x)], H§,”(-x +i0)-H.‘,”(—x -i0)=—2[J_.,(x) +e"”“Jv(x)], H$2’(-x +i0)—H$2’(-x —i0)=2[Jv(x) +e"“.IV(x)], where x>O,characterizing thebehavior ofthecylinder functions onthe cut[-oo,0]. 12.Verify that Iv(-x +i0)-Iv(—x -i0)=2isinvrcI.,(x), K.,(—x +i0)—Kv(—x -i0)=—-r:i[I_v(x) +I.,(x)], where x>0. Comment. The formulas given inProblems 11-12 take aparticularly simple form ifv=n(n=0,i1,12,...). 13.Verify theexpansion fa-Iv (I) dt =2§ Jv+2k+1(Z), RCV > -1. O lC=D Hint. Usetherecurrence relation (5.3.6) toshow that both sides have the same derivative. 14.Derive therecurrence relation fa1~J.(¢)at=z“Jv+1(z) -(6-v-1)r1~*-1J...(1)d1, Re(11+v)>-1.O O Hint. Apply (5.3.5) intheform v+1 _.d V+11J.(t)-El: J.,.,1(t)], andthenintegrate byparts. 15.Using theresult ofProblem 14,show that theevaluation ofintegrals of theform J:t"‘J.,(t)dt, R6»>-1,m=o,1,2,... reduces totheevaluation oftheintegral I:J....(r>dt. whose value wasfound inProblem 13. Comment. Ifv=i(m -1),i(m -3),i(m -5),...,then theco- efﬁcient ofthelast integral vanishes, and the original integral can be expressed inclosed form interms ofBessel functions. I42 CYLINDER FUNCTIONSZ THEORY CHAP. 5 16.Verify theformula“ 0°H 1- I-§)dx=-—-i-» Reu>§, Re(v—11)>-1. ° 2“1“/>+ Ni+1;'1: 17.Verify theformula fwe'“”Jv(bx)dx= , Rev> -1, a>O, b>0. 0 b"\/a2 +b2 18.Show that theBessel function Jo(x) satisﬁes thefollowing integral equal tion: 21°'J°(x)=;L J0(y)dy, 0<x<66. 19.Theintegral Bessel function oforder visdeﬁned bytheformula Jiv(z)=fziﬁgdt, |argz]<rt. Show thatJi.,(z) isanentire function ofvandananalytic function ofzinthe plane cutalong thesegment [—oo, O](infact, anentire function ofzfor v=i1,i2...). Verify theformulas vJiV(z) = at-1, (iv)O vJiv(z) =f2J.,_1(t) dt—J.,(t) -1, Rev >0,|arg2|<7:. 0 Hint. Usetheresults ofProblems 14and16. 20.Prove thefollowing expansions oftheintegral Bessel functions: 66 _16 21¢ 11.<z)=1<>g§ +Y+ 14<<4.|argz|<6 _ _ 1 66 (_1)k(z/2)2lc+n. _ .l1,,(z) -—:1+kg)-Wk+n)k!(n +k)!’ ]z|<oo, n-1,2,... Hint. Substitute (5.3.2) into Problem 19,formula (iv). 21.Derive theasymptotic formula 1.6.).(g,)1'2__.._sir1<»¢ -in-1->. 22.Prove theintegral representation [Ai(x)]’ =47%;-low 1., 18+xt)tdt, x>0 forthesquare oftheAiry function oftheﬁrstkind. Hint. UseProblem 7,formula (iii). 47G.N.Watson, op.cit.,p.391. CYLINDER FUNCTIONS: APPLICATIONS 6.l. Introductory Remarks Asalready noted inSec. 5.1,thecylinder functions have avery wide range ofapplications tophysics and engineering, which cannot even be touched upon inabook ofthissize. Instead, weconﬁne ourselves toadis- cussion ofafewselected problems ofmathematical physics involving cylinder functions,‘ where theselection hasbeen made with theaimofillustrating theapplication ofthetheory ofChapter 6.Wearemainly concerned with thesolution ofboundary value problems forvarious special domains. In addition toseveral examples ofanelementary character, weinclude some that aremore complicated, e.g.,theDirichlet problem forawedge (seeSec.6.5). 6.2.Separation ofVariables inCylindrical Coordinates Consider thepartial differential equation l82u 8uV2=-—- — .. ua2at2+bat+cu, (621) where V2istheLaplacian (operator), tisthetime, anda,b,caregiven con- stants. Avariety ofimportant differential equations occurring inmathe- matical physics (e.g., inelectrodynamics, thetheory ofvibrations, thetheory ofheatconduction) arespecial cases of(6.2.l). Theboundary conditions im- posed onthefunction uoften require theuseofasystem ofcylindrical 1Weassume thatthereader hasalready encountered thesimplest problems ofthis typeinaﬁrstcourse onmathematical physics. I43 I44 CYLINDER FUNCTIONS; APPLICATIONS CHAP. 6 coordinates r,<p,z,related totherectangular coordinates x,y,zbythe formulas x=rcoscp, y=rsin<p, z=z, where O<r<oo,—1c<<p<n, —oo<z<oo. Incylindrical coordinates, equation (6.2.l) becomes 1a a 182 ea 162 a 75001:) +72aq>:+ azgzﬁézg ‘Lbail+‘”’ (612) andhasinﬁnitely many solutions oftheform u=R(r)Z(z)<D(<p)T(t), (6.2.3) where each ofthefunctions ontheright depends ononly onevariable. Sub- stituting (6.2.3) into(6.2.2) anddividing byRZ<1>T ,weobtain 1d dR ld2<1> 1d2Z 11d2T Rrir lrdr) +r2(D d<p2 +2dzz —C=Tl???‘ +bT)' (614) Since thevariables r,rp,zandtareindependent, both sides of(6.2.4) must equal aconstant, which wedenote by-—x2. This leads totwoequations 1d2T dTF725 "l"b-2? -l->t2T= 0 (62.5) and 1d dR 1d2<D 1d2Z '§;Jrl'71;l+"2+;@T,>2‘=""2?.1Z—2" Thesame reasoning shows thatboth sides ofthelastequation must equal a constant, which thistime wedenote by—>\2, obtaining theequations d2ZF-(A2+¢)z=0 (62.6) and 1d dR ld2<D '2li<va;l’a;l *‘*2*"2’l="6%‘Again, both sides ofthelastequation must equal aconstant, denoted byp.2, which implies d2 "ZED? +lL2(I) =0 and 1ddR 2H;(r-5)+(7?+>8-;l%)R=0. (6.2.s) The process just described iscalled separation ofvariables, andleads to sec.6.2 CYLINDER FUNCTIONSZ APPLICATIONS I45 inﬁnitely many solutions oftheform (6.2.3), depending ontheparameters x,1,pi,which cantakerealorcomplex values? Thus, determining thefactors intheproduct (6.2.3) reduces totherela- tively simple problem ofsolving theordinary differential equations (6.2.5-8). Theﬁrstthree ofthese equations canbesolved interms ofelementary func- tions, butifweintroduce anewvariable proportional tor,thefourth equation becomes Bessel’s equation, whose solutions involve cylinder functions. The required solution ofthegiven physical problem isobtained bysuperposition oftheparticular solutions (6.2.3), where thespeciﬁc conditions oftheproblem dictate thechoice oftheparameters x,A,itandthecorresponding solutions of(6.2.5-8). Finally, wecallattention totwoimportant special cases ofequation (6.2.l), obtained bymaking certain choices oftheconstants a,bandc: 1.Laplace’s equation Vzu=0(corresponding tothechoice a=b=c=0). This equation hasparticular solutions oftheform u=R(r)Z(z)(<p), (6.2.9) ld dR uz _+<12 — —-O,where (6.2.l0)2 2 Q-i2z=0, LP,-+p.2<1>=0.dz dtp Inthespecial casewhere theconditions oftheproblem aresuch thatu isindependent oftheangular coordinate cp,wehave u=R(r)Z(z) (6.2.l1) where 1d dR d2Z T73; (Y 'l")\2R =0, F —)\2Z =0. 2.Helmholtz’s equation Vzu+kzu=O(corresponding tothechoice a=b=0,c=—k2). Inthiscase, application ofthemethod of separation ofvariables leads toparticular solutions oftheform u=R(r)Z(z)<I)(<p), (6.2.13) ;;(.§)+(A2_§)R=0, d2Z dzq)E;-'—O\2—k2)Z=0,where (6.2.14) 2Without lossofgenerality, wecanassume thateach oftheparameters x,X,y.belongs toanarbitrarily chosen half-plane, since changing thesignofx,A,udoes notaﬁ'ect the “separation constants”—x“, -73, it’. l46 CYLINDER FUNCTIONSi APPLICATIONS CHAP. 6 6.3.TheBoundary Value Problems ofPotential Theory. TheDirichlet Problem foraCylinder Afunction u=u(x,y,z)issaidtobeharmonic inadomain -rifuandits ﬁrstandsecond partial derivatives with respect tox,yandzarecontinuous andsatisfy Laplace’s equation Vzu=0in1.Consider theproblem ofﬁnding afunction uwhich isharmonic in1-andsatisﬁes oneofthethree boundary conditions ul,=f, (6.3.la) Bu710=/, (6.3.1b) (Z-Z+hu)0=f, h>0, (6.3.1¢) where 0'istheboundary of-r,fisagiven function ofavariable point of0,3 and8/on denotes thederivative with respect totheexterior normal to0'. This problem iscalled theﬁrst boundary value problem ofpotential theory or theDirichlet problem iftheboundary condition isoftheform (6.3.la), the second boundary value problem ofpotential theory ortheNeumann problem if itisoftheform (6.3.lb), andthethird ormixed boundary value problem of potential theory ifitisoftheform (6.3.lc). These problems play avery im- portant roleinmathematical physics.‘ Wenow consider theDirichlet prob- lemforthecasewhere 1isacylinder oflength landradius a. Letr,cp,zbeacylindrical coordinate system, with z-axis along theaxisof thecylinder andorigin inonefaceofthecylinder (seeFigure 18).Tosatisfy theboundary condition (6.3.la), weﬁrstsolve twosimpler problems cor- responding totheboundary conditions u|,=a =O, ul,=0 =fo, ulz=, =f,, (6.3.2a) u|,=a =F, u|z=0 =u|z=, =O. (6.3.2b) (Intheﬁrstcase, fvanishes onthelateral surface ofthecylinder, andinthe second case, fvanishes ontheends ofthecylinder.) Obviously, thesumof thesolutions satisfying theboundary conditions (6.3.2a) and(6.3.2b) will then satisfy themore general boundary condition (6.3.la).5 3IffE0,theboundary condition issaid tobehomogeneous, andotherwise inhomo- geneous. Here itisassumed thatuiscontinuous intheclosed domain 1:+e(cf.Sec. 8.1). 4Foramore detailed formulation ofboundary value problems, andforconditions guaranteeing theexistence anduniqueness ofsolutions under various assumptions con- cerning thedomain -randtheboundary function f,seethebooks byFrank andvonMises, Tikhonov andSamarski, Courant andHilbert, andSmirnov (Vol. IV),cited intheBiblio- graphy onp.300. 5Itshould benoted thatinmany problems involving inhomogeneous boundary conditions, repeated useofthesuperposition method leads tosolutions ofexcessively complicated form. This canoften beavoided byusing another method, duetoG.A. Grinberg. Selected Topics intheMathematical Theory ofElectric andMagnetic Phenomena (inRussian), Izd.Akad. Nauk SSSR, Moscow (1948). sec.6.3 CYLINDER FUNCTIONSI APPLICATIONS I47 Forsimplicity, wetemporarily assume thattheboundary conditions are independent oftheangular coordinate <p,sothat fo=f0(r), ﬁ=ﬁ(r), F=F(z)- Then thesolution uwillalsobeindependent of<p,andtherefore, according to(6.2.ll, 12)theparticular solutions ofLaplace’s equation take theform u=R(r)Z(z), where R(r)andZ(z) satisfy thedifferential equations 1d dR d2Z;EQE%¢m=d 7?-vz=a mm) Solving these equations, weﬁndthat R=AJ0(7\r) +BY0()\r), Z=Ccosh X2+Dsinh7\z, (6.3.4) where J0(x) and Yo(x) areBessel functions oforder zero, oftheﬁrstandsecond kinds, respectively. Z First weconsider theboundary conditions (6.3.2a). Since J0(>\r) —>1,Yo(>\r) —>ooasr—>0,and since the solution Rmust satisfy thephysical requirement ofbeing bounded ontheaxisofthecylinder, theconstant Bmust equal zero. Then thehomogeneous boundary condition becomes AJ0(>\a) =O, andhence theadmissible values oftheparameter Aare ,-(7 —~\ )\,,=x,,/a, where thex,,arethepositive zeros ofthe Bessel function J0(x) [seeSec.5.13]. Thus weobtain the FIGURE 13 following setofparticular solutions ofLaplace’s equation: u=u,,=[Mncosh <x,,E)+N,,sinh(xnE)]J0<x,, I), n=1,2,...a a a (6.3.5) Bysuperposition ofthese solutions, wecanconstruct asolution ofour problem. Infact, suppose each ofthefunctions f0(r) andﬂ(r) canbeex- panded inaFourier-Bessel series (seeSec.5.14), i.e., fo(")= (x.§),/.0)= §)- (6.16)I E-_-l where 2 G. 1;...= r/.(r>1o(x.§)dr, p=0,1. (6-3-v Then theseries 0., sinh<x,,17%) sinh (xnIi) r 2;f 1 ﬁ" 1Jld)u= 0,, +_ 0x,,— >(6.3.8) = sinh(xn—) sinh(xn—)a a I48 CYLINDER FUNCTIONS! APPLICATIONS CHAP. 6 whose terms areoftheform (6.3.5), clearly satisﬁes both Laplace’s equation andtheboundary conditions (6.3.2a).° Next weconsider theboundary conditions (6.3.2b). Inthiscase, wemust setC=Oandchoose 1:5’?!-1 n=l,2,... ifthehomogeneous boundary conditions aretobesatisﬁed. Then thesolu- tions of(6.3.3) take theform R=A14?) +BK,($), (6.3.9)IZTCZ Z=DSin where I0(x) andK0(x) areBessel functions ofimaginary argument (seeSec. 5.7). Since K0(mrr/l) —>ooasr—>0,wemust alsosetB=O.Therefore the particular solutions ofLaplace’s equation arenow 71712!‘ HTIZ u=u,,=M,,1,(T) sin n=1,2,... (63.10) Applying thesuperposition method justdescribed,” weﬁndthatthesolution ofLaplace’s equation satisfying theboundary conditions (6.3.2b) isgiven by theseries °° Iolﬁlcll .mrzu=25-5; SlnT, (63.11) ’<>lTl>¢ where -theF,,aretheFourier coefﬁcients ofF(z)inaseries expansion with respect tothefunctions sin(nnz/l): F,,=%F(z)sinLl”dz. (63.12) Remark I.Thesolution oftheNeumann problem andthemixed prob- lem, involving theboundary conditions (6.3.1b) and(6.3.la), isobtained in thesame way, butnow wemust useDini series (seeSec. 5.14) instead of Fourier-Bessel series. Remark 2.Togeneralize ourresults tothecase ofboundary conditions involving theangular coordinate <p,weconstruct particular solutions ofthe 6Here wehave inmind formal solutions, whose validity needs subsequent veriﬁca- tion. Asomewhat more rigorous point ofview isadopted inChap. 8(cf.p.208). 7Often called theFourier method, ortheeigenfunction method. SEC.6.4 CYLINDER FUNCTIONS! APPLICATIONS I49 more general form (6.2.9), satisfying theequations (6.2.l0). Thevalues ofthe parameter uarenowdetermined byimposing thecontinuity conditions 8u buu|,,,=_,, =1,llw=na aw: -7‘ =TPw=n' This isequivalent tothephysical requirement thatthesolutions beperiodic incp,andgives pt=m(m=0,1,2,...).Therestoftheanalysis differs only slightly from thatjustgiven, andleads tothefollowing particular solutions ofLaplace’s equation rcosmou=u,,,,,=[MM cosh(x,,,,, 5)+ N,,,,,sinh<x,,,,, Z)]J,,,<x,,,,, —). ,(6.3.14)a a as1nmq> u=u,,,,,=M,,,,,1,,(L') sing °9sml’, (63.15)l lsinmcp corresponding to(6.3.2a) and (6.3.2b), respectively, where thenumbers x,,,,,(m=0,1,2,...;n=1,2,...)denote thepositive zeros oftheBessel function J,,,(x). Then theboundary value problems aresolved bysuper- positions ofthese solutions intheform ofdouble series, with coefﬁcients ob- tained byexpanding thefunctions f0 :fO(ra (P)! fl=ﬂ(r7(P): F=F(Zs inappropriate double series. Example. Find thestationary distribution oftemperature uinacylinder of length landradius a,withoneendheldattemperature uo,while therestofthe surface isheldattemperature zero. Thedesired solution isfound atonce from (6.3.8) bysetting f,=uo, ﬁ=0,andusing (5.3.5) toevaluate theintegral (6.3.7): ...,sinh(it, J(,<x,, ”=2”"Z. <6-3-16>"a 6.4The Dirichlet Problem foraDomain Bounded byTwo Parallel Planes Using thesuperposition method, wecanalsosolve theboundary value problems ofpotential theory forthedomain consisting ofthelayer between twoparallel planes (seeFigure 19).Lettheboundary conditions beofthe form (6.3.la), andconsider thecaseofrotational symmetry, where thefunc- tions foandj",appearing intheconditions ul:-:=0 =.fI)> ulz=l =fl I50 CYLINDER FUNCTIONS2 APPLICATIONS CHAP. 6 depend only onthevariable r.Afunction which isharmonic inthedomain 0<z<landsatisﬁes theconditions (6.4.l) canbefound byintegration with respect toAofthefollowing particular solutions ofLaplace’s equation: u=u,,=[M,_cosh Az+N),sinhAz]J0(Ar), A20. (6.4.2) Infact, assuming thateach ofthefunctions f,and1’,canberepresented asa Fourier-Bessel integral (5.14.l1), weﬁnd that theformal solution ofthe problem isgiven by _°° sinhA(l—z) sinhAzll—‘L )\J0()\V)[fo';( 'l'_fj_;, db, where f,,_,=lowrf,,(r)J(,(Ar) dt,,6=0,1. (64.4) Z Z 0 ” FIGURE 19 Theboundary value problem forthehalf-space z>0canbesolved in thesame way. Infact, thesolution turns outtobe u=fooAJ(,(Ar)f,,e"""‘ (11,0 where /.=r/((110m) dt. iftheboundary condition isoftheform ul2=0 6.5. The Dirichlet Problem foraWedge Inthecaseofawedge-shaped domain, bounded bytwointersecting planes (seeFigure 20),theboundary value problems ofpotential theory canalsobe solved bythesuperposition method, with thehelp ofcylinder functions. To obtain asuitable setofparticular solutions ofLaplace’s equation Vzu=0, sec.6.5 CYLINDER FUNCTIONS: APPLICATIONS |5| weintroduce acylindrical coordinate system whose z-axis coincides with the lineinwhich thetwoplanes intersect, andweset A=ic, Oé0'<00, p.=ir, O<1<oo I inthedifferential equations (6.2.10).Then, according toSec. 5.7, thesolutions of these equations become R=AI,,(o'r) +BK,,(o'r), (P2 <1)=Ccosh -rcp+Dsinhrcp, (P, Z=ECOSo'Z-l-FSino'z, 0 X where I.,(x) and K,,(x) aretheBessel “GU” 2° functions ofimaginary argument, and A,B,...,Farearbitrary constants. Because oftheasymptotic behavior of thefunctions I,,(<:r) andK,,(er) asr—>oo(seeSec.5.11), wemust setA=O, which leads tothefollowing setofparticular solutions: )COS 0'Z sin<11’ (65.1) 0<e<oo, 0<1<oo.u=um=[Mm cosh up+NMsinh1'<p]K"(o'I‘ Wenow show how touse(6.5.l) tosolve theDirichlet problem forthe domain between thetwoplanes <p=<p1and<p=(P2.8Forsimplicity, suppose thefunctions fp=j§,(r, z)appearing intheboundary conditions ul<D=Q>p =fm P=1:2 areeven functions ofz,which implies thatthesame istrueofthesolution u=u(r,cp,z).9Assuming thateach ofthefunctions f,,canbeexpanded ina Fourier integral f,=f,,(r,z)=log,,(o,r)66$0'2dc, (65.3) where1° g,,(6,r)=Z£0f,,(r,z)66$0'Zdz, (65.4) BItwillbeassumed that indices areassigned toq>1,Q2insuch away that thedomain under consideration corresponds totheinterval q>1<qa<<p,. 9Thecase where theﬂ, areoddfunctions ofzishandled inthesame way. Then the solution inthegeneral case isrepresented asthesum ofthesolutions ofthetwosimpler problems with thefollowing even andodd boundary conditions: ul@=¢|1 =‘2l:fr("» Z)ifP(r! _Z)l- 1°G.P.Tolstov, op.cit.,p.190. I52 CYLINDER FUNCTIONSZ APPLICATIONS CHAP. 6 wetrytorepresent thesolution ofourproblem asadouble integral oo no ‘h _ u=I6666zd6J[G,(6,6)L51“(‘P2W0 6 S1I1h(‘P2 —<P1)T_ (6.5.5)Slnh <p— )1 +G2(°, T) Kt1(°") dr, formed byintegrating solutions ofthetype (6.5.1) with respect tothepara- meters <1and-r.Clearly, thefunctions G,,(c, 1-)must satisfy therelation g,,(6,r)=foo(;,(6,z)1<,,(6r)-dz, 0<r<66, (6.5.6)O andhence arethecoefﬁcients ofthefunctions g,,(cr, r),expanded asintegrals with respect tothefunction K,,(er). Insome cases, wecanuseformula (5.l4.l4) toﬁndthefunctions G,,(o, 1:). Infact, ifwewrite X=or, €=°'P, A/;¢f(X) =g(¢,F), (5.l4.l4) becomes g(o,r)=F2,fewz1<,,(6r) 611111TCTd-rjawg(o',6) d6.(6.5.?) Theexpansion theorem (6.5.7) isvalid ifg(o',r),regarded asafunction ofr, ispiecewise continuous andofbounded variation inevery ﬁnite subinterval [r1,r2],where O<r,<r2<oo,andiftheintegrals 1/2 1 co I |g(o, r)]r‘1 log7dr, I|g(e,r)|r‘1'2 dr (6.5.8)0 1/2 areﬁnite [cf.(5.l4.l5)]. Provided that thefunctions g,,(a,r) hasthese properties, acomparison of(6.5.6) and(6.5.7) shows that o,,(6,1')=é661611TH‘I0°g,,(6,r) dr, (65.9)0 andthen (6.5.5) gives aformal solution oftheproblem. However, itoften happens that theﬁrst oftheintegrals (6.5.8) isnotﬁnite, since g,,(e, r) generally approaches anonzero limit g,‘,(c, 0)asr—>O.Toavoid thisdiffi- culty, weintroduce themodiﬁed functions gZ‘(<I,r)=gz(<1.r)—g6(<1,0)@“”, P=1,2, (6-5-10) andassume, asisusually thecaseinphysical problems, thattheconditions forapplying formula (6.5.7) aresatisﬁed byg§,"(c, r).Wethen have g;<(6,r)=fog)c;;:(6,z)1<,,(6r) d-r, (6.5.11) sec.6.6 CYLINDER FUNCTIONS! APPLICATIONS I53 where o,t(6,z)=éz61611TITlowg;';(6,r) dr. (6.5.12) Ontheother hand, itiseasy toprove theformula“ ilowK,.,(x)dr=6-x, >6>0, (6.5.13) which implies g,,(6,0)6—1" =%tg,,(o', 0)fewK,,(6r) d-r. (6.514) Adding (6.5.11) and(6.5.14), weﬁndthedesired representation ofg,,(e, r) asanintegral with respect toK,,(c:r). Comparing theresult with (6.5.6), we ﬁnally obtain G,,(c, 1:)=Gj,§‘(e, 1:)+ég,,(c, O). (6.5.15) andthen thesolution isgiven by(6.5.5), asbefore. 6.6.The Field ofaPoint Charge near theEdge ofa Conducting Sheet Wenow illustrate themethod developed inthepreceding section, by ﬁnding theelectrostatic ﬁeldduetoapoint charge qlocated near thestraight lineedge ofathinconducting sheet held atzero potential. Toavoid com- plicating thecalculations, weassume thatthecharge qisatapoint Ainthe same plane astheconducting sheet. Choosing acoordinate system whose z-axis coincides with theedge ofthesheet andwhose x-axis passes through thepoint A(seeFigure 21),werepresent thepotential alloftheelectrostatic ﬁeldasthesumofthepotential altoduetothesource andthepotential udue totheinduced charges: 1=to+"1“"0= (“-1)Then theproblem reduces tothespecial caseofthegeneral problem ofSec. 6.5which corresponds tothefollowing choice ofangles andboundary condi- tions : <P1=0, <P2=2", f1(', Z)=f2('z Z)=*' ' (5-6-2) 1‘Use(5.l0.23) toexpand thefunction e“"°°=“°‘inaFourier integral with respect tocosrot,obtaining e-6.6661.<1= K,,(x) cos-rotdr, x>0, 0 andthensetat=0. I54 CYLINDER FUNCTIONSZ APPLICATIONS CHAP. 6 I 4 (=0-,0 0 FIGURE 21 Using theintegral representation given inProblem 6,formula (i),p.140, weﬁndthat 2==> 2g,,(6,r)=-iflo-ﬁ dz=-f1<,[6(r +6)],(6.6.3) where K0(x) isMacdonald’s function. Inthepresent case, 2g,,(6,0)=-f1<,(6d), andhence, according tothemethod ofSec.6.5,wemust ﬁrstdetermine the quantity o:(6,T)=_‘1-21611111TCTIwKi_i-_-°[°(’ J’ally‘K°(°")e_°'1<.,(6r)dr. (6.64)0 Since theevaluation oftheintegral in(6.6.4) isquite complicated, weomit the details andmerely givetheﬁnal result: 4G:(6,z)=“-3[1<,(6d) -1<,,(6d)]. (66.5) Substituting (6.6.5) into(6.5.15), weobtain 6.6».1)=-K..(6d). (6-6.6) andthen formula (6.5.5) gives 6=-‘gla”6666z<16In°?i°Sh(Y1_“PPK-(6d)K<(or)dr.(6.6.?)0 0cos 1-c-r " ‘T Theintegral in(6.6.7) canbeexpressed inclosed form interms ofele- mentary functions, andtheﬁnal result ofthecalculations turns outtobe u=_i___q€.____ \/r2 +a2+2arcos<(1—garc tan )TE SEC.6.7 CYLINDER FUNCTIONSI APPLICATIONS I55 (weomit thedetails)?’ Itfollows from (6.6.8) that (L=_ arc tan 1c\/r2+a2+2arcos<p+z2 \/r2+a2+2arcos<p+z2 (6.6.9) Finally, weobserve thatthesurface charge density onthesheet isgiven by thequantity” 164» qZ 1 "Wow: <6-61°) 6.7.Cooling ofaHeated Cylinder Asanexample oftheapplication ofcylinder functions tothenonstation- aryproblems ofmathematical physics, wenow consider theproblem ofthe cooling ofaninﬁnitely long cylinder ofradius a,heated tothetemperature uo=f(r) [risthedistance from theaxis] andradiating heat into thesur- rounding medium atzero temperature. From amathematical point ofview, theproblem reduces tosolving theequation ofheat conduction 662-‘;=kvzd, (6.7.1) subject totheboundary condition 8u(5+hu)Ta=0, (67.2) andtheinitial condition ult=o ="0=f(") (6-7-3) where k,c,p,Aandh=A/khave thesame meaning asinSec.2.6.Separating variables in(6.7.1) bywriting u=R(r)T(t), weﬁndtheequations db§+n2T=0, %E(r%{)+x2R=O, where -562istheseparation constant andb=cp/k, with solutions R=AJ0(xr) +BYO(w.r), T=Ce“’°2’”’. 1’Itshould benoted that inthepresent case, theformula 2 Q K6[<=(r+an=E1<..<<m>K..<-1) <1- allows ustoderive thesolution (6.6.7) without recourse tothegeneral method ofexpan- sionasanintegral with respect tothefunctions K,,(or). Toobtain thisformula, set <1)=rrinformula (42), p.55oftheBateman Manuscript Project, Higher Transcendental Functions, Vol.2. 1°G.Joos, op.cit.,p.267. I56 CYLINDER FUNCTIONS; APPLICATIONS CHAP. 6 Since J0(xr) —>1,Y0(ur) ->ooasr—>0,andsince Rmust satisfy thephysical requirement ofbeing bounded ontheaxisofthecylinder, theconstant B must equal zero. Itfollows from (6.7.2) thattheparameter xmust satisfy theequation hJ0(xa) —xJ1(xa) =O. (6.7.4) Ifwewrite x=xa,then (6.7.4) becomes haJ0(x) —xJ1(x) =O, (6.7.5) which hasonly realroots, symmetrically located with respect totheorigin (seeSec.5.13). Let0 <x1<-~~<x,,<---bethepositive roots ofequation (6.7.5). Then theadmissible values oftheparameter xarex,,=x,,/a, and hence theappropriate setofparticular solutions of(6.7.1) is u=u,,=M,,J0(x,, g)e"‘5‘/“Q”, n=1,2,... Superposition ofthese solutions gives "=.Z.M.1<»I><t zil”"‘i"“’% (6.16) where, because oftheinitial condition (6.7.3), thecoeﬁicients Mnmust be chosen tosatisfy therelation f(r) =£1M,,J0(x,, 5)» 0<r<a. (6.7.7) This isjusttheproblem ofexpanding f(r)inaDini series, which canbesolved byusing formulas (5.l4.9—10). Thus wehave 2 “ rMn = I:/l(V)Jo(Xn dr, andthesolution ofourheatconduction problem isgiven bytheseries (6.7.6), with these values ofthecoefﬁcients. 6.8Diffraction byaCylinder Finally, wegiveanexample illustrating theapplication ofBessel functions ofthethird kind. Consider thediffraction ofaplane electromagnetic wave byaninﬁnite conducting cylinder ofradius a.Let(r,<p,z)beasystem of cylindrical coordinates such that thez-axis coincides with theaxisofthe cylinder andtheangle cpismeasured from thedirection ofpropagation ofthe incident wave. Weassume that thetime dependence isdescribed bythe factor e‘°",where 65istheangular frequency oftheincident radiation, and that theelectric vector oftheincident wave isparallel totheaxisofthe SEC. 6.8 CYLINDER FUNCTIONSI APPLICATIONS cylinder. Then theproblem reduces toﬁnding thecomplex amplitude ofthe secondary ﬁeldEsatisfying Helmholtz’s equation 18 8E 182E7505;) +;$+k*=E_0, (6.s.1) theboundary condition E|,=,+E,6~*rw=s <1=0 (6.s.2) andtheradiation conditions l _3E .E=o(__). lim6/,(— +116E)=0, (6.s.3)\/r " Hm 8 where k=6)/cisthewave number, andE0istheamplitude oftheincident plane wave.“ Applying themethod ofseparation ofvariables, weﬁndthattheparti- cular solutions of(6.8.1), which must alsobeperiodic in<p,areoftheform E=E,=[M,,H§,‘>(kr) +N,,H§,2’(kr)] 2:. n=0,1,2,...,(68.4) where H,§1’(kr), H,‘,2>(kr) aretheHankel functions introduced inSec.5.6.It follows from thesymmetry condition thatEisaneven function ofcp,and hence weneed onlyconsider solutions containing cosncp.Moreover, examin- ingtheasymptotic behavior oftheHankel functions atinﬁnity, weseethat theradiation conditions willbesatisﬁed onlyifM,=0(noincoming waves). Therefore thesolution ofourproblem must have theform E=EN,,H,‘,2)(kr) cosn<p. (6.8.5) n=0 Itfollows from theboundary condition (6.8.2) that 2N,,H,§2>(ka) cosmp+E6e‘”“‘°°s “’=0- (63.6) n=0 Setting z=kaandt=—ie“" informula (6.8.4), weobtain 6-"ta R=J0(ktl) +2Z(-1)"J,,(1<d) 666ncp, (6.8.?)n=1 which, together with (6.8.5), implies NOH§,2>(ka) =—E(,J0(ka), N,,H§,2’(ka) =—2E0(—i)"J,,(ka). Therefore therequired solution isgiven by E=-E, H§,2>(kr) +2if-1)" H§,2>(kr) 666ncp]-(6.s.s) 1‘SeeA.N.Tikhonov and A.A.Samarski, Dtﬂerentialgleichungen derMathe- matischen Physik, VEB Deutscher Verlag derWissenschaften, Berlin (1959), p.497. I58 CYLINDER FUNCTIONS: APPLICATIONS CHAP. 6 PROBLEMS 1.Inpolar coordinates r,<p,thefreetransverse vibrations ofastretched mem- brane (with equilibrium position intherep-plane) aredescribed bytheequa- tion15 l32 ,, . veto.6.1)=E (1) where I3 3 IZ322 i‘ 1-i i I V—r8r(Wu) +r28:92 Solve theequation ofmotion (i)forthecase ofacircular membrane ofradius a,subject totheboundary condition u[,=a=0 (fastened edge) andtheinitial conditions ul.=6=/6). =66). 2.Solve Problem 1with thesame boundary condition, butwith themore general initial conditions 16 6 u|.=6=/6.6). =.6166). 3.Inpolar coordinates r,q>,thefreetransverse vibrations ofanelastic plate (with equilibrium position intherrp-plane) aredescribed bytheequation 182,,t ..V‘*u(r. 11>,I)=—13% (11) where V2hasthesame meaning asinProblem 1,andV4=V2(V2). Solve the equation ofmotion (ii)forthecaseofacircular plate ofradius a,subject to theboundary conditions BuIll,-=a =0, 67,10‘ =0 (clamped edge), andtheinitial conditions 6 61.-.=/6). 5‘0=go).t: 15Forthederivation ofequation (i),andequation (ii)below, seee.g., I.M.Gelfand andS.V.Fomin, Calculus ofVariations (translated byR.A.Silverman), Prentice-Hall, Inc.,Englewood Cliffs, N.J.(1963), p.162ff.Here wedonotspecify thephysical mean- ingoftheconstant b.Byfree vibrations, wemean vibrations intheabsence ofexternal forces. 1°Fordetailed solutions ofProblems 1-2,seeG.P.Tolstov, op.cit.,p.288If. PROBLEMS CYLINDER ruucrrous: APPLICATIONS I59 Hint. Separate variables in(ii)bywriting u=R(r)T(t). The radial equation thenbecomes ' 1d d1ddR(V —K4R =0, (Ill) where >44istheseparation constant. The general solution of(iii)which re- mains ﬁnite atthecenter oftheplate is R(r) =AJo(ur) +BIo(xr). Ans. - umo=§<#%i~km@?fvmmw@»HLmmwp "0 a2 x2b2t G, +%Sin —"t;2—LPg(P)Rn(P)dP]’ where Rm)=Io(x)J.,(x -Jo(x)1o(x thenumbers 0<x1<---<x,,<---arethepositive roots oftheequation R§,(a) =0,andRnERM. AZ 4.Find thestationary distribution oftemperature uina cylinder oflength land radius awhose ends areheld at ‘ temperature zero, while therest ofthesurface isheld at 1{ temperature uo. 2 5.Find thestationary distribution oftemperature uin theinhomogeneous cylinder shown inFigure 22,made upoftwoadjacent cylindrical sections with different thermal 11 1 conductivities k1andk2,ifthelateral surface isheld at temperature uo,while theends areheld attemperature »~""‘“~\ zero. Hint. Ifuland U2denote thetemperatures inthesec- tions labelled 1and2,respectively, then theboundary con- FIGURE 22 ditions are\‘ u1|r=a :u2ir=a =“O, u1iz=—l1 =”2|z=lg =Os 311 8u “M=@#@ “aM=“5a; 6.Suppose anaxially symmetric temperature distribution uit=0 =f(r) isestablished attime t=0inaninﬁnitely long cylinder ofradius a,which transfers noheat through itssurface. Find thesubsequent evolution intime ofthetemperature distribution. I60 CYLINDER FUNCTIONS! APPLICATIONS CHAP. 6 7.Find thepotential Ki)oftheelectrostatic ﬁeld inside aclosed cylindrical sur- face oflength landradius a,whose base andlateral surface areheld atthe potential V,while thetopsurface isheld atzero potential.“ 8.Find thestationary distribution oftemperature uinthehalf-space z>0, subject totheboundary condition I'<(1 ui::=0=/<r>={gf ’r>a. Ans. u(r,Z)="oilfa’e-*ZJo<»r>J1<~1) d».0 9.Find thepotential aboftheelectrostatic ﬁeld Z inthespace between twogrounded plane elec- trodes z=iaduetoacharge qatthepoint r=O,z=O. J a Hint. Useformula (5.2.4). : q "’(”‘)vﬁ?FIGURE 23 °° cosh 7\z - -M___ atx.qfa6 cosh laJ°(V)d 10.Find thestationary distribution oftemperature uintheinﬁnite wedge of thickness Ishown inFigure 23,ifthefacecp=onisheld atthetemperature u|w=m=f(r)sin-nil?’ n=1, while therestofthesurface isheldattemperature zero. Ans. 2 nrcz °° -ru(r,qa,z)=;sin—l—J‘ {f(O)+; sinh 11:-r U °° _ nrcp dp sinh cpr nrrr XL [f(P) —9""°'lf(0)iK11(T)% Kn(T)d‘F- 11.Solve thepreceding problem foranarbitrary temperature distribution uiq>=oc =f(r) Z)- "Onecanthink ofthetwoparts ofthesurface asinsulated from each other byan inﬁnitely thin gasket. SPHERICAL HARMONICS: THEORY 7.l. introductory Remarks Byspherical harmonics wemean solutions ofthelinear differential equation 2 (1-z2)u”-221/+[V(\I+1)-ﬁt =0, (7.1.1) where zisacomplex variable, andp.,vareparameters which cantake arbitrary realorcomplex values. Equation (7.l.l) isencountered inmathe- matical physics when using systems oforthogonal curvilinear coordinates to solve theboundary value problems ofpotential theory forcertain special kinds ofdomains (e.g., thesphere, spheroid, torus), anditisthesimplest of these domains (i.e., thesphere) which gives risetotheterm “spherical har- monics.” Inthespherical case, thevariable ztakes realvalues intheinter- val(—1,1),and theparameters itandvarenonnegative integers, but boundary value problems with more complicated geometries lead tothe consideration ofmore general values ofz,rtandv.1Formost applications, it issuiﬁcient toassume (aswewilldointhisbook) that ziseither areal variable intheinterval (—1,1)oracomplex variable intheplane cutalong thesegment [—oo, 1],while visanarbitrary realorcomplex number and it=misanonnegative integer (m=0,1,2,...).Thereader willﬁndamore general treatment inthereferences onspherical harmonics cited intheBib- liography onp.300,especially thebooks byHobson, Robin andLense. 1SeeChap. 8,where weconsider problems inwhich thevariable zandthepara- meters u,vtakevarious realorcomplex values. l6l I62 SPHERICAL HARMONICSZ THEORY CHAP. 7 7.2.TheHypergeometric Equation andItsSeries Solution Before presenting thetheory ofspherical harmonics, itisappropriate to consider theproblem ofsolving thelinear differential equation z(1—z)u” +[Y—(oi+{5+1)z]u’ —ocﬁu=O, (7.2.1) where zisacomplex variable, andll,B,Yareparameters which cantake various realorcomplex values. Equation (7.2.1) iscalled thehypergeometric equation, andcontains asspecial cases many differential equations encountered intheapplications. Reducing (7.2.l) tostandard form bydividing itbythe coefﬁcient ofu”,weobtain anequation whose coefﬁcients areanalytic func- tions ofzinthedomain 0<|z|<1andhavethepoint z=0asasimple poleoraregular point, depending onthevalues oftheparameters oz,[5andY. Itfollows from thegeneral theory oflinear differential equations that(7.2.1) hasaparticular solution oftheform u=z‘Zc,,z", (7.2.2) k=O where coaé0,sisasuitably chosen number, andthepower series converges for|z|<1.2 Substituting (7.2.2) into(7.2.l), weﬁndthat Zc,,z‘*"‘1(s +k)(s+k-1+Y)-Zc,,zs*"(s +k+oc)(s+/<+is)=0,k=0 k=0 which gives thefollowing system ofequations fordetermining theexponent s andthecoefﬁcients ck: c0s(s —1+Y)=0, c,,(s+k)(s+k—-l+y)—c,,_1(s+k—l+<x)(s+k—l+B)=0, k=l,2,... Solving theﬁrstequation, weobtain s=0ors=1—Y.Suppose Yaé0, —1,—2,...andchoose s=0.Then thecoefﬁcients ckcanbecalculated from therecurrence relation (k—1+ot)(k——l+B)Ck= Ck_1, k=l,2,..., Ifwesetco=l,thisimplies ¢,,=%%, 1<=0,1,2,..., where wehave introduced theabbreviation (>\)o=1, (>\),,=>\(7\+1)---(A +k-1), k=1,2, (7.2.3) 2E.A.Coddington, op.cit.,Chap. 4. SEC.7.2 SPHERICAL HARMONICSI THEORY I63 asinfootnote 17,p.121. Thus, ifYat0,-1,—2,...,aparticular solution ofequation (7.2.l) is _ _ .._0°(°‘)i¢(l3)i¢ ku-ul-F(ot, B,Y,z)-kgom z, |z[<1, (7.2.4) where theseries ontheright isknown asthehypergeometric series.“ Thecon- vergence ofthisseries for|z]<1follows from thegeneral theory oflinear differential equations.‘ However, byusing theratio test, itcaneasily be proved without recourse tothistheory thattheradius ofconvergence ofthe series (7.2.4) isunity, except when oneoftheparameters oc,Bequals zero ora negative integer, inwhich casetheseries reduces toapolynomial. Similarly, choosing s=l—Yand assuming that Yaé2,3,4,...,we obtain _(k—Y+°°)(k—Y+l5) _ck- k(k+1_Y) c,,_1, k-l,2,..., or _(1—Y+¢)t(1~Y+ﬁ)t _c,,- k!(2_Y)k ,/<_0,1,2,..., ifwesetco=l.Thus, ifYaé2,3,4,...,aparticular solution of(7.2.l) is °°l— l— ,,=,2=,._.kZo ,. 7.2.5 =21-*F<1—Y+¢,1-Y+@;2—Y;z>, ()|z|<l,|argz| <TC. Therefore, ifY750,l,2,...,thetwosolutions (7.2.4~5) exist simultaneously andarelinearly independent? Then thegeneral solution of(7.2.l) canbe written intheform u=AF(ot, t5;Y;z) +Bz1*YF(l —Y+oc,l—-Y+L5;2—Y;z), (7.2.6) where |z|<l,|argz|<rt,andA,Barearbitrary constants. However, ifY isaninteger, thismethod leads toonly oneparticular solution, andtoﬁnda second solution wemust modify themethod, thereby obtaining asolution which ingeneral contains logarithmic terms.“ Bychanging variables in(7.2.l), wecan obtain anumber ofother differential equations whose solutions can beexpressed interms of 3IfYequals zerooranegative integer, thenthecoefﬁcients ckbecome inﬁnite, starting from acertain value ofk,andasolution oftheform (7.2.2) cannot beconstructed if s=0.However, itiseasytoseethatthissituation does notarise ifs=1—Y. *E.A.Coddington, op.cit.,Theorem 3,p.158. 5Toprove thelinear independence, consider theasymptotic behavior ofthesolutions asz—>0.Thetwosolutions coincide ifY=1. “E.A.Coddington, op.cit.,Theorem 4,p.165. I64 SPHERICAL HARMONICS2 THEORY CHAP. 7 hypergeometric series. Thus, forexample, setting z=12,wearrive atthe differential equation z(1—mg; +2[Y—g—(<1+(5+-3:2]g’—4oc[3tu =0,(7.2.7) with particular solutions u=u1 =F(ot,B;Y;t2), Y;é0, -1,—2,..., (7.2.8) u=u2=z2'2YF(l—Y+a,l—Y+l3;2—Y;t2) [t]<l, |argt|<r:, Y;é2,3,4,..., (7.2.9) which fornonintegral Yconstitute apairoflinearly independent solutions of (7.2.7) inthedomain 0<|t|<l. 7.3. Legendre Functions Thesimplest class ofspherical harmonics consists oftheLegendre poly- nomials considered inChapter 4,which aresolutions ofequation (7.l.l) for it=0andnonnegative integral v=n(n=0,1,2,...).Thenext class of spherical harmonics, inorder ofincreasing complexity, consists ofthe Legendre functions, which aresolutions of(7.l.l) forp.=0andarbitrary real orcomplex v,i.e.,solutions oftheequation (l—z2)u” —2zu' +v(v+1)u=O, (7.3.l) known asLegendre’s equation. Todetermine these functions, weﬁrstnote that(7.3.1)canbereduced tothehypergeometric equation bymaking suitable changes ofvariables. Inparticular, thesubstitution t=%(l-—z)converts (7.3.l) intotheequation r(l—0% +(1-20% +v(v+1)u=0, (7.3.2) which isthespecial caseof(7.2.l) corresponding to ot=—v, [3=v+1, Y=l, while thesubstitution t=2'2,u=z""1v converts (7.3.l) intotheequation dz 3 5d l[(1-07;’+H»+5)-(V+5H7’; -+ +5))=0,(7.33) which isthespecial caseof(7.2.l) corresponding to a=i+1 B:l'_|_l,Y:v+§. 2 ’ 22 2 SEC.7.3 SPHERICAL HARMONICSZ THEORY I65 Therefore itfollows from theresults ofthepreceding section thattwoparti- cular solutions of(7.3.l) are u=u1=F(——v,v+l;l;!--5-5), Iz-—l|< 2, (7.3.4) X/r:F(v+l) v v 1 31u=u2= F(i+l,§+§,v+§,?), |z|>1, |argzI <11:,vat—-l,—2,..., (7.3.5) where F(<x,B;Y;z)isthehypergeometric series. These solutions arecalled theLegendre functions ofdegree voftheﬁrst andsecond kinds,"’ denoted by P,(z) andQ,,(z), respectively. Thus wehave lPv(z)=F(—v,v+1;l;%), |z- l]<2, (7.3.6) \/¥=r(v+1) 1. »131Qv(Z)= F(i+I,§+§,v+§,?)’ [2]>1, |argz| <1:,v75—l,-2,... (7.3.7) Thefunctions P,,(z) andQ,,(z) aredeﬁned incertain restricted regions of thecomplex z-plane, but,aswenowshow, theycanbecontinued analytically into larger regions.“ Tomake theanalytic continuation ofP,,(z), the Legendre function oftheﬁrstkind, weusetheformulae /2 2 at _ 1 ;c0s1n2" qad(P=% k=0,1,2,... (7.3.8) towrite (7.3.6) as _°°(—\')1.(v +1)». 1-Z"Pv(Z) _'kg ( 2 ) O __2 “’(—),,(v+1) l—z" "/2. ‘5.2., ve>..ki LITI““2"°*""*° (739) 2"'2 11— .=;f0 F(—v,v+ l;§;—2——€s1n2<p)dcp, 7The term degree isappropriate here, since fornonnegative integral v=n,P,,(z) is actually apolynomial ofdegree n,infact, thenthLegendre polynomial (see Sec. 7.9). 8Wepoint outthat inthischapter, unlike Chapter 9,thesymbol F(ot, B;Y;z)always denotes thesumofthehypergeometric series, andhence thevariable inthefourth posi- tion always has absolute value <1.This restriction disappears ifweinterpret F(ot,I3;Y;z)asthehypergeometric function. Infact, prior knowledge ofthetheory of thehypergeometric function leads toconsiderable simpliﬁcation ofthetheory ofspherical harmonics. 9Formula (7.3.8) isanimmediate consequence ofProblem 3,p.14. I66 SPHERICAL HARMONICSZ THEORY CHAP. 7 where reversing theorder ofsummation andintegration isjustiﬁed because theseries isuniformly convergent inthevariable <p.Thehypergeometric series intheright-hand sideof(7.3.9) canbesummed inﬁnite form. Infact, wehave theidentity F(_V’v +l;%;_W):(\/1+ W+X/w)2""1 +(\/1+ w+\/w)‘2"'1 2\/1+W =f,,(w), |w|<l (7.3.l0) which isproved bynoting that thefunction fv(w) isanalytic inthedisk ]w|<landsatisﬁes thedifferential equation“ w(l+w)f,,” +(%+2w)f,,’ —v(v+l)f,,=0. (7.3.1l) Butreplacing wby—wconverts (7.3.ll) intothehypergeometric equation withparameters oz=—v,[3=v+l,Y=11;.Then, since equation (7.2.l) has aunique solution which isanalytic inthedisk |w|<landapproaches unity asw—>0,itfollows that fv(W) EF(—v,v +1;%;—W), asasserted. Wenow substitute (7.3.l0) into(7.3.9), obtaining theintegral representa- tion 2"/2 —l.P.<z)=;f.(‘T wet)de fortheLegendre function oftheﬁrstkind. Inderiving thisformula, itwas assumed that |z—1|<2,buttheintegral intheright-hand sidedeﬁnes an analytic function forevery zinthecomplex plane cutalong thesegment [—oo,——1].Infact, foranysuch z,thevariable —1.w=Z—2—s1n’<p, 0< <5-6 belongs tothew-plane cutalong [—oo, -1]. Since j§,(w) isanalytic inthis plane, ourassertion follows bytheusual theorem from complex variable theory.“ Thus theanalytic continuation ofP,,(z) isgiven bytheformula 2"'2 z-1 .2P,(z)=dofv(Ts1n cp)dt, |arg(z +1)|<TE.(73.12) 1°Thepoint w=0isaregular point ofthefunction f,(w), since fv(W) takes its original value after making acircuit around this point. Toverify (7.3.l1), itiscon- venient toﬁrstshow that s/Wm/Es/1 +wI\/1+wﬂ,]’}’ -(V+i)=f.=0, andthen carry outthedifferentiation. 11E.C.Titchmarsh, op.cit.,p.99. SEC. 7.3 SPHERICAL I-IARMONICSZ THEORY Thefunction deﬁned by(7.3.12) isanalytic inthez-plane cutalong [—oo,1] (seeFigure 24),where itisasolution ofthedifferential equation (7.3.1), byan obvious application oftheprinciple of analytic continuation." Inparticular, (7.3.l2) implies P,,(1) =1. (7.3.l3) Aswillbeshown below, every solution of(7.3.1) which islinearly independent _ , 1 _ ofthesolution u=P,,(z), approaches ‘I O I inﬁnity asz->l,andtherefore the Legendre function oftheﬁrstkind can also bedeﬁned asthesolution of FIGURE 24 (7.3.l) which approaches unity asz->1. Since ﬁ,(w) isanentire function oftheparameter v,itfollows from (7.3.l2) thatthesame istrueofP,(z). Moreover, itiseasily veriﬁed that f—v-1(W)=fv(W), andhence P_,,_1(z) =P,,(z) (7.3.14) forarbitrary realorcomplex v. Tomake theanalytic continuation ofQv(z), theLegendre function ofthe second kind, westart with theformula — v 1vco dt _\/TI:F(\I-I-I) (id-i)k(i-I-1)k 7315 1,21.+.+=/.(/3-_—1“ 1,+§)(3+§)(3+§) ’(" ) I”22424k lc Rev> -1, k=0,l,2,..., which iseasily proved bymaking thesubstitution t=s“1andusing formulas (l.5.2), (l.5.6) and(l.2.3) from thetheory ofthegamma function. Then, using (7.3.l5) andthedeﬁnition ofQ,,(z), andassuming that |z|>l, |argz|<rc, Rev>—l, I”Letf(z)beanalytic inadomain D,andsuppose Lf(z) =0forallzinasmaller domain D*contained inD,where Lisalinear differential operator whose coefﬁcients areanalytic inD.[Inthepresent case, dz dL= '-'Z2)?" 2ZE -I-V(V + Then Lf(z) =0forallzinD.Cf.footnote 6,p.3. I68 SPHERICAL HARMONICSZ THEORY CHAP. 7 wehave \/Er(v +1)E+1I,,I2+iI,, ik Q”)=I‘(v+%)(2z)“*‘ '¢=° (v+;)kk! Z <~)<*~>_1i24,,2 4,,1f<><> dz211:_('22))/+1k_=0 (V+ k! Z 1t2k+v+3/2‘/t _1 " 3 5 (73.16) w "_I_=—1— _ °°(2+4I1.I2+4I,, 1<)1. 1Z3,()22"* tv+%\/I _ ii-—- =0 (V+ zt2k IC Z<2z)"+1I. FIE+"‘+K”*TF2)71-3/=~/_,-1’ where reversing theorder ofsummation andintegration canbejustiﬁed byan absolute convergence argument. Therestofthederivation isbased onthe formula .1 s_ 3__1l+\/l—w““/1_—gv(W). (73.17)A‘ P-Il8< AwI\J<U1 ua >1! ‘5.- -5;!»-3~< Toprove (7.3.l7), itissufficient toshow thattheright-hand sidesatisﬁes equation (7.2.l) forthevalues“ v 3 v 5 30(=i-I-Z9 ‘\{=V-I-5’ Z=W. Together, (7.3.l6) and(7.3.l7) imply I °° 1 dtQvtz)=7... g.(W)—i...,,V7 _,- 7.3.18|z|>l, |argz|<r:, Rev>—l. ( ) 1°Tosimplify thecalculation, which isabittedious, itisconvenient toﬁrstshow that _1 _1_2 <v1T7.g.)' = 11/%<vTTwg.)'1'= Then multiply theﬁrst equation by(v+%)\/I —-wandthesecond byw,carry outthe differentiation, and add theresulting equations. Formula (7.3.l7) canalso bederived from thesecond oftheformulas (9.8.3) bysetting ot=;+%’ z=w. sec.7.3 SPHERICAL HARMONICSZ THEORY I69 Wenowassume temporarily thatzisarealnumber greater than 1,and introduce anewvariable ofintegration bysetting zt=1+(z—l)cosh2\l1. Then (7.3.l8) takes theform“ Q.<z)=I0” cosht1»)dt. where t/1_ t/7h.<w)= larswl<-=.larg(1+w>|<7--1+w (73.19) Although thisformula forQV(z) hasbeen derived under theassumption that z>1,itisnothard toseethattheintegral ontheright hasmeaning ina larger region. Infact, forzintheplane cutalong [—oo, 1]and1.11inthe interval [0,00],“ theintegrand iscontinuous in1.1»forevery zandanalytic inzforevery ti».Moreover, ifRev>—l,theintegral converges uniformly inevery region O<p<|z——l|<R<oo, |arg(z—l)|<1-c—8, andhence, bytheusual argument,“ represents ananalytic function inthe plane cutalong [—oo,1].Thus theanalytic continuation ofQ,,(z) isgiven by theformula °° z—l 2Q.,(z) =faM7 cosh 1l»)d¢, |arg(z—1)]<rt,Rev > 1. (7.320) Toobtain theanalytic continuation ofQ,,(z) forthecase Revs—1,we ﬁrstobserve that Qv(z) satisﬁes therecurrence relation 2v-{-3 v+2 Q.(z)=TzQ...<z) -3-,Q...<z). (7.321) which canbeveriﬁed bydirect substitution oftheseries (7.3.7). Ifpisany 1*Inthecourse ofthecalculations, weusethefamiliar identity X/1ﬁ:~/A+\/A2-B+A/A-\/A2—B_ 2 2 15Forthese values ofzand<11,thevariable w=——-zg1cosh’\l; belongs totheplane cutalong [—oo,0],where h\,(w) isanalytic. 16E.C.Titchmarsh, op.cit.,pp.99-100. I70 SPHERICAL HARMONICSZ THEORY CHAP. 7 positive integer, wecanuse(7.3.21) towrite thefunction Q,,(z) with arbitrary index v75—l,—2,...intheform Qv(z) =<1p(Z,V)Qv+P(z) +bt=(Z,v)Q.+p+1(Z), (7-3-22) where a,,(z, v)andb,,(z, v)arepoly- nomials in2.Then, choosing pso large thatRev>—(p +1),wecan use(7.3.20) tomake theanalytic con- tinuation ofeach oftheLegendre functions intheright-hand side of _1' O T T(7.3.22), andsubstituting thecorres- ponding expressions into (7.3.22), weobtain afunction which isanaly- FIGURE 25 ticinthez-plane cutalong [—oo, 1] (seeFigure 25).Itfollows that Q,,(z) isanalytic inthiscutplane, forarbitrary complex vaé—l,—2,...Like P,,(z), thefunction Q,(z) satisﬁes thedifferential equation (7.3.l) [cf.foot- note 12,p.167]. Moreover, (7.3.20) implies lim Q(z)=oo. (73.23) v2~1+ Comparing (7.3.23) and(7.3.l3), weseethat Pv(z) and Qv(z) arelinearly independent solutions of(7.3.l). Wenowstudy Q,(z) asafunction ofthedegree v,andshow thatforevery ﬁxed z,theratio ___Q_!(L)_ 24q.<z)-H,+1) <7-3.) isanentire function ofv.For|z[>1,thisfactisanimmediate consequence of(7.3.7). Togiveaproof which isvalid forevery zintheplane cutalong [—oo,1],weusetheintegral representation (7.3.20) and therecurrence relation qv(Z)=(Zv+3)zqv+1(Z) —(v+2)2qv+2(Z), (7-3-25) implied by(7.3.21). Itfollows from (7.3.20) thatq,,(z) isananalytic function ofvinthehalf-plane Rev>-1." Repeated application of(7.3.25) leads to theexpression qv(Z) =°‘P(V1 Z)qv+p(Z) +B170’! z)qv+v+1(Z)> (7-3'26) where pisapositive integer, andoc,,(v,z),B,,(v,z)arepolynomials inv.Itfol- lows thatq,,(z) isanalytic inthehalf-plane Rev>—(p +l).Since pcanbe 1’Note thath,(w) isanentire function ofv,while theintegral (7.3.20) isuniformly convergent invintheregion Rev2——l+8,where 8>0isarbitrarily small. There- fore theusual theorem concerning analytic functions deﬁned byintegrals isapplicable. sac.7.4 SPHERICAL I-IARMONICSZ THEORY l7I chosen arbitrarily large, weconclude thatqv(z) isanentire function ofv. Therefore, according to(7.3.24), Q,,(z) isameromorphic function ofv,with simple poles atthepoints v=—1,—2,... Thegeneral solution uofthedifferential equation (7.3.l) canbewritten asalinear combination ofLegendre functions oftheﬁrstandsecond kinds, i.e., u=AP\,(z) +BQ,(z), (7.3.27) where |arg(z—l)|<rt,v#—l,—2,...Intheapplications, itisoften necessary toﬁndageneral solution of(7.3.l) forthecase where xisareal number intheinterval (—1,1).Since Pv(z) isdeﬁned forsuch x,weneed only construct asecond linearly independent solution. Itisnothard toseethat such asolution isgiven bythefunction Qv(X) =%lQv(X +i0)+Qv(X-i0)], (7-3-28) equal tohalfthesumofthevalues ofQ,,(z) ontheupper andlower edges of thecut(cf.Sec.7.7).“ Thus, ifz=x(-1 <x<1),thegeneral solution of (7.3.l) is u=AP,(x) +BQ,,(x), v76—l,—2,... (7.3.29) 7.4.Integral Representations oftheLegendre Functions TheLegendre functions have various integral representations interms of deﬁnite integrals andcontour integrals containing thevariables zandvas parameters. Asarule, themost general representations ofthistype involve contour integrals, butforpractical purposes, representations involving inte- grals along segments oftherealaxisareofgreatest importance. Forthis reason, wewillonlyconsider representations ofthistype, referring thereader elsewhere forintegral representations ofother kinds.“ Webegin byderiving anintegral representation ofthefunction Pv(z). Assuming thatz=cosh at(ct>0)andintroducing anewvariable ofintegra- tionin(7.3.12) bysetting sinhS=sinhgsincp, weﬁndthat P,,(cosh 1)=3fai§ d6 (7.4.1)it0\/2cosha—2cosh(-) 1‘IntheGerman literature, thesymbols P,(z) and Q,,(z) areused todenote thesolu- tions of(7.3.l) for-1<z<1,andthecorresponding Gothic letters areused forall other cases. 1°E.W.Hobson, op.cit., and E.W.Barnes, Ongeneralized Legendre functions, Quart. J.Math., 39,97(1908). I72 SPHERICAL HARMONICS: THEORY CHAP. 7 foranyrealorcomplex value ofthedegree v.Writing (7.4.l) intheform 1 oz e—(v+%)9 P"‘°°“‘°‘> =¥- d°I.andthen setting e°=cosh ot+sinhotcosalt, wearrive atanother integral representation oftheLegendre function ofthe ﬁrstkind, i.e., I" d ”v<°°sh“>=ti, <7“) where visarbitrary. Replacing vby—v—1in(7.4.2) andusing (7.3.14), we obtain P,(coshGt)=iIH(coshoi+sinhoncos1.)“dt. (74.3)0 Two other useful integral representations ofthefunction P,(cosh oz)can 7!/ ( r /3 1 ro\—a -a 9 +a FIGURE 26 bederived from (7.4.1) byusing contour integration, provided that —1<Rev<0.Webegin byconsidering theintegral If e(v+1/2)! dl Wc\/2cosha—2cosht ’ evaluated along thecontour Cconsisting ofthesegments (—oo, —ot—-p), (—ot +p,at—p)and(G+p,oo)oftherealaxis,twosemicircles ofsmall radius pbypassing thetwobranch points t=iot,andthelineImt=-rt(seeFigure 26). Letf(t)bethesingle-valued branch of\/2cosh at—2cosh tsuch that thevalues ofargfalong thesegment (—o< +p,—ot—p),thesegment (oz+p,oo),thelineImt=1-:andthesegment (—oo,on—p)are0,-1:/2, 0and 1-c/2,respectively. Then f(t)isanalytic inside C,andif-1<Rev<0,the integrals along thesegments Ret=iN, needed toclose thecontour, sac.7.4 SPHERICAL HARMONICSI THEORY I73 approach zeroasN—>oo.Therefore, passing tothelimit asp—>O,andtakingb.account ofthechange ofargfalong thepath ofintegration, weotam 1 on e(v+ 1/2)9 1 00 etv+%)9 -I' d6TI—oc\/2COSI'lO!.—2COSI10 T".1\/2COSh(i—2COShot 1 -00 e(v+ 1/2)(9+1ti) 1 —oc e(v+ %)8 TIso\/2cosh6 +2cosha Tr!_w\/2cosh6 —2COShot which after some simple transformations becomes P,(cosh0t)=Z6”1/weIIn?°-‘L d6W o\/2cosh9+2cosha +iIwi-_- de, —1<Rev<0."1oz\/2cosh6—2coshot (7.4.4) Replacing vby—v—1in(7.4.4) andrecalling (7.3.l4), weﬁndthat P,(cosh <1)=%e—<v+1/pm fw df)0 cos cos ot(7.4.5) 2J'°° sinh (v+%)(") rciO,i i de, \/2cosh 6—2cosh on where again -1<Rev<O.Adding (7.4.4) and(7.4.5), andthen subtract- ing(7.4.5) from (7.4.4), weobtain 2P,(cosh U.)=ﬁcos(v+%)rcIn_ d6,TI 0\/2COSh9+2COShot _4i. °° cosh (v+%)(-) 0T;sm(v +%)n.Ib \/2cosh6 +2COShotd6 +iI”lQ+_%)°_ dt,,"1M\/2cosh6 -Zcosha which imply thedesired integral representations P,(cosh 0!.)=2cos(v+%)11:Junﬂi d6,TI 0V2cosh 6+2cosh at (7_4_5) ot>0, -1<Rev<0, £1) ' 1 P,,(cosh ti.)=gcot(v+%)7'tI d6,Ti on\/2cosh 6—2cosh at (7_4_7) ot>O,——1<Rev<(). Next wederive integral representations ofQ,(z), theLegendre function I74 SPHERICAL HARMONICS: THEORY CHAP. 7 ofthesecond kind. Assuming thatz=cosh on(oc>0)andintroducing anew variable ofintegration in(7.3.20) bysetting sinhg=sinhgcoshti, weﬁndthat °° —(v+ 1/2)8 V II =I — 7.4.8 Q(cOs OI) ll\/2cosh6 —2COShot ( ) forRev>-1.Then writing e°=cosh at+sinhatcosh <p, wereduce (7.4.8) totheform Q,(cosha)=£w , ot>0, Rev>—l. (7.49) Formulas (7.4.1—9) were derived under theassumption thatat>0,i.e., thatz=cosh or>1,but,according totheprinciple ofanalytic continuation, theyremain valid inanyregion ofthecomplex at-plane where both sides ofa given formula represent ananalytic function. Forexample, (7.4.2) holds in theregion Recosh at>O,while (7.4.6) holds inthewhole z-plane cutalong [—oo,—1]. Finally, wederive anintegral representation ofthefunction P,,(z) which isvalid intheinterval —1<z<1.Inthiscaseweset z=cosl3(0<I3<r:), sing=sin§sin<p informula (7.3.12), obtaining 2I‘ cos(v+%)(iP. =-ii_____ d6 7.4.10(cow) r=.I0\/2cos(i —2cost3 ( ) forarbitrary values ofthedegree v. 7.5. Some Relations Satisﬁed bytheLegendre Functions Thedifferential equation (7.3.l) does notchange ifwereplace vby—v—1 orzby~z,andhence ithassolutions P_,,_ 1(2), Q_,,_ 1(2), P,(-—z) and Qv(—z), aswellasP,,(z) andQ.,(z). Since every three solutions ofasecond- order linear differential equation arelinearly dependent, there must becertain functional relations between thesolutions justenumerated. Thesimplest such relation istheformula P—v-1(2) =Pvfl), (7-5-1) SEC. 7.5 SPHERICAL I-IARMONICSZ THEORY proved inSec.7.3.Toobtain arelation connecting P,,(z), Qv(z) andQ_,,_ 1(2), weassume temporarily that z>1and -1<Rev<0.Inthiscase, —l<Re(—v—1)<0,andusing formulas (7.4.7—8), wehave Q,,(cosh <1)-Q_,,_1 (cosh at)=rtcotvrcP,(cosh ll), or sinvrc[Q,,(z) -Q_v_1(z)] =rtcosvrcP,,(z). (7.5.2) Formula (7.5.2) remains valid forallzintheplane cutalong [—oo,1],since inthisregion both sides areanalytic functions ofz.Moreover, forallzinthe cutplane, both sides of(7.5.2) areanalytic functions ofv,except when visan integer, and therefore (7.5.2) holds forallvatO,i1,i2,...Setting v=n-%(n=O,i1,i2,...)in(7.5.2), weﬁndthat Qn—‘/2(2) =Q—n—V2(z)' Wenowderive another relation between thesolutions of(7.3.l), assuming temporarily that|z|>1and[argz|<TC.Then formula (7.3.7) gives Qv(-z) =—e*‘”“Q,,(z), vaé—l, -2,..., (7.5.4) where theupper signcorresponds toImz>0andthelower signtoImz<0. Using theprinciple ofanalytic continuation, wecandrop thecondition |z|>1,thereby establishing thevalidity of(7.5.4) forarbitrary zintheplane cut along [-oo,1]and arbitrary vaé-1,-2,...Finally, combining (7.5.2) and(7.5.4), weobtain —sin vrc[e*‘”“Q,,(z) +e*‘”“Q_v_1(z)] =rtcosv7:P,,(-z), andthen using (7.5.2) toeliminate Q_v_1(z), weﬁndthat §%@om=m@eW-nod ow) wherev aé—1,—2,...,and theupper signischosen ifIm z>0andthelower signifImz<0. Therelations (7.5.1—5) play animportant roleinthetheory ofspherical harmonics. Inparticular, itfollows from (7.5.5) that 2I ' —v:rti i%Eoo+o=ame —aen2_ (7.5.6)I . - i%Eoo—o=awW-men if-1<x<1.This implies Q.,(x +i0)—Q,(x -i0)=—ir:P,,(x), —l<x<1, (7.5.7) andshows whythecutmust beextended tothepoint z=Iinthecaseofa Legendre function ofthesecond kind. I76 SPHERICAL HARMONICSI THEORY CHAP. 7 7.6.Series Representations oftheLegendre Functions TheLegendre functions deﬁned inSec.7.3areanalytic functions ofthe complex variable 2intheplane cutalong [-oo,-1]inthecaseofPv(z), and along [—oo,l]inthecaseofQ,(z). Inrestricted regions ofthese cutplanes, theLegendre functions canberepresented byhypergeometric series with various choices ofat,[3,Yand2,examples ofwhich aregiven bytheseries (7.3.6—7). Asimple method forconstructing allexpansions ofthistypeisdue toBarnes,” andisbased ontransformations ofthecontour integrals used to deﬁne theLegendre functions, butmost ofthese results canbeobtained by more elementary means. Webegin byderiving formulas suitable forrepre- senting theLegendre functions inthedomain I2]>1,[arg2|<1:.According to(7.3.7), wehave Q(z)= F(Z+1X+l-v+2-L) (751)“ I‘(v+%)(2z)"*1 2 ’22’ 2’22 '' for2inthisdomain andarbitrary vaé—l,-2,...Toobtain thecorre- sponding series expansion oftheLegendre function oftheﬁrstkind, weassume temporarily that2visnotaninteger andusetherelation (7.5.2), which can then bewritten intheform tanvrc P.<z>=—n-1Q.<z) -Q-._1(Z)l- (7.62) Substituting theseries (7.6.l) into(7.6.2), andusing formula (l.2.2) totrans- form theratios ofgamma functions weobtain I\><[\,)>-atD11_Po+%> .1_g,__.__,._Pf’)‘v;1~(.+1)(2Z)FI2 2 F("\' ""2) _._ V Y+ (2z) 1F(5+1.5 +2,v+2,2,, |2|>1,|arg2|<1:.(7.6.3) Thecondition imposed ontheparameter vcanbereplaced bytheweaker condition 2v;é2p+1(p=0,il,i2,...),since both sides of(7.6.3) re- main analytic atpoints v=p.Therefore formula (7.6.3) holds forany V7‘; i723i%,... Toderive expansions oftheLegendre functions which hold inthepartof 2°E.W.Barnes, op.cit.The reader familiar with thetheory ofthehypergeometric function canderive theformulas ofthissection asspecial cases ofthegeneral relations ofSecs. 9.5-6. Acompilation ofrepresentations oftheLegendre functions interms of hypergeometric series isgiven intheBateman Manuscript Project, Higher Transcendental Functions, Vol.1,pp.124-139. sac.7.6 SPHERICAL HARMONICS: THEORY I77 thecutplane where [2]<1,weﬁrstnote thatthesubstitution t=22trans- forms thedifferential equation (7.3.l) into dz 13d 1¢(1-z)ﬁ‘+(5-51)i‘+§(§+§)u=0, (7.6.4) which isthespecial caseofthehypergeometric equation (7.2.1) corresponding tothevalues v v I I “='2’f*=2+2’ 1/-2‘ According toSec. 7.2,thegeneral solution of(7.6.4) forI2|<1canbe ~_<I\)<written intheform (v 1 v1) (1 _ ) where AandBarearbitrary constants. Inparticular, ifthevalues ofthese constants arechosen tobeA=P,,(0), B=P,j(0), then uEP,,(z), andto obtain thedesired expansion, weneed only calculate thevalues ofthe Legendre function P,,(2) anditsderivative atthepoint 2-0. With thisaim, weset2=0intheseries (7.3.6), obtaining P,(0)=F(-v,v +1;1;%)=k_§0 %, _ 1 ir(/<-v)1“(/<+v+1) ‘r(-v)r(v +1),=, 2'</<12 __Slrlv7'tﬁ: F(k—v)F(k+v+ I.) _ TE,=, 2"k!2 ’ where wehave used formula (l.2.2) from thetheory ofthegamma function. Ifwetemporarily assume that -1<Rev<0,then (seeSec.1.5) 1{:(v1)+1) =B(k -v,v+1)=L1t”'"'1(l —t)“dt, k=0,l,2,..., andhence _Sinv7r eF(k+v+l) 1 VP,(O)_T,;0mI0 tr1(1-1) dt _ Sinvrc 1_v_ V 0°F(k+v+I)(£)k ‘" TCI,’ 1(I“’)"’,Z0 kIF(v+l) 2 Sinvrt 1_v_ V _t _v_--7-for 1(1-t)(l 2) ‘dt, I78 SPHERICAL HARMONICSZ THEORY CHAP. 7 where thereversal oftheorder ofsummation andintegration isjustiﬁed by anabsolute convergence argument. Setting 1-t=\/§,weﬁndthat 2"sinvrr1 2"sinvrcF(—V)F(2 +P,,(0) =-if s‘/2<"'1>(1- s)'”'1ds =—— ———--—--1 " ° T‘ P1_I (22)or \/P,(0)=ii. (7.6.6) -<1)1“+1)22 2 where wehave used formulas (1.2.2—3). Since both sides of(7.6.6) areentire functions ofv,ourresult holds forarbitrary values ofv.Using (l.2.2), we canalsowrite (7.6.6) intheform P6+%)... P,,(0) =—-\'—-—— cos?- (7.6.7) 1/;;1—(5+1) Once wehave found P,,(0), wecaneasily deduce P,§(0) byusing therecur- rence relation (7.8.5). This gives V ‘15) =VPv_1(0) ZV— SIII Z2739 \/1-:F(§ +5) or 2r(%+1) W P§(0) =——-T sin3. (7.6.8) \/nI‘(§ +5) where wetakeaccount offormula (l.2.1). Combining (7.6.5, 7-8), weobtain thefollowing series expansion oftheLegendre function oftheﬁrstkind, valid for[z|<1andarbitrary v: P11+1)Pv(z)= 22¢<>sEF(3 +1,—3;1;z2)\/_F(v+1) 22222 1=5(7.6.9) +i——- sin 2 — —, -_ I 2 2 \/T:I‘(% ‘I'|\>_<I\><2r(3+1)2 +1;g-Z2) SEC. 7.6 SPHERICAL HARMONICSZ THEORY Thecorresponding expansion fortheLegendre function ofthesecond kind is obtained from (7.6.9) and(7.5.5). After some simple transformations, we ﬁndthat Pe+)d. Q\/(Z) =e¥vatf/2 _i_?___ZF _____’_ +1;_;Z2 41+1)2 2/>IQ l\)<l\J< l\-lb) \_/ (7.6.1o)1P3 -)_. <2+2\/; v 1 v12+lViF§+i,—i;i;2) 1 21"(+1)2 where |2|<1,v75-1,-2,...,andtheupper signischosen ifImz >Oand thelower signifImz<0.Aformula ofeven greater practical interest is theseries expansion ofQ,,(x), obtained from (7.6.l0) and(7.3.28): l\J__<l\)<PG+I)\/TCCOS% 1 3 _g______i ____ ._. 2QV(x) — + +192:-xi) 2 2 1t+9dm4 ....en»v F(2+2’ 2;2;"2I2ri+1) -1<x<1, vaé-1,-2,... Toobtain another important class ofexpansions ofLegendre functions, wetemporarily assume that 2isarealnumber greater than 1andthat Rev>-1.Writingz =cosh ot(ot>0)andusing theintegral representation (7.4.9), weobtain 66 dtp foo dq’ V h Z _ Z Q(cos a) 10(cl ztp 2?)+1 20(cosh on+sinhotCOSh <p)“" 8cosh __e_a sinh E" :e—(v+1)ufw dﬁp 0 -26¢ 2%”H1 2+2‘Pl—e tanh E cosh V5 =e—(v+1)6tj_w dq’ 5:F(" +k‘I’1)e—2koz tanhzkf 0coSh2v+2%k=0 F(v +l)k! 2 .. ..rhe?_ —(v+1)ot F(V +k+ —2lczz an 2-e ——i e i dcp,Igio F(v+l)kI II,cOSh2v+2% I80 SPHERICAL I-IARMONICS2 THEORY CHAP. 7 where thereversal oftheorder ofsummation andintegration iseasily justi- ﬁed. Then setting t=tanhz (qa/2), weﬁndthat 2162...,tanh 2d 1k_l/ yd F(k+4}-)F(v +1)ii (P= I 2(I —I) 7: 9 0coSh2v+2% 0 2 which implies °°P(v+k+1)F(k +1%)Qv(c0sh 0,)=e—(V+1)L! kg‘) e-zra Z-...,.r<»+1>Pe) ""6+1>.<1).8‘>P<»+%) .2,/<!<v+-2->..e " Therefore wehave Q,,(cosh 6.)= e-<v+1>“F(v +1,4;v+%;8-2“), (7.6.12) or,ifwereturn tothevariable 2, Q<2)=ﬁllfiiz —~/Z2-1)""F{» +1.-1» +9.12—V?—1)2}.V F0’+2) (76.13) Let2beacomplex number belonging tothedomain |arg(2-l)|<1-c. Then w=2—\/22—l=2—\/2-l\/2+1 belongs tothedomain [w|<1,Iargw|<7:,andisananalytic function of2 (wechoose thebranch of\/22 —lwhich ispositive when zisrealandgreater than 1).Since both sides of(7.6.l3) areanalytic functions, thisformula, just proved forrealz>1,remains valid inthewhole domain |arg(2—l)|<rt. Using theprinciple ofanalytic continuation, wecanalsoeasily getridofthe condition Rev>—1,replacing itbythesingle requirement thatvat-1, —2,...Therefore (7.6.l3) holds throughout thedomain ofdeﬁnition of Q,(z), which explains theparticular importance ofthisformula. Toderive aseries expansion ofthefunction P,(z) from (7.6.l3), weuse therelation (7.6.2). Assuming temporarily that2visnotaninteger, weﬁnd after asimple calculation based on(1.2.2) that F(v +1) i Pv(Z) = tanvn(z- V22-l)""1F{v +1,-};v +%;(2 —\/22-l)"’} v 2 i__ i + (z _I‘/Z2 _1)_vF{_Vs _V;(Z_’\/Z2 —1)2}s |arg(z-l)|<1:.(7.6.14) SEC.7.7 SPHERICAL HARMONICSI THEORY |8| Thecondition imposed ontheparameter vcanbereplaced bytheweaker condition 2vaé2p+l(p=0,il,12,...),since both sides of(7.6.l4) re- main analytic atthepoints v=p.Therefore formula (7.6.l4) holds forall vaéif1%,...andforallzintheplane cutalong [—oo, 1].Forv=ii 1-%,...,theformula becomes indeterminate, andapassage tothelimit isre- quired'to obtain thecorresponding analytic expression forPv(z). 7.7.Wronskians ofPairs ofSolutions ofLegendre’s Equation Letu1(z) andu2(z) beapair ofsolutions ofLegendre’s equation, with Wronskian W{u1(z), u2(z)} [seeSec.5.9]. Then §u1—fwn+wv+nm=0, éw-wm+w+mFa andsubtracting theﬁrstequation multiplied byuzfrom thesecond equation multiplied byul,weobtain %KP-ﬁWW£Lw@H=Q which implies CW{"1(-Z), "z(Z)} =If?" Inparticular, choosing u1(z)= Qv(z), u2(z) =Q_V_1(z), assuming tem- porarily that2visnotaninteger, andletting |z|—>ooinformula (7.6.l), we ﬁndthat V-1‘ +1 \/EP(-V)112(1)=-1B_i_;)—(2Z)"[1 +0(lZl‘2)], ’\/gv V m@=-q%§§%%§u+mMoL a@=%%¥§Q?u+mMpi Therefore MmaMm=§%¥%%§%3§lu+wMp1 =-nww§u+mmpi I82 SPHERICAL HARMONICSI THEORY CHAP. 7 where wehave used formulas (1.2.l-2) from thetheory ofthegamma function. Acomparison ofthese results shows thatforourchoice ofulandu2,thecon- stant Cequals 1rcotwr,andhence WQ#LQ~@@%¥¥%§> wee—1n<» 010 Formula (7.7.l) isvalid forarbitrary vaé0,i1,12,...,since both sides arestillanalytic atthepoints v=n—%(n=O,i1,12,...).Itfollows from (7.7.l) that forallnonintegral v,Qv(z) and Q_,,_1(z) areapair of linearly independent solutions ofequation (7.3.l), except forthecaseofhalf- integral v,where theWronskian vanishes andQv(z), Q_v_1(z) areconnected bythelinear relation (7.5.3). Next letul=Pv(z), uz=Qv(z). Tocalculate theWronskian ofthispair ofsolutions, weuse(7.6.2), assuming once again that2visnotaninteger. This gives t 1 mmamm=%¥W@@Q+@»q:? |arg(z—1)|<1:. (7.7.2) According totheprinciple ofanalytic continuation, (7.7.2) isvalid for arbitrary v;-é—1,—2,...,andtherefore thefunctions Pv(z), Qv(z) areapair oflinearly independent solutions ofequation (7.3.l) foranyvsuch thatboth functions aremeaningful. Similarly, using therelation (7.5.5), weﬁndthat W%@KPm=—§%EWMAQW=~g%E§7’ |arg(1i2)]<Tc,(7.7.3) forarbitrary values ofv.Thus thesolutions Pv(z) andPv(—z) arelinearly independent ifvisnotaninteger. Finally wepoint outthat intheinterval -1<x<1wehave theformula WMﬂQ®F7%? am) where Qv(x) isthefunction deﬁned by(7.3.28), andvaé—1,—2,... The results obtained inthissection show that thegeneral solution of Legendre’s equation (7.3.l) canbewritten inanyofthethree equivalent forms u=APv(z) +BQv(z), ]arg(z—1)| <1:,vaé-1, —2,..., (7.7.5) u=CPv(z) +DPv(—z), |arg(1iz)|<rr,vaé0,il,i2,..., (7.7.6) u=EQv(z) +FQ_,,_1(z), larg(z—l)|<1:,2vaé0,il,i2,..., (7.7.7) sac.7.8 SPHERICAL I-IARMONICSZ THEORY I83 where A,B,...,Farearbitrary constants. Thesame formulas canbewritten forrealz=xintheinterval (—1,1),ifQv(x) istaken tobethefunction deﬁned by(7.328). 7.8. Recurrence Relations fortheLegendre Functions TheLegendre functions satisfy simple recurrence relations connecting func- tions with consecutive indices. Toderive these relations, wesetz=cosh ot (oz>0),assuming forthetime being that zisareal number greater than 1.Then, using theintegral representation (7.4.1), wehave P,,+1(cosh oi)+P,,_1 (cosh oz) _if“ cosh (v+Q6cosh 6do TI0\/2cosh<x —2cosh 0 4°‘COShotc0sh(v ++})6 2tr=7;L d6 —aft) \/2cosh<x— 2cosh6cosh(v +=}_;)6d6 4 °‘—-—————i—— .=2cosh otPv(cosh <1)—mfo \/2COShot —2cosh6ds1nh(v +%)6 =2cosh<1P,(cosh<1)-4fasinh(”+"96sinh9d6O (2\'+1)“ \/2cosh oz—2cosh 6 on Q9_ _19 =2cosh otPv(cosh or)—(zviDWI cOSh$ 7)d6 o cos or—cos =2cosh onPv(cosh on)-—%i [PVH (cosh on)—Pv_1 (cosh 11)], which implies (V+l)Pv~i-1(2) _(2v +1)zPv(Z) +VPv—l(Z) = According totheprinciple ofanalytic continuation, formula (7.8.1) holds for arbitrary zintheplane with acutalong thesegment [—oo,—1].Inthesame way, weﬁndthat PVH (cosh oz)—Pv_1 (cosh <1) :if“ sinh(v+%)6sinh6 de W0\/Zcosha —2cosh6 =—gr‘ Sinh(v +~})6d\/2 cosh on—2cosh90 =(Zv+1)3f“\/2¢osha- 2c0sh0cosh(v +§)6d(-1.7T0 I84 SPHERICAL 1-IARMONICSZ THEORY CHAP. 7 After differentiation with respect to01.,thisbecomes 2°‘ cosh (v+l)6P‘, C0811 —P(,_ OS] +1( oz) 1(c la) (V )7!o\/2coshoc—2cosh6 =(2v+l)Pv(cosh on), Of P$+1(Z) —Pl-1(1) =(2\'+1)Pv(Z)> (7-3-2) where theresult holds inthewhole plane cutalong [—oo,-1]. Therestoftherecurrence relations satisﬁed bythefunction Pv(z) canbe deduced from formulas (7.8.l—2). Forexample, differentiating (7.8.l) with respect tozandusing (7.8.2) toeliminate ﬁrstP,j_1(z) andthenP5,1(2)from theresulting equation, wearrive attherelations P,j+1(z) —zP,§(z) =(v+l)Pv(z), (7.8.3) zP§(z) —PQ_1(z) =vPv(z). (7.8.4) Moreover, replacing vbyv—1in(7.8.3) andeliminating P{,_1(z), wehave (1—z2)P§(z) =vPv_1(z) —vzP\,(z). (7.8.5) Recurrence relations forQ,,(z), theLegendre function ofthesecond kind, canbeobtained injustthesame way, starting from theintegral representation (7.4.8). Itturns outthatthese recurrence relations areexactly thesame asfor thefunction P,,(z): (V+1)Qv+1(-Z) -(2v+1)zQv(z) +vQ,,_1(z) =0, (78.6) Q$.1(z) —Q3-1(1) =(Zv+1)Qv(Z)v (7-8-7) Q£.1(Z) —zQ$(z) =(v+1)Qv(Z): (7-8-8) zQ¢(z) —Q6-1(2) =VQv(Z)a (7-8-9) (1—Z’)Q£(z) =vQv-1(2) —\'ZQv(Z)- (7-810) Formulas (7.8.6—l0) hold foranycomplex zintheplane cutalong [—oo,1] andforarbitrary v#—1,—2,...21Itiseasily veriﬁed that these formulas remain valid forthefunctions Q,,(x) deﬁned by(7.3.28). 7.9. Legendre Functions ofNonnegative Integral Degree and Their Relation toLegendre Polynomials Animportant class ofspherical harmonics, frequently encountered inthe applications, consists oftheLegendre functions ofnonnegative integral 21Note that vQ,,_1(z) ->1,Q(,_1(z) —>é asv-+0. SEC.7.9 SPHERICAL HARMONICSI THEORY I85 degree v=n(n=0,1,2,...).Since forv=n,equation (7.3.l) coincides with equation (4.3.8), which hastheLegendre polynomial ofdegree nasa particular solution, itisnatural toexpect thatthere isasimple connection between thisclass offunctions andtheLegendre polynomials. Toestablish theconnection, weﬁrstobserve that substitution ofv=0,1into (7.3.l0) gives f0(w) =1,f1(w) =1+4w, and then (7.3.12) implies P(,(z) —1, P1(z) =2.Since therecurrence relation (7.8.l) fortheLegendre functions coincides withtherecurrence relation (4.3.l) fortheLegendre polynomials, it follows thatthefunctions P,,(z) ofnonnegative integral degree v—n(n=0, 1,2,...)areidentical with theLegendre polynomials considered inChap. 4. The Legendre functions ofthesecond kind ofnonnegative integral degree v=ncanalso beexpressed inclosed form interms ofelementary functions. Toprove this,wesetv=0,1in(7.3.7), assuming temporarily that zisapositive number greater than 1.After some simple calculations, this leads to °° 1 1 1 2+1 Qo(Z) =2TWT =5108?], “=° +12 Z 0.9.1) °° 1 1 z z+l 91(2)- -WK "1, where, according totheprinciple ofanalytic continuation, theformulas (7.9.l) arevalid inthewhole z-plane cutalong [—oo,1].Thecorresponding expressions fortheremaining functions Q,,(z) canbederived from (7.9.l) and therecurrence relation (7.8.6). Byusing mathematical induction, itiseasily veriﬁed thattheresult canbewritten intheform Q,,(z) =%P,,(z) 10%: -—f,,_1(z), n=0,1,2,..., (7.9.2) where P,,(z) istheLegendre polynomial ofdegree n,andf,,_1(z) isapoly- nomial ofdegree n—1[f_1(z) E0].Formula (7.9.2) shows that the Legendre functions ofthesecond kind ofnonnegative integral degree have logarithmic singularities atthepoints z=i1.Bearing inmind that z+1 l+x_.lOg:=iOg-1'-';—}+7'l-‘l, forz=x1-i0(—1 <x<1),andusing thedeﬁnition (7.3.28) ofQ,,(x), we ﬁndthat l 1+ 1+ Qo(X) I§1°g%’ Q1(X) =;10g% —1, (7.9.3) Q.<x>=log§f—,’§—/.._1<><). which, inparticular, shows that Q,,(x) —>iooasx->i1. I86 SPHERICAL HARMONICSZ THEORY CHAP. 7 7.l0. Legendre Functions ofHalf-Integral Degree Another special class offunctions encountered inpractice consists ofthe Legendre functions ofhalf-integral degree v=n——§(n=0,1,2,...).” This class offunctions isalso oftheoretical interest, since thecase v=n—§ occupies aspecial position inthetheory ofspherical harmonics, andmany formulas need modiﬁcation when v=n—ii.Inthepresent section, we assume thatthevariable zisgreater than 1,setting z=cosh at((1>0).This isthecaseofgreatest practical interest (cf.Sec.8.11). Toobtain ageneral formula forthefunction Q,,_1/, (cosh oz),weuse (7.6.l2), which forv=n—-1-becomes Q,,_1/, (cosh oz)= ) e“""‘/¢>°‘F(n +<5,1};n+1;e'2°‘), (7.10.1) where on>0,n=0,1,2,...Asimilar representation ofP,,_1/,(cosh at)can- notbewritten down directly from (7.6.14),since thisformula becomes indeter- minate forv=n—-1;.However, therequired expansion canbededuced from therelation (7.6.2) byusing L’Hospital’s rule topass tothelimit v->n—~}.This gives I (710.2) Writing formula (7.6.12) intheform Q,(cosh oz)=ICED €'0r(2lc+v+1), (7_10_3) weﬁndthat 8Q,(cosh at)=iF(k+v+1)F(k +%) av noF(k+v+%)F(k +1) ><[rl»(k+v+1)—~.I.»(k+v+%)—<z]e"°“2"*"*‘), (7.10.4) 5Q_v_1 (cosh ac):_5: I‘(k—v)F(k +%) at P1/<—»+»1>P<k+1) ><[tl»(k —v)—tl»(k—v+5-)~o<]e“°“2"“’>, (7.10.5) where <,b(z)isthelogarithmic derivative ofthegamma function (seeSec.1.3). Ifwesetv=n—%(n=1,2,...),theﬁrstnterms oftheseries (7.10.5) become indeterminate, since F(k—n+1)=oo, \l1(k—n+l)=00, k=O,l,...,n—l. 2’Because of(7.5.1, 3)there isnoneedtoconsider thecasen=—1,—2,...separately. SEC.7.10 SPHERICAL I-IARMONICSZ THEORY I87 However, using formulas (1.2.2) and(l.3.4), weobtain -\l»'(k—\'+%)__ ,,_ _ _vllnflgll/Q . —-( "F(n k), k——O,1,...,n -1, which implies 8Q_v_1 (cosh 01)] av v=n—‘/2 "'1(—1)"“"I‘(n —k)F(k —n+-fr) ,,_ : 1" —oc(2 n%) Z, F(k+1) (kJ"ii)" I onPk Pk % —ot2k+n+1-1;§§§§}%%§%nw+a-1w+u—ue< @ (7.l0.6) ifweintroduce anewsummation index intheseries <1) 2“,lc=n byreplacing kbyk+n.Forn=0theﬁrstterm in(7.l0.6) must beset equal tozero. Moreover, itfollows atonce that zp Iu+ +wu+aZ <7-1°”)><[4:(k+n+-1,»)-¢(/<+n+1)-ot]e'°“2"*"*‘/1). Substituting (7.10.6—7) into(7.10.2), andnoting that (—1)"""1‘(k -n+1)=mg), according to(l.2.2), weﬁndthat P,,_1/,(cosh <1) _e°‘<"-‘/2) -21F(n-/<)r(1<+-1)e_,,,, 'n.nPw+nnn+t-m fm”@wFM+n+%Wk+%“Lwe,Z,r(/< +rt+1)F(k+1) (7'1°'8) ><[2ot+1l2(k+1)— ¢(k+&) +\lz(k+n +1)-tl»(k+n+1})]e"2’"*, where on>0,n=0,1,2,...,andtheﬁrstterm must beomitted ifn=O. Formula (7.l0.8) isthedesired series representation ofthefunction I88 SPHERICAL 1-IARMONICS2 THEORY CHAP. 7 P,,_1/,(cosh 01).Toﬁndthevalues ofthelogarithmic derivative ofthegamma function appearing in(7.l0.8), weuseformulas (1.3.6—9). Thus wehave 1 1 11%)=-Y-2log2, (110.9) 1 1¢(m+.1.)= —'\{—2IOg2-I"2<I +5+--~+2-F3). where Y=0.57721566. ..,andn=1, 2,... Integral representations oftheLegendre functions ofhalf-integral degree canbeobtained bysetting v= =1;intheappropriate formulas ofSec.7.4. Inaddition, there aresome special integral representations valid only for thisclass ofspherical harmonics. Forexample, Q,,_1/,(cosh<1)=f"€°"5-"‘*°__¢1,@, n=0,1,2,...,(110.10)0\/2cosh01 —2cos<p which iseasily proved byexpanding theright-hand sideinaseries ofnegative powers ofcosh 01,carrying outtheintegration andcomparing theresult with (7.3.7).23 Finally, wepoint outthattheLegendre functions ofhalf-integral degree canbeexpressed interms ofthecomplete elliptic integrals oftheﬁrstand second kinds 11/2 dc? It/2 ' =yo :J;J —k2S1112 (P with modulus Osk<1,afactofsome interest, since there exist detailed tables ofK(k) andE(k).2‘* Toderive these expressions, weusetheintegral representations (7.4.1) and(7.10.10) andreduce theresulting elliptic integrals tothestandard form (7.10.11). Forexample, wehave25 P_1/2 (cosh 01)=L K(tanh g)» Q_1/,(cosh 01)=2e‘°"2K(e‘°‘), 1-:cosh g (7.10.12) andsoon. 2“Seefootnote 17,p.121,andusetheeasily veriﬁed formula " H n(n+2k)!JIOCOSIKPCOS +2kQdQ= > k=0,I,2,... 2*A.Fletcher, Atable ofcomplete elliptic integrals, Phil. Mag., 30,516(1940). 25Toprove theﬁrst formula, make thepreliminary substitution sinhg =sinh 3sin<p in(7.4.1), andthen usethefourth entry inTable 4,p.319oftheBateman Manuscript Project, Higher Transcendental Functions, Vol.2.Toprove thesecond formula, usethe sixth entry inthesame table. sec.7.11 SPHERICAL 1-IARMONICSZ THEORY I89 7.Il. Asymptotic Representations oftheLegendre Functions for Large |v| The study oftheasymptotic behavior oftheLegendre functions as |2|->ooforﬁxed visanelementary problem, whose solution isanimmediate consequence ofthevarious series representations ofPv(z), Q,(z) given above. Alesstrivial problem, andoneofgreat practical importance, istoﬁnd asymptotic representations oftheLegendre functions aslv|—>ooforﬁxed z. Inthissection, itwillbeassumed thatzisarealnumber greater than 1and [argvl<{ﬁr—8(see, however, therema konp.192). Forasymptotic for- mulas valid under more general assumptions concerning zandv,werefer thereader tothespecial literature onspherical harmonics.“ Toderive anasymptotic representation ofP,(z), webegin with the integral representation (7.4.l), which wewrite intheform P,(cosh 01)=ifz (2cosh on—2cosh 6)'1’2e‘"+ '/1”d6O +£1012 coshot-2cosh 0)-We-<"+‘/wede =j,+jg.0 (111.1) Making thesubstitution t=01—6intheintegral f1,weobtain e(v+1/2)0c 01e-(v+1/2): 1 ht h -1/2d /“ml, ‘tan2°“ ’e(v+1/2)a one—(v+1/2)! d ore—(v+1/2)t TC(2sinh01)”{I-O (sinh t)1’2 I0(sinh t)1’2 I —1/2 00e—(v+!/2)t X —tanhicoth Oi) — dl—'J; em‘/Q“ 7112 = Ifs+f4—/sI- (- -) Theintegral /3canbeexpressed interms ofthegamma function, andinfact Vfa :21/2 J“)e—(v+1)t(1 _e-2:)-1/2 dz.:2-1/2B(%, 0 (TC 1/2 ) PG+1)< 1~.>+D11 2°E.W.Hobson, op.cit., E.W.Barnes, op.cit., and G.N.Watson, Asymptotic expansions ofhypergeometric functions, Trans. Camb. Phil. Soc., 22,277(1918). The lastreference gives themost detailed treatment oftheproblem. I90 SPHERICAL HARMONICSI THEORY CHAP. 7 (seeSec.1.5),which implies /3=(§)1/2[1 +o(|v|—1), (111.3) because oftheasymptotic behavior ofthegamma function forlvl—>oo, largvl <%1c—8(seeSec.1.4). Toestimate theintegral f4,weusetheinequality (1-x)‘1'2-1< x(1—a)“1'2, 0<<a<1,51 which implies Ixt —1/2 m t (1—tanh Ecoth oz) —1<21/2cosh Etanh 5coth 01, 0<<oz. From now on,weassume that O<ot0< Q<ot1<00. Itfollows that Ifsl < <21/2cosh 55coth 1xIaem“ Sm6+%)t(sinhf)_1'2 tanhé dt0 21/2coshg@011101.,I0°e-<1"5*“6+‘/1>‘(sinh 1)—1/2tanhgat0 0(1)fooe"""@1511/2 at=O(|v|"3'2, (111.4)0 where weuse(l.5.l). Finally wehave W <1) ifs] sI e-(lv| s1nb+ 1/,)t(sinh t)-1/2 dtg(“sinh 00-1/ZJI e-(|v| s1n6+1/2)tt1/2 dt ct a <(01,sinh01,)-1/2 Fwv"S1“611/2at=O(|v[“3/2). (111.5)0 Itfollows from (7.11.2-5) that e(v+%)oc fr= 2 I1+0(|\'l"1)I- (7-11-6) Toestimate f2isaneasier matter. Weseeatonce that andhence|/1<-I-fa(2cosh01-- 2cosh 0)-1/2<10 2 Tl?0 <Tlnfoa (2cosh on—2cosh 6)“/2 cosh 2d6=$1 jg=0(1). (7.ll.7) sac.7.11 SPHERICAL HARMONICSI THEORY l9I Combining (7.11.6-7), weobtain thedesired asymptotic representation ofthe Legendre function oftheﬁrstkind: e(V +1/z)0t Pv(Cosh G)= ‘I’0(iVI_1):|, (111.8) R |v|->oo, ]argv|<g—8, 0<o1o< <ot1<0O. Toderive anasymptotic representation ofQ,,(z), under thesame assump- tions, webegin with theintegral representation (7.4.8), making thesub- stitution 6=01+t: h e—(v+1/pa coe—(v+1/,)t 1 h ht —1/2 d Qv(°0$ ¢)— fo +cot octan 2-) t e—(V-I-I/2)d 00 e—(V+1/2)! =—-.———i e dt(2sinh01)”{I0 (sinh t)1’2 we—(v+1/,)t 1 1 h ht —1/2 d "1. "l+°°‘W“2)I‘ie—(v+1/2):! =@ [fa"I"/el (7-11-9) The integral fahasalready been estimated in(7.11.3). Toestimate the integral fa,Weusetheinequality l—(l+x)“’2<%x, x>0, which implies 1—(1+coth 01tanht)'1’2 <<}coth 01tanh t, t2O. Therefore lfsl<0(1)f: e-W“"611/2 dz=O(|vl'3l2), (111.10) provided that 012010>0.Combining these results, weobtain thedesired asymptotic representation oftheLegendre function ofthesecond kind: 1/2 Qv(c()5h 0;)= e—(v+ 1/z)<x[l +0(|v|-1)], (7.l1.ll) |v|—>0O, |8.1‘gv|<g—3, O<o10<01<oo. I92 SPHERICAL HARMONICSI THEORY CHAP. 7 Remark. Bysimilar methods, onecanderive asymptotic representations ofP,(z) andQ,(z) forthecase where zbelongs totheinterval (—1,1)and argv=0.Itisfound that” P,(cos0)=(V%6)1'2 sin[(v+%)0+g..].[1 +0(|v|-1)], Q.1@os0)= @0810+%)6+11:1-11+0<|v|"*>1. CD v—+o0, 8< <1c—3. (7.ll.I2) 7.l2. Associated Legendre Functions Thenext class ofspherical harmonics, inorder ofincreasing complexity, consists oftheassociated Legendre functions, which aresolutions ofthedif- ferential equation 2 (1—22);/’ —2zu' +[v(v+1)— u=O, (7.l2.l) forarbitrary vandintegral m=O,1,2,....These functions generalize the functions P,,(z) and Q,,(z) considered inSecs. 7.3-11, andreduce tothese functions form=0. Todeﬁne theassociated Legendre functions, weassume that zisan arbitrary complex number belonging totheplane cutalong [—oo,1],andwe introduce anewfunction vrelated toubytheformula u=(22-l)”"2v =(z—1)'"'2 (2+1)"‘/2v. Then equation (7.12.1) takes theform (1—z2)v” -—2(m +l)zv’ +(v—m)(v +m+l)v=O. (7.l2.2) Letwbeasolution ofLegendre’s equation (1—z2)w” —2zw’ +v(v+l)w=0. (7.l2.3) Then itiseasily veriﬁed that thefunction o=w"") satisﬁes equation (7.12.3).28 Itfollows thatthesolutions of(7.12.1) aregiven by P112)=(Z2—Ir"/2 Q{,"(z)=(Z2-1)"/2 ,lm =0,1,2,..., (712.4) where P,(z) andQ,(z) aretheLegendre functions deﬁned earlier. Thefunctions 2”SeeJ.Lense, Kugelfunktionen, second edition, Akademische Verlagsgesellschaft, Geest &Portig K.-G., Leipzig (1954), p.168ff.,andE.W.Hobson, op.cit.,p.293ff. 2"UseLeibniz’s rule(D.V.Widder, op.cit.,p.483) tocalculate thederivatives (z2v”)‘"" and(Z1/)<'">. s1=.c.7.12 SPHERICAL HARMONICSI THEORY I93 P§"(z) and QT(z) arecalled theassociated Legendre functions oftheﬁrst andsecond kinds, respectively. Itfollows from (7.12.4) andtheresults of Sec.7.3thatP;"(z) andQ(,"(z) areentire functions ofzintheplane cutalong [—oo, 1].Moreover, P§"(z) isanentire function ofv,while Q$‘(z) isamero- morphic function ofv,with poles atthepoints v=—1,-—2,... Intheapplications, itisoften necessary toﬁndthesolution ofequation (7.l2.l) forrealz=xbelonging totheinterval (~—1,1).Tothisend,weﬁrst notethatvalues oftheassociated Legendre functions ontheupper andlower edges ofthecutare Pm(x + :ei:(m7ti/2)(l _X2)»:/2 ’V — xm Q2110‘ i,I-0)=eihrmi/2)(l _x2)m/2 Then weintroduce twonewfunctions P{,"(x) andQL"(x) bywriting P,§"(x) =em"/2P;"(x +i0)=e""""2P;"(x —i0) =1-1>'"<1-X2)“ (_1)m _ _ _ _ (7.12.s) QL"(X) =—T l@"'""”2QZ."(X +10)+@'"'"’2Ql"(X —10)] =1-1>'"<1 -X2)” where —1<x<1,visarbitrary [except thatv 75—1,—2,...inthecase of QQ"(z)], m=0,1,2,....,andQv(x) isthefunction deﬁned by(7.3.28). The functions P(,"(x) andQ;"(x), which areeasily seen tosatisfy equation (7.12.l) forrealz=x(-1 <x<1),willsimply becalled theassociated Legendre functions fortheinterval (—1,1).” Inthespecial casewhere v=nisanonnegative integer (n=0,1,2,...), P,(z) —P,,(z), where P,,(z) istheLegendre polynomial ofdegree n.Then, according to(4.2.1), wehave 1 dm+n m ___ 2_ m i_ __ n P"(Z)_(Z1)/22"n!dz”‘*"(z2 1)’ (712.6) m=0,1,2,..., n=0,1,2,..., andobviously P{,"(z) E0ifm>n.Ifmsn,thefunction PZ,"(z) istheproduct 29Some authors deﬁne P;"(x) andQ;"(x), -1<x<1bytheformulas Pew=<1—x’)'"” Q2"(x)=<1—x“>“'*"—m,%;L” differing from (7.12.5) bytheconstant factor (-1)”‘, afactwhich should bekept inmind when consulting handbooks andtables involving these functions. I94 SPHERICAL HARMONICSZ THEORY CHAP. 7 of(22-1)""2andapolynomial ofdegreen -m.Intheinterval -1<x<1, theanalogue offormula (7.12.6) is Y!l'I'7L P,',"(x)=(-1)'"(1 -x2)“/2271-n! 2%,(x2-1)". (712.7) Ifwesetv=(d/dz)"‘Pv(z) in(7.12.2) andmultiply theresult by(22-l)""2, weobtain therecurrence relation Pl"”(Z)+ P1""1z>-0—m)(v+m+1>Pz"<z>=0. m=0,1,2,...,(7.l2.8) which canbeused tocalculate thefunction P{,"(z) stepbystep, starting from Pl’(Z)=Pv(z), PKZ) =(Z2 __]_)1/2P\’/(Z) = Pv_1(Z) + Pv(Z). Injustthesame way, weﬁndthat Q7”(z)+ Q3"“(z)—1»—~01»+m+note)=0. m=0,1,2,... (7.12.9) Similarly, using thedeﬁnitions (7.12.5), wecaneasily deduce recurrence rela- tions forthefunctions P.§"(x) andQ’J‘(x), obtaining PL"*2(x) + P$*1(x) +(v-m)(v +m+1)P{,"(x) =O, S_xi (712.10) QL"”(X) + Q1"+1(><> +0-m)(v+m+1)QL"(X)=0. where -1<x<1,visarbitrary [except thatvaé-1,-2,...inthecase ofQ;"(x)] andm=0,1,2,... The associated Legendre functions also satisfy recurrence relations of another type, involving functions with thesame superscript mbutdifferent subscripts v.Toderive these formulas, which generalize thecorresponding formulas ofSec.7.8,weﬁrstdifferentiate (7.8.2) mtimes with respect toz anduse(7.12.4), obtaining P,j",*11(z) -P,',"_*f(z) =(22-l)1/2(2v +1)P;"(z). (7.12.11) Then, differentiating (7.8.l) mtimes with respect tozandagain using (7.12.4), weﬁndthat (v+1)P5"+1(Z) —(21+1)ZPl"(Z) —(Zv+1)m(Z2 —1)"2Pl""(Z) +\'Pl"-1(1) =0, which together with (7.12.11) implies (V—"1+1)Pl"+1(Z) —(2\'+1)ZPl"(Z) +(v+m)Pl"_ 1(1)=0, m=0,1,2,... (7.l2.12) SEC. 7.12 SPHERICAL I-IARMONICSI THEORY This recurrence relation istheﬁrstofthetype mentioned, andreduces to (7.8.l) form=0.Toobtain twoother such recurrence relations, wedif- ferentiate (7.8.2) and(7.8.3) mtimes withrespect tozandreplace (d/dz)"‘P,(z) by(zz-1)‘"'l2P;"(z), obtaining dPm..<z> dm".1(1) Z"dz-dz-,.’"_11P1"..<z)- P:"_.(z>1 -<21+1>P1"<z>. (712.13) "P{';‘,1(Z)- zdpffl +,.”f11zP1"(z>- P1"..1111-0+m+1>Pz"(z). (7.l2.l4) where m=0,1,2,...Subtraction of(7.12.14) from (7.12.13) then gives /id":(i) -Qmﬁli) -% [zP(,"(z) -1>y_,(z)] =(v-m)P{,"(z). (7.12.1s) Form=0,formulas (7.12.13—15) reduce toformulas (7.8.2—4), respectively. Finally, replacing vbyv—1in(7.l2.l4) andusing (7.12.l5) toeliminate (d/dz)P,',"_1(z), weobtain thefollowing generalization offormula (7.8.5):3° (Z2-1)‘l£-%g_-Z) =vzP{,”(z) -(v+m)P;'=_,(z), m=0,1,2,...(712.16) Recurrence relations forthefunctions Q’J‘(z) canbederived inexactly the same way, starting from formulas (7.8.6—l0), andobviously must beidentical with thecorresponding recurrence relations forthefunctions P§"(z). Inthe caseoftheassociated Legendre functions fortheinterval (-1, 1),recurrence relations canbederived byusing (7.l2.5). Forexample, wehave (v-m+l)P,',",,1(x) -(2v+l)xP§"(x) +(v+m)Pv_1(x) =O, (x2-1)‘%) =vxP§"(x) -(V+m)P,’,"_1(x) =0,m=0,1,2,..., andsoon. Aclosely related result istheformula giving theWronskian ofthepairof solutions P(,"(z), Q(,"(z) ofequation (7.12.1). Toderive thisformula, weﬁrst differentiate each oftheequations (7.l2.4) with respect toz,andthen use (7.12.4) again toeliminate thederivatives. This gives ‘2%'.Z(l =m [(22-1)1/2P;"+1(z) +mzP;"(z)]. 7.12.17) ,,, <Q?)- [<22-l)"’Q3"*‘(Z) +mZQl"(Z)l- 5°InHobson’s treatise (op.cit.,p.290), thisformula isgiven incorrectly. I96 SPHERICAL HARMONICS2 THEORY CHAP. 7 Substituting (7.12.17) intotheexpression fortheWronskian, weobtain W{Pr<z>. Qua}= lQL"*‘(z)P€"(Z) -P;"“(Z)Q€‘(Z)l- Next weobserve that(7.l2.8) and(7.l2.9) imply theidentity QT“(Z)P€"(Z) —PI"+‘(Z)Q$"(Z) =(v+"t)(l" —V—l)[QL"(Z)P€"“(Z) —P€"(Z)QL""(Z)l, andtherefore theWronskian becomes W{Pl“(Z), QI"(Z)} =(v+"t)(l" —v—1)W{P$"'1(-Z), QI"“‘(Z)}, m=l,2,... Repeatedly applying thisformula andusing (7.7.2), weﬁndthat W1P:"(z>. Qua}=“Ff,Q‘1”)1)F§51’_‘,)”), _1,.- or,after taking account of(1.2.2), W{P{,"(z), Q{,"(z)} =%%:{_B (712.18) where [arg(z-1)]<n:, v;é—l,—2,..., m=O,1,2,... This result generalizes (7.7.2) andshows that P,’,"(z), Q{,"(z) areapair of linearly independent solutions ofequation (7.12.l), except when v=0, 1,...,m-1,inwhich caseboth sides of(7.12.l8) vanish identically. Thus, apart from thisdegenerate case, thegeneral solution of(7.12.1) canbewritten intheform u=AP.§"(z) +BQ{,"(z). (7.12.19) Itfollows from (7.12.18) andthedeﬁnition (7.12.5) oftheassociated Legendre functions fortheinterval (-1,1)that W1P1"<><>. Q$”(X)}- W1P1"<><+10).QC"(><+10>} +W{P€"(X —1'9),Ql"(X—i0)} F(v+m+1) 1=Tm Y (7.12.20) Wealsoobserve thatthedifferential equation (7.12.l) does notchange ifwe replace vby-v-1orzby-z,andhence ithassolutions PT,_1(2),Q'1v_ 1(2), P§"(—z) andQL"(-z), aswellasP(,"(z) andQ(,"(z). Since every three solutions ofasecond-order linear differential equation arelinearly dependent, there must becertain functional relations between thesolutions justenumerated. These relations canbeobtained directly bydifferentiating each oftherelations SEC.7.12 SPHERICAL HARMONICSZ THEORY I97 (7.5.1-2,4—5) mtimes with respect toz,and then using thedeﬁnitions (7.12.4). This gives P'L',_,(z) =P,’,"(z), (7.12.21) sinv1r[Q§‘(z) —Q’1.,_1(z)] =11:cosv-n:P(,"(z), (7.12.22) Q€"(—Z) =—@*”’“Q2"(Z). (7-12-Z3) Py(Z)@*"ri -P;"(-Z) =isinvvrQZ‘(z), (712.24) where m=0,1,2,...,andtheupper sign ischosen ifImz>Oandthe lower signifImz<0. Theassociated Legendre functions canberepresented byhypergeometric series insuitably restricted regions ofthez-plane cutalong [—oo, 1].The problem ofderiving allexpansions ofthistype liesbehind thescope ofthis book. Atthispoint weconsider only thesimplest examples, referring the reader interested inamore detailed treatment tothesources cited infootnote 20,p.176. Anexpansion ofP,',"(z) valid inthedomain [z-1|<2,|arg(2-l)|<1-: canbeobtained bym-fold dilferentiation oftheseries (7.3.6). First wenote that d .. _no(°‘)n(F)n k_1_ on(°¢)n+1(I5)r<+1 k a’*<“’l*"t"‘>r.Z.r1”>:*1¢T"" _LB w R; 1.2, 0+1)./<1 " ‘"‘) =0iY—BF(oc+ l,l3+ l;Y -I-l;x) forIx]<1,since (71),,,1 =7.01+1),,bydeﬁnition. Repeated application of thisformula gives %F(a,B;Y;x)= F(a+m,l3+m;y+m;x), m=O,1,2,...d (7.12.26) Itfollows that Pc"<z>-122—1>'"/2§,F(-1.» +1;1;‘-5-f) _(2—1)"!/2(_1)m(_V)m(V +l)m _ —__Z 2m (Um F(m-v,v+m+1,m+1,i2 Z). Moreover, according to(1.2.2), F+ 1 I”+1)“= F(m—v) F(v+1) <-"lm=Tm =1'9"‘i’ I98 SPHERICAL HARMONICSZ THEORY CHAP. 7 andhence mI F(v+m+l) ,_M, Pd’) 2'"F(m+1)r(v-m+1)(Z 1)I W227) XF(m—v,v+m+1;m+1;%), where |z-1|<2,[arg(z-1)]<rt,visarbitrary, andm=0,1,2,... This expansion generalizes formula (7.3.6), towhich itreduces form=0. Toobtain thecorresponding formula fortheinterval -1<x<1,weuse (7.l2.5) and(7.l2.27), obtaining Pm(x) :m(-—l)"=1"(v +m+1) (1_x2)m,2 " 2r(m+l)F(v-m+1) (712.28) ><F(m—v,v+m+ 1;m+ 1;?)- Next wederive theformula generalizing thebasic expansion (7.3.7) ofthe function Q,(z). Using theduplication formula (l.2.3), wewrite (7.3.7) inthe form ...,rk+"-¥)r(k+%1)=_ —(2k+v+1) 1 Q”) = k!F(k+V+%) Z ’IZI<’ andthen differentiate thisseries mtimes with respect toz.According to (1.2.l, 3),wehave[\,)>-nPi‘ ,,l\/lZ7 % Z-(2k+v+1)Z771 =(—1)"‘(2k +v+l)(2k+v+2)---(2k+v+m)z~<*r+v+'"+1> =(_1)m F(2k+v+m+1)Z_(,,,+,,m,,, F(2k+v+1) 2m1,(k+v+m+2)1,(k+v+m+1) =(_1)m 2 2 Z-(2k+v+m+1) P(k+3%2)1‘(k + ’ andtherefore dmd€:'n(Z) =(__1)m2m —1z—(v+m+ 1) ,F(,,,gm, ,V-_»;1_;1) 1 XIE0, kIF(k +v+ Zzk 1/_I‘( +m+1)_ = T%_)_ Z(v+m+l) v+m+2 +m+1 31 SEC.7.12 SPHERICAL HARMONICSZ THEORY I99 which implies (-1)"'\/Em +m+1 M )QT(Z)= (Z2— 1) (V+’)Z (112.29)v+m+2v+m+1_ 3_1 XF—7r_"—7—_"+?;’ where ]z|>1, |arg(z—1)|<rc, m=0,l,2,..., v;é—1,—2,.... Weconclude thisoutline ofthetheory oftheassociated Legendre func- tions byciting thefollowing integral representations which generalize the corresponding formulas ofSec.7.413‘ M=1“(v+m+l)(z2_1)“/2 P”(Z) 2M/?=r(m +{~)F(v-m+1) 7! iii V"M><fa(Z+'\/Z2-1cos¢) sin2'"t].»d¢, Rez>0,m=o,1,2,..., 0.12.30) P(,"(z) =F—i———€_\;Ij(-V’: ‘*1-)1)fox(z+\/27-:-“I coslilycosmil»dab, Rez >O,m=0,1, 2,...,(7.12.3l) m __<—wHo+m+n 1P"(cosii)_\/Er(m +%)r(»_m+1)(2sin6)” B cos(v+=})0 Xiaawrnamﬁﬁ’ O<[3<1-c, m=O,l,2,... (7.l2.32) PROBLEMS 1.Prove theformulas Pv(—x +i0)—Pv(—x —i0)=2isinvr:Pv(x), Qv(—x +1'0)—Qv(—x —1'0)=2isinvrcQv(x), where x>1. 31The parameter visarbitrary in(7.l2.30—32). For these and many other integral representations, with suggestions astoproofs, seetheBateman Manuscript Project, Higher Transcendental Functions, Vol.1,p.155ff. 200 SPHERICAL HARMONICS2 THEORY CHAP. 7 2.Derive thefollowing representations oftheLegendre function Pv(z) in terms ofhypergeometric series: v+1 v__P\,(z) ‘Z -5,1,1 Z2), |1—22]<1,|arg(z +1)[ <1:, Pv(z)= —v;1%), Rez>0. Hint. Apply themethod used toderive (7.6.9). 3.Derive thefollowing formulas: Pv(z)=(Z+\/Z2-1)vF(_v,%; 1;ail z+ z— ’ 2(\/22 -1)IDi 1, —' 1 , Z+X/Z2 _1 < |arg(z )|<11: 1 2\/Z2 -1 : __»\/_§_Tv ___ _. ._i____ ’P\,(z) (z z l)F( v,2,1, z_\/ziul) 2\/Z2 -1 , < 1, |arg (Z — <77.‘, z—\/z2— l— 1 lP\,(z)=z"F(—-%»—-2-—v;l;l —?), Rez2>E, |argz| <1r. Hint. Expand theintegrand of(7.4.3) inseries ofpowers ofsinz(11:/2) cosz (111/2) andcostl»,andthen integrate term byterm. 4.Derive thefollowing formulas — (z —l)""1F(l +v,l +v;2 +2v;%),Q“(Z)"2"+11“(v +7 |z—1| 2,|g(—1)|<1r, v¢—1,——2,..., > 3.1‘ Z Q<z)=i,,-<z+1>" F1+»,1+»; +»;1+Z» V 2v+l1"(V +7 [z+ 1|>2, |arg(z+1)| <1r,vaﬁ —1,—2,...,\/n1“(»+1) __1< 222) ) \/'1"(v+1) _,+ v+1v+1 31Qv(Z) = 2 — /2“ 1)F<—-2is——2i;V + a), [22—1[>1,|arg(z—1)|< Tc,v¢-1,—2,.. Hint. Apply themethod used toderive (7.6.9). 5.Prove theformula Qv(Z)=A/Q’ <22-1)-“(Z —we-l)w _\/T1xF1,1-v+2._L_Z__,22’ 2’ 2\/Z2-1 z—\/z2—-1 < 2\/z2—li1,|arg(z —l)|<Tc,v¢—1,~2,.. PROBLEMS SPHERICAL HARMONICS2 THEORY 20l Hint. Introduce thenewvariable ofintegration t=0—oiin(7.4.8), and thenexpand inpowers of1—e“. 6.Prove thatifvisnotaninteger, then theasymptotic behavior asz—>—1of theLegendre function oftheﬁrstkind anditsderivative isdescribed bythe formulas” Pv(z)zm%‘10g%1. P\’,(z)Z z->-1. 7.Using theresult ofthepreceding problem and thefunctional relations connecting theLegendre functions oftheﬁrst andsecond kinds, show that foranyv,thefunction Qv(z), [arg(z-l)|<717hasalogarithmic singularity atz=1,while thefunction Qv(x), —1<x<1haslogarithmic singularities atboth endpoints oftheinterval (-1, 1). 8.Derive theintegral representations Pv(Cosh 1)=f.(Tl_F'T5 fan6"°°sh°‘Io(!Sinh 0t)i"dt, o Q.,(cosh Ot)=ﬁglfﬁ fooe““Sh°‘K0(t sinh oc)t”dt, 0 Pv(COS = 3 In €‘t cos eJQ(t S111 e)lv dt, 0 where ilmaisg! Rev>—1, 0<6<rc, andJo(x), Io(x) andKQ(x) areBessel functions. 9.Derive theintegral representations PV_,/2(coSha) =$A/éjt e_,¢0=haK7“(;Qdt, {Rev]<5,or>O, Qv_1/,(cosha) = e"°°=“°‘I;’/i(t;)dt, Rev >—%, at>0, O where I.,(t) andK.,(t) areBessel functions ofimaginary argument (see Sec. 5.7).” 10.Prove theformulas 1 m m 1 m 2 ( I_1P.<x)P.(x> dx=0,f_11P.<x>12 dx= m=0,1,2,..., l=m,m+ l,..., n=m,m+ l,..., generalizing theresults ofSec.4.5. 32Apossible approach istousetheexpansion ofP.,(z) given byE.W.Hobson, op. cit.,p.225. 3“Proof oftheformulas given inProblems 8-9canbefound inWatson's treatise (op.cit.,p.387). 202 SPHERICAL I-IARMONICSI THEORY CHAP. 7 Comment. These formulas playanimportant roleinthetheory ofseries expansions withrespect tothefunctions P,'{‘(x). 11.Prove thefollowing addition theorem fortheLegendre polynomials: P,,(zz' —1/z2 -—l\/2'2 —lcos cp) =P.<z>P.(/> +2§1(—1)'"((—--:1’; PII‘(z)P.’{‘(Z') Cosme- Prove theanalogous theorem fortheLegendre functions :3‘ P.,(zz’ —\/z2 —1\/z’2 -—lcos cp) =P.1z>1>.</) +2i1-1)" P1<z)Pc"</> cos"W.m=1 |arg(z—l)|<1:,|arg(z’—l)|<1:,Rez >0,Rez’>0. 12.Prove thattheLegendre functions ofcomplex degree v=-1}+iTsatisfy theintegral equation cosh 11:1 °°P_ (y) P-1/2+¢.(X)="ﬁ,—-fl %j,_t%,—dy, 1<X<°°~ 13.Derive thefollowing integral representation ofthesquare ofthefunction P—1/2+i'r(-x): P_12T 2= , 1€ <00. ii /+i(x)] xcosh 1:1 °° P_1/,+,,(y) TU 1 2 14.Derive thefollowing asymptotic formulas fortheLegendre functions of complex degree v=-1}+iv“ e19 P_§’2+11-(COSe)% , T—>0O, 8<6<r:—3, TIT \/213-1/2+“-(COSh0€)% S1H(dT+%W), 'l'——>0O, 8SOL<l1<O0. \/-rrrslnllot 15.Prove that 1m _ 1"(2m +1)F(m +n+1) idX2P2"(")d" “22"“I‘(m-n+1)r(2m +Zn+2) 16.Prove theformulas fl Pz..(X) -11/coshzot —x2 ii PW) dx=2P2..(0)Q2..(¢0Sh<1)- (ii)dx=2iP2,,(O) Q2,.(i sinh oz), (i) -1\/sinh” at+x2 3"Fortheproof ofthese andsimilar formulas, seeE.W.Hobson, op.cit.,Chap. 8. 35These formulas areimportant inconnection with theproblems ofmathematical physics considered inSecs. 8.5,8.9,12-13. They arespecial cases ofgeneral asymptotic formulas given inBarnes’ paper (op.cit.). PROBLEMS SPHERICAL HARMONICSZ THEORY 203 Hint. The substitution oi—>a —lirriconverts (i)into (ii).Toprove (i), expand '\/cosh2 at-—x2inapower series andintegrate term byterm, using theresult ofthepreceding problem. Also anticipate formula (9.5.2), and use(7.3.7) and(7.6.7). SPHERICAL HARMONICS: APPLICATIONS 8.I.Introd uctory Remarks Thepresent chapter isdevoted tothestudy ofsome boundary value prob- lems ofmathematical physics which canbesolved bytheuseofspherical harmonics. Except forSec.8.14(dealing with Helmholtz’s equation), wewill beconcerned exclusively withpotential theory, i.e.,withsolutions ofLaplace’s equation. Infact, wewillconﬁne ourattention totheDirichlet problem, which, according toSec.6.3,canbestated asfollows: Given adomain 1:with boundary 0',andafunction fdeﬁned on0,ﬁndthefunction usuch that1)uis harmonic in11andcontinuous intheclosed domain -r+cr,and2)ucoincides withfon<1.Inthecaseofanunbounded domain, thisstatement oftheprob- lemmust besupplemented byacondition characterizing thebehavior ofthe function uatinﬁnity. Aneffective general method forsolving boundary value problems isto ﬁndasystem Soforthogonal curvilinear coordinates oi,[5,Ysuch that l.Thesurface ocorresponds toaconstant value ofoneofthecoordinates <1,B.Y; 2.Variables canbeseparated inLaplace’s equation, after ithasbeen transformed tothesystem Sbyusing theformulas x:x(a> B:Y)’ y:y(a's BaY)’ Z:z(°L> grY)‘ Ifsuch acoordinate system Scanbefound, then asolution oftheproblem canusually beobtained bysuperposition ofparticular solutions ofLaplace’s 204 SEC. 8.2 SPHERICAL HARMONICSZ APPLICATIONS equation written inthesystem S(cf.Sec.6.3). Inthisregard, weremind the reader ofthefollowing factfrom advanced calculuszl Ifthesquare ofthe element ofarclength inthesystem Sisgiven by use=hgdot’+hﬁdﬁz +11$dyz, (s.1.2) interms ofthemetric coefﬁcients h,,,h[,,hY, then inthesystem S,the Laplacian operator takes theform _1ah,,/ta. ah,h.,8u ah,.hB6u V2“"h.h.h.i@<»(h.’@e) Weir. ﬁr)+@1(h. 81)]<8“) 8.2.Solution ofLaplace’s Equation inSpherical Coordinates Oneofthemost important systems oforthogonal curvilinear coordinates permitting separation ofvariables inLaplace’s equation isthesystem of spherical coordinates r,6,cp,related totherectangular coordinates x,y,zby theformulas x=rsin6coscp, y=rsin6sin<p, z=rcos 6, (8.2.l) where CD 0<r<oo, Os <rc, —n<<p<1'c. The corresponding triply orthogonal system ofsurfaces consists ofthe spheres r=const, thecircular cones 6=const andtheplanes rp=const passing through thez-axis. Moreover, thesquare oftheelement ofarclength is dsz=drz+r20'62+r2sin26d<p2, (8.2.2) andhence, according to(8.l.2), themetric coefﬁcients are h,=1, he=r,ha,=rsin6, andLaplace’s equation takes theform [cf.(8.l.3)] 1a a 1a.a 1a2v2u=__(r28’:)+ 2. (s1n68g)+ 2., ‘i=0. (23.2.3)r28r rsin666 rsin68), (8.2.4) then variables canbeseparated, sothat theproblem ofdetermining each factor in(8.2.4) reduces tothesolution ofanordinary differential equation. 1F.B.Hildebrand, op.cit.,p.302. 206 SPHERICAL HARMONICSZ APPLICATIONS CHAP. 8 Infact, substituting (8.2.4) into(8.2.3), multiplying byr2sin”6anddividing by R®, weﬁndthat ld 2dR l d. d6) .2 ld2<DTiara +€SmEs1n6% sin6=—6T2-, which ispossible only ifboth sides equal aconstant, which wedenote by11.2. This leads totwoequations dzfl) TP2 -l-tkzq) =0, 1i(r,dR)_ {L2_1(dSin6do)_ Rdr dr~sin26 ®sin6 d6 d0 Thesame reasoning shows thatboth sides ofthelastequation must equal a constant, which thistime itisconvenient todenote byv(v+l).Asaresult, weobtain theequations(s.2.s) ld .d6) 2Si?) 86 (S111 6 'l'[v(v + — ®=0, %(r2gz)_v(v+l)R=0. (8.2.?) Thus, determining thefactors intheproduct (8.2.4) reduces totherela- tively simple problem ofsolving theordinary diﬁerential equations (8.2.5—7). Thecorresponding particular solutions (8.2.4) ofLaplace’s equation depend ontwoparameters 11andv(ingeneral, complex)? which canbeused tocon- struct solutions ofboundary values problems ofmathematical physics involv- ingvarious special domains (spheres, cones, etc.). Theparameters 11,vand thecorresponding solutions ofequations (8.2.5—7) must bechosen insuch a waythateach particular solution (8.2.4) isharmonic inthegiven domain, and anappropriate superposition ofparticular solutions solves thegiven boundary value problem. 8.3. The Dirichlet Problem foraSphere Asasimple example oftheapplication ofthesuperposition method, we consider theinterior Dirichlet problem foraspherical domain. Tokeep things assimple aspossible, weassume thattheboundary function fandthe solution uareindependent oftheangle q>.Choosing theorigin atthecenter ofthesphere (ofradius a)andthez-axis along theaxisofsymmetry, wecan formulate ourproblem asfollows: Find thefunction u=u(r,6)such that1)u 2Without lossofgenerality, wecanassume thatRep.20andRev 2—§,since replacing uby-11orvby—v—1does notaffect theseparation constants 11.2and v(v+1). SEC. 8.3 SPHERICAL I-IARMONICSI APPLICATIONS isharmonic inthedomain r<aandcontinuous intheclosed domain r<a, and2)usatisﬁes theboundary condition u|,=,, =f(6), where f(6)iscontinuous intheinterval 0<6<1:.“ Therotational symmetry oftheproblem corresponds tosetting (D=lin (8.2.4) andit=0in(8.2.6). Then (8.2.6) reduces tothedifferential equation (7.3.l) fortheLegendre functions ofargument x=cos6,which for —l<x<1hasthegeneral solution [cf.(7.3.29)] G)=AP,(cos6)+BQV(cos6), (8.3.l) where Pv(x) andQ\,(x) areLegendre functions oftheﬁrstandsecond kinds, andvisanarbitrary complex number such that Rev 2~-}."‘ Since the variable x=cos6actually ranges overtheclosed interval [—1,1],andsince as x->1,Q,,(x) —>oowhile Pv(x) remains bounded [cf.(7.3.13, 23)andProblem 7, p.201] wemust setB=0ifthesolution istoremain bounded inside the sphere. Moreover, since PV(x)—> ooasx-> -1unless visanonnegative integer [cf.(4.2.6) andProblem 6,p.201], thesame reason compels usto choose v=n(n=0,1,2,...).Therefore, theonly solutions of(8.2.6) for it=0which remain bounded intheclosed interval 0<6s1:correspond to nonnegative integral vandareoftheform o=AP,(cos0), n=o,1,2,..., (s.3.2) where P,,(x) istheLegendre polynomial ofdegree n.Asfortheradial equa- tion(8.2.6), itisanEuler equation, with general solution (forvaé—%)5 R=Cr"+Dr“"‘. (8.3.3) Inthepresent casev=n,andtherequirement thatthesolution bebounded atthecenter ofthesphere compels ustochoose D=0.Itfollows that R=Cr“, n=0,1, 2,..., (8.3.4) andhence theappropriate setofparticular solutions ofLaplace’s equation inside thesphere is u=u,,=M,,r"P,, (cos6), n=0,1,2,. .. (8.3.5) Wecannow solve ourboundary value problem bysuperposition ofthe solutions (8.3.5). Infact,suppose theboundary function f(6)canbeexpanded inaseries ofLegendre polynomials (seeSec.4.7), i.e., f(a)=f,,P,,(cos e), 0<<TE, (83.6) CD 3The statement oftheproblem must besuitably modiﬁed iffhasdiscontinuities. 4Asalready noted (seefootnote 2),thisistheonly case that need beconsidered. 5E.A.Coddington, op.cit.,Theorem 1,p.147. SPHERICAL HARMONICSZ APPLICATIONS CHAP. 8 where f,=(n+g)f0"f(e)P, (cos1))sined6, (8.3.?) andsuppose theseries (8.3.6) converges uniformly intheinterval [0,1:].Then, choosing M,,=f,,a‘" andsumming thesolutions (8.3.5), weobtain theseries u=2 P,,(cose), (s.3.s)n=0 which, according toHarnack’s theorem onsequences ofharmonic functions,“ converges uniformly for0sr<atoaharmonic function with boundary values ulr= a: i.e.,(8.3.8) solves theDirichlet problem forasphere.” Remark 1.Thesolutions oftheNeumann problem andthemixed prob- lem,involving theboundary conditions (6.3.1b) and(6.3.lc), canbeobtained bysimilar methods. Remark 2.Inthecaseofthemore general problem where f=f(6,<p)isa function ofboth angular coordinates, itturns outthattheappropriate setof particular solutions ofLaplace’s equation inthedomain r<ahastheform“ u=u,,,,,=[MM cosmgo+N,,,,,sinm<p]r"P,T (cos6), (839) m=0,l,2,..., n=m,m+1,m+2,..., H interms oftheassociated Legendre functions P1,"(cos6).Moreover, by replacing thefactor r"in(8.3.9) or(8.3.5) bythelinear combination Cr"+Dr‘"‘1, weobtain particular solutions which canbeused tosolve boundary value problems foraspherical shell, orforthedomain lying out- sideasphere (inthelatter case, wemust setC=0toprevent thesolution from becoming inﬁnite asr—>oo). 8.4. The Field ofaPoint Charge inside aHollow Conducting Sphere Asanapplication oftheresults ofthepreceding section, consider the problem ofdetermining theelectrostatic ﬁeldduetoapoint charge qinside a 6SeeW.J.Sternberg and T.L.Smith, TheTheory ofPotential andSpherical Har- monics, University ofToronto Press, Toronto (1952), pp.216, 247, andR.Courant and D.Hilbert, Methods ofMathematical Physics, Vol.2,Interscience Publishers, New York (1962), p.273, where theresult iscalled Weierstrass’ convergence theorem. 7One canalso solve theinterior Dirichlet problem forasphere inthecase where f(6)isonly piecewise continuous. Seetheanalogous treatment oftheinterior Dirichlet problem foracircle, given inA.N.Tikhonov andA.A.Samarski, op.cit.,pp..284, 301. BSeeE.T.Whittaker andG.N.Watson, ACourse ofModern Analysis, fourth edition, Cambridge University Press, London (1963), p.392. sac.8.4 SPHERICAL HARMONICSZ APPLICATIONS 209 hollow conducting sphere ofradius a,held atzero potential. Choose the origin 0atthecenter ofthesphere, andletthez-axis passthrough theposi- tion Aofthecharge, which isatdistance bfrom O(seeFigure 27).Toeliminate the z singularity atA,wewrite thepotential gl:of theelectrostatic ﬁeldasasumofthepotential ofthesource and thepotential uofthe secondary ﬁeldduetothecharges induced on theinner surface ofthesphere, i.e., QY?Dg‘=~.¢=g+m wan where p=AP=\/r2+bz-2brcos6 isthedistance from Atoavariable point P, with coordinates r,6.9Since 1.]:must vanish onthesurface ofthesphere, determination ofthefunction u=u(r,6)reduces tosol- vingtheDirichlet problem withtheboundary conditionFIGURE 27 _=-—===i===== a smulna \/a2 +b2—2abcos6f() (i ) The right-hand side of(8.4.2) caneasily beexpanded inaseries of Legendre polynomials, andinfactthere isnoneed toevaluate theintegral (8.3.7). Instead, weuseformula (4.2.3) which immediately implies ﬂ_a=__Q§%€YP(ws®. man 1- 1|.an: Moreover, since b<aitfollows from theestimate (4.4.2) that theseries (8.4.3) isuniformly convergent intheinterval [0,TC].Therefore, according to Sec.8.3,thefunction uisgiven bytheformula q” b"u=-En20 P,,(cos6). (8_4_4) Using (4.2.3) again, weﬁndthatthesumoftheseries (8.4.3) is lu=—(1 _i__ =in (8,4_5) 41 br br2P1—23cos6+——2G a where 2 qE=q% v=%. @=Vﬁ+b”—%7wm. 9Since theproblem isrotationally symmetric, uisindependent oftheangle cp. 2l0 SPHERICAL HARMONICS: APPLICATIONS CHAP. 8 Thus thepotential ti»canbewritten asasum I qq=--,, 8.4.6 1»P+P () where theﬁrstterm isthepotential ofthecharge qintheabsence ofthecon- ducting sphere, andthesecond term isthepotential oftheimage charge q’at theimage point A’,which takes account oftheinﬂuence ofthesphere.” 8.5. The Dirichlet Problem foraCone Theability toseparate variables inLaplace’s equation written inspherical coordinates alsoallows ustosolve boundary value problems forthedomain bounded bythesurface ofaninﬁnite circular cone. Choose theorigin atthevertex ofthe cone, andletthez-axis liealong theaxisof symmetry ofthecone (seeFigure 28).Then theequation ofthecone is6=60(60<Tc), and theDirichlet problem forthecase of\\ \ id l‘ , axially symmetric boundary conditions can 0 1 bestated asfollows: Find thefunctions /l u=u(r,6)such that l)uisharmonic inthe ’ domainO <r<oo,O<6<60andcontinuous intheclosed domain O<r<oo,0<6<60, and 2)usatisﬁes theboundary condition u|8=8o =f(r) and thecondition atinﬁnity u|,_..,, ~>0uniformly in6,11where f(r) iscontinuous intheinterval 0Sr<oo and./<r>1.-.. -0.Inapplying themethod ofseparation ofvariables tothisproblem, we must setB=0in(8.3.l), ifthesolution istoremain bounded ontheaxisof thecone. However, inthepresent case, there isnoreason tochoose vto beanonnegative integer, since P,,(cos 6)isbounded forarbitrary vif 0<6<60. Infact, with some extra restrictions onthefunction f(r), the problem canbesolved bychoosingFIGURE 28 v=-—%-l-it, 1'20, which corresponds tothefollowing setofparticular solutions ofLaplace’s equation: u=u,=[M,cos(1logr)+N,sin(Tlogr)]r‘1’2P_;/2+“ (cos6).(8.5.l) Here M,andN,arearbitrary continuous functions (120),andthesolutions 1°Note thatp’=A’P, i.e.,p’isthedistance between theimage point A’andthe variable point P(seeFigure 27). 11Thesecond condition isnecessary fortheuniqueness ofthefunction u.SeeA.N. Tikhonov andA.A.Samarski, op.cit.,p.288. sac.8.5 SPHERICAL I-IARMONICSI APPLICATIONS 2|l depend continuously ontheparameter -r.Using (7.3.6), weﬁnd that the Legendre functions ofcomplex degree appearing in(8.5.l) have theseries expansion P_./,,.,, (cos6)=F(-,1.+i-r,%—it;1;sinz (8.52) i+T2 .6 (L+'t2)(2 +T2) . 6 =I+%T)TSlH2§+ S1H4i+--- Itfollows from (8.5.2) thatP_1/, H,(cos6)isrealandsatisﬁes theinequalities Qcc1<P-1/.+..(<><>S9). 0< <r=.(8.53) P-1/2+t1(cO5 0)<P—1/¢+i1(¢O5eo), 0$ 5eo- Now suppose thatf(r)issuch that<p(r)=r1’2f(r) hasaFourier expansion oftheform” g(r)=r1l2f(r) =J00[G,(-r) cos(rlogr)+G,(-r) sin(-rlogr)]dr, 0 0<r<oo, G,(-r) =%f0w f(r)r“1l2 cos(-rlogr)dr,G,(r) =iJ:f(r)r‘1’2 sin(-rlogr)dr, (8.5.4) where theintegral isuniformly convergent inevery ﬁnite subinterval [r1,r2] such that0<r1<r2<oo.Then, choosing G¢(T) G11)M,=-_-_. N,=2-__P— 1/2+i\' (COS 60) P— 1/2+iT (cos 00) in(8.5.l), andintegrating with respect totheparameter -rfrom 0tooo,we obtain thefunction P_i/2 -(cos6)u=rrl/2 J'0°° [GC('r) COS(Tlogr)+Gs(1') Sin('1'logr)] dr, (8.5.5) which gives thesolution ofourproblem, atleast formally. 1’Theexpansion (8.5.4), which reduces tothestandard form oftheFourier integral ifwe make thesubstitution logr=E(—oo<E<oo),isvalid iff(r) iscontinuous andof bounded variation inevery ﬁnite subinterval [r1,rz],where 0<r1<V2<oo,andifthe integral fa“1/<r)|r-"2 dr0 isﬁnite. SeeE.C.Titchmarsh, Introduction totheTheory ofFourier Integrals, second edition, Oxford University Press, London (1950), Theorem 3,p.13. 2l2 SPHERICAL HARMONICSZ APPLICATIONS CHAP. 8 Example. Find theelectrostatic ﬁeld duetoapoint charge qontheaxisof ahollow conducting cone, heldatzeropotential, ifthecharge isatdistance a from thevertex ofthecone. AsinSec.8.4,wewrite thepotential 11.1asasum 1»=%+1., (s.s.6) where p=\/r2 +a2—2arcos6.Then usatisﬁes theboundary condition =11’)=- 1*“) _ O Using theintegral representation (7.4.6), weﬁndthat °° (rlogr)Ge =_iJ d (T) TFo\/r\/r2+a2—2arcos60 r __ qJ“ cos(~rlogr) Q M0ar __ qJ“ cos[r(s+loga)] ds "Va —w\/2coshs-2cos60(s.s.s) __2qcos(-rloga)J“ cos-rs ds Tn/E 0\/2coshs—2cos60 ___q_cos(rloga) __ 6—‘ coshnq: P—1/2+11( cos 0)’ andsimilarly, '1G41)=-V”; Pu/....(-cos 6.). Thus thesolution oftheproblem isgiven bytheintegral _ q°°P—1’2+i1(cO5 9) _ COS[1102(r/e)lll—— — J0 P_i@+;,( COS 60) —iiSh WT d1‘. Itisnothard toseethatthisintegral isabsolutely anduniformly convergent forr1<r<r2,0<6<60, where0 <r,<r2<oo.Infact,itfollows from (8.5.3) thattheintegral inquestion ismajorized bytheintegral” °° d1‘ _I 60LP_1/, H,(-cos 60)gt-_-r -Ecos5- (8.5.10) 1°Toverify (8.5.l0), setB=1-:in(8.12.8). sac.8.6 SPHERICAL HARMONICS2 APPLICATIONS 2l3 Using thisresult, wecanprove thatformula (8.5.9) actually gives thesolution ofourproblem.“ 8.6. Solution ofLaplace’s Equation inSpheroidal Coordinates Wenowturntoother systems oforthogonal coordinates permitting sepa- ration ofvariables inLaplace’s equation, andleading toparticular solutions which canbeexpressed interms ofspherical harmonics. Webegin ourdis- cussion byexamining twocoordinate systems suitable forsolving boundary value problems forspheroidal domains.“ First weconsider prolate spheroidal coordinates 01.,(3,<9,related totherectangular coordinates x,y,zbythe formulas x=csinhocsiniicosgo, y=csinhasinpsinqo, z=ccoshacosi-3, (8.6.1) where O<ot<OO, Oé <11, —*rr<<p$1r, 'CD andc>0isascale factor.“ Then every point ofspace ischaracterized bya unique triple ofnumbers at,B,cp.Thecorresponding triply orthogonal system ofsurfaces consists oftheprolate spheroids oi=const with fociatthepoints (0,0,ic), thedouble-sheeted hyperboloids ofrevolution B=const, which areconfocal with thespheroids, andtheplanes cp=const passing through thez-axis (seeFigure 29).Asimple calculation shows thatthesquare ofthe element ofarclength is ds2=c2(sinhz at+sing13)(dotz+d62) +c2sinh’ asin28d<p2. (8.6.2) Therefore themetric coefﬁcients are ho,=h0=c\/sinhz on+sin’(5, h=csinhonsinB, andLaplace’s equation takes theform [cf.(8.2.3)] Vzu= 1 [18(sinh at +-L 2(sin(5-83)c2(sinhz at+sinz6)sinhatat. Zia sin13as op l 1 62u +lsinhz at+sin”6)Eiqazl —0'(8'6'3) 1‘Inexamining theconvergence ofintegrals involving Legendre functions ofcom- plex degree v=-1+i-1:, itisuseful torecall theasymptotic formulas proved in Problem 14,p.202. 15The terms spheroid and ellipsoid ofrevolution aresynonymous, and spheroidal coordinates might becalled degenerate ellipsoidal coordinates, since cross sections ofthe coordinate surfaces normal tothez-axis arecircles rather than ellipses (concerning ellipsoidal coordinates, seeE.W.Hobson, op.cit.,Chap. ll). 1°Ifapoint hascylindrical coordinates r,qaandz,thenz+ir=ccosh(oz+i[3). 2l4 SPHERICAL HARMONICS: APPLICATIONS CHAP. 8 1/ FIGURE 29 Now suppose welook forsolutions of(8.6.3) which have theform u=A(a)B(B)(<p). (8.6.4) Then thevariables separate, justasinSec.8.2,andthefactors A,B,<1)satisfy theordinary differential equations d2(I>T? +(PG) =0, (8.6.5) + mLB% (sin(1%) [v(v+1)- B=0, (see) 2 ﬁt7‘;(sinh1%)-[v(v+1)+§,51’,ﬁ(]A =0,(8.61) where p.andvareparameters whose choice isdictated bytheconcrete condi- tions oftheproblem. Forexample, intherotationally symmetric casewhere uisindependent ofthevariable cp,wesetpt=0,<1)=1,while inthemore general casewhere udepends on<p,weset(.1=m(m=0,l,2,...),since u must beperiodic incp. Next weconsider oblate spheroidal coordinates ot,6,cp,related tothe rectangular coordinates x,y,zbytheformulas x=ccoshasinl5cos<p, y=ccoshotsin(5sin<p, z=csinhacosB, (8.6.8) where 17 0<ot<OO, O< <11, —1c<<p<T:. '® 1”Ifapoint hascylindrical coordinates r,tpandz,wenowhavez+ir=sinh(ot+i6). SEC. 8.7 SPHERICAL HARMONICSI APPLICATIONS Inthiscase, thetriply orthogonal system ofsurfaces consists oftheoblate spheroids at=const, thesingle-sheeted hyperboloids ofrevolution B=const andtheplanes <p=const (seeFigure 30).Thesquare oftheelement ofarc length andLaplace’s equation nowtake theform dsz=c2(coshz at—sin”B)(da2 +d6”) +c2coshz atsinz(5dcpz, (8.6.9) l 1 8 8u l8.du 2 ___ __ i _ i i _ _ Vu—c2(cosh2 on—sinz6)lcosh at86¢(cosh oi86:)+sin606ismB66) l 1 éizu+ -E-E) 5?]_0.(8.6.10) I a=const '{5"‘§*'>2 =COIIST FIGURE 30 Separating variables, instead of(8.6.5—7) weﬁndthefollowing system of equations fordetermining thefactors A,Band(D: 2 % +u2 =0, (8.6.ll) §1LEZ1%(sin(5i1Tl]g) +[v(v+1)_$8 =0, (8.6.i2) $53 (cosh8%)-[v(v+1)-$128 =0.(8.613) 8.7.The Dirichlet Problem foraSpheroid Using theparticular solutions ofLaplace’s equation Vzu=0found in Sec.8.6,wecanconstruct functions harmonic intheinterior orexterior ofa ZI6 SPHERICAL HARMONICS2 APPLICATIONS CHAP. 8 spheroid, thereby solving theboundary value problems ofpotential theory fordomains ofthistype. Tokeep things assimple aspossible, weconsider theDirichlet problem, assuming thattheboundary function fandthesolu- tionuareindependent oftheangle <p.Webegin with thecase ofaprolate spheroid. The rotational symmetry oftheproblem corresponds tosetting (D=1in(8.6.l1) and11=0in(8.6.l2—l3). Then equation (8.6.l2) reduces tothedifferential equation fortheLegendre functions ofargumentx =cosli (cf.Sec.8.3), whose only bounded solutions intheclosed interval [0,TC]are oftheform B=cP,,(66sB). n=0,1, (87.1) where P,,(x) istheLegendre polynomial ofdegree n[cf.(8.3.2).]1‘1 Todeal with equation (8.6.7), weobserve that (8.6.7) transforms into equation (8.6.6) under thesubstitution (5=ia.Therefore thegeneral solution of(8.6.7) for(I=0,v=nisoftheform A=MP, (cosh M)+NQ,, (cosh at). (8.7.2) Ifat=<10istheequation ofthespheroid onwhich theboundary conditions arespeciﬁed, then theinterior domain corresponds tothevalues 0<at<a0 andtheexterior domain tothevalues 610<at<00.19 Since P,(cosh at)->l, Q,(cosh 61)~>ooasat—>0[cf.(7.3.l3, 23)andProblem 7,p.201], wemust setN=0when dealing with theinterior problem, andhence theappropriate setofparticular solutions ofLaplace’s equation consists ofthefunctions u=ti,=M,,P,,(66511o<)P,,(cos11). n=0,1,2,...(87.3) Ontheother hand, fortheexterior problem weneed solutions which are harmonic outside thespheroid andvanish atinﬁnity (cf.Sec.8.5). According to(7.6.l, 3),thisrequires setting M=0,sothattheappropriate particular solutions ofLaplace’s equation arenow oftheform u=u,,=N,,Q,, (cosh ot)P,,(cos{3), n=0,l,2,... (8.7.4) Next weconsider thecaseofanoblate spheroid. Since equations (8.6.6) and(8.6.l2) areidentical, theonly difference between thiscaseandthecase ofaprolate spheroid isthatequation (8.6.7) isreplaced byequation (8.6.13). Therefore wehave thesame admissible values oftheparameter vasbefore, i.e.,v=n(n=0,1,2,...),andthefactor B(t-1)isagain given by(8.7.1). Since equation (8.6.l3) transforms into equation (8.6.12) under thesubstitution [5=%n-—iot,thegeneral solution of(8.6.l3) forthecase p.=0,v=nisof theform A=MP,,(i sinhfl)+NQ,,(i sinhoi), (8.7.5) 1“This assertion holds forboth theinterior andtheexterior problem. 19Thisistrueforeither aprolate oranoblate spheroid. sEc.8.7 SPHERICAL HARMONICS2 APPLICATIONS 2l7 corresponding tothefollowing particular solutions ofLaplace’s equation: u=u,=[M,,P,,(i sinh11)+N,,Q,,(i sinhot)]P,, (cos6). (8.7.6) Wenowshow thatN,,must besetequal tozeroifthesolutions (8.7.6) are tobeharmonic inside thespheroid. Theproof ofthisassertion islesstrivial than inthecaseoftheprolate spheroid, since both solutions P,,(isinhll)and Q,,(isinhoi)arebounded inthewhole interval 0<oi<0:0.Infact, wemust now examine thebehavior ofgrad unear thesingular curve ofthetrans- formation (8.6.8), i.e.,thecurve at=0,B=1:/2onwhich theJacobian 8(x,y,z)/6(o1, B,cp)vanishes. Itisanimmediate consequence of(8.6.9) that (grad “)2=c2(coshzcit —sin”(3)llZ:)2 + (8'7'7) ifweassume thatuisindependent oftheangle cp.Thedenominator inthe right-hand sideof(8.7.7) vanishes onthecurve at=0,6=rc/2,andtherefore anecessary condition forgrad utobeﬁnite isthattheexpression inbrackets should alsovanish forat=0,[5=1-1:/2,i.e.,that N,=0,since (8.7.6) and (7.6.9—10) imply l(%)2+ =<"‘>""”*-Moreover, thiscondition isalsosufﬁcient. Infact, if u=u,=M,,P,,(i sinha)P,,(cos6), n=0,l,2,..., (8.7.8) then (%)2 + =M,§[PZ(i sinha)P,§2 (cos6)sin’B —P,’,2(i sinha)Pf (cosB)cosh’ 01.]. The expression inbrackets isapolynomial incos(3which vanishes if cos6=1isinhonandhence isdivisible bycosh” at—sin2B.Itfollows that grad uiswell-behaved onthecurve at=0,(3=7':/2,sothat(8.7.8) gives the appropriate solutions ofLaplace’s equation intheinterior ofanoblate spheroid. Inthecaseoftheexterior problem, wemust setM,=Oasbefore, which gives thesolutions u=11,,=N,,Q,,(isinh ot)P,,(cosB). n=0,1,2,...(8.79) TheDirichlet problem foraspheroid cannowbesolved bysuperposition ofthesolutions (8.7.3—4) and(8.7.8—9). Forexample, consider theinterior problem foraprolate spheroid, andsuppose theboundary function f=f(6) canbeexpanded inaseries ofLegendre polynomials '@ /(8)-i/..P.(c@s1-1). 0<<c."=° (8.7.10) /..=tn+1)fol/(or. (cos11>sin8dt. 2|8 SPHERICAL HARMONICS: APPLICATIONS CHAP. 8 which isuniformly convergent intheclosed interval [0,1"c]. Then, using Harnack’s theorem onsequences ofharmonic functions (mentioned onp.208) weseethattheseries u="inf51(°il“")) P(cosis), (81.11)="P,,(cosh 0:0" with terms oftheform (8.7.3), converges uniformly for0<<motoa harmonic function with boundary values uioc=oz0 =f(@), andhence solves thegiven boundary value problem.9 Remark 1.Thesolutions oftheNeumann problem andthemixed prob- lem,involving theboundary conditions (6.3.lb) and(6.3.lc), canbeobtained bysimilar methods. Remark 2.Inthecase ofthemore general problem where f=f(B,<p)is afunction ofboth coordinates [5and<p,itturns outthattheappropriate set ofparticular solutions ofLaplace’s equation forprolate andoblate spheroid are . P’,{‘ hu=u,,,,,=[MM cosmqo+N,,,,,sinm<p]P}," (cos(5)Qm$22113, (8.7.12) _ _ . m P{,"(isinh<1)u-um,-[M,,,,, cosmq;+N,,,,,SlI1m<p]P,, (cos(3)Qﬂisinh(Z),(8.7.l3) respectively, where m=0,1,2,...andn=m+l,m+2,...Theupper rowin(8.7.l2—l3) corresponds totheinterior problem andthelower rowto theexterior problem. 8.8. The Gravitational Attraction ofaHomogeneous Solid Spheroid Asasimple example oftheresults ofthepreceding twosections, wenow calculate thegravitational potential ofahomogeneous solid prolate spheroid ofmass manddensity p.Letthepotentials inside andoutside thespheroid be denoted byii»,andi.l.»_,,,respectively. Then, asiswell known,“ theproblem reduces toﬁnding thesolution oftheequations V24», =——4TCp, V241,, =0, (8.8.l) which satisfy theboundary conditions __ a¢i _akife _ ‘pile _‘Leia: 50 _50’ kpeiw _0: where 0isthesurface ofthespheroid and8/6n denotes thederivative with 2°W.J.Sternberg andT.L.Smith, op.cit.,p.134. SEC.8.8 SPHERICAL HARMONICSZ APPLICATIONS 2|9 respect totheexterior normal to0.21Solving thisproblem isequivalent to solving theequations V24/* =0, Vzilie =0, ifwerepresent ti»,intheform ofasum slit=410+41*, (3-3-3) where i,b*isharmonic inside thespheroid and1.1»,isaparticular solution of Poisson’s equation, e.g., ‘~l-'0=—T=P(X2 +y2)- (33-4) Using (8.6.l) tointroduce spheroidal coordinates cc,B,cp,andapplying thesuperposition method totheparticular solutions (8.7.3—4), wewrite the functions <l.i*and41,,intheform '~.IJ*=5:M,,P,,(cosh a)P,, (cosB), "=° (s.s.s) 4.»,=2N,,Q,,(cosh<x)P,,(cos(3). Todetermine thecoefﬁcients M,,andN,,,wehave theboundary conditions 8» 84" =le , (s.s.6)oz=ozo"Pi|a=a0 =¢e|a=aoa Ea=uo 6“ where moisthevalue ofthecoordinate oncorresponding tothesurface ofthe spheroid.” Noting that ___ 2'2 -2__2T‘PC2 -2 _ilio- Tcpc sinh ixsin(3— T sinh on[PO(cos(5) P2(cosB)], (8.8.7) andcomparing coefficients inboth sides ofeach oftheequations (8.8.6), we ﬁndthat 22.M0-—7%” sinhz <10=NOQO (cosh oto), 2 MZPZ (cosh <10)+gal sinhz oto=NZQ2 (cosh Oto), 2 —ii-Ttéi cosh :10=NOQ3 (cosh 0:0), (8.8.8) 42 ,M2P§ (cosh oto)+TC?“ cosh one=N2Q2 (cosh <10), 2‘Theﬁrstoftheequations (8.8.l) isknown asPoisson’s equation. Asusual, we assume that41,,alt,andtheir ﬁrstandsecond derivatives with respect tox,y,zarecon- tinuous. 2’Ifaisthesemi-major axis andcthedistance from theorigin tothefocus ofthe spheroid, then coshoco=$- SPHERICAL I-IARMONXCSZ APPLICATIONS CHAP. 8 and M,,P,, (cosh oco)=N,,Q,, (cosh ao), n=1,3,4,5,...,(8.89) M,,P,’,(cosh1,)=1v,,Q;,(cosh<10), n=i,3,4,5,.... Itfollows from (8.8.9) thatM,,=N,,=0forallndifferent from 0and2. Therefore, using (8.8.8) tocalculate thenonzero coefﬁcients Mo,N0,M2,N2, wecanwrite thesolution inclosed form, susceptible todirect veriﬁcation. After some simple calculations, during which weuse(7.7.2) andthefor- mula m=%rcpc3 cosh 110sinhz oo,, wearrive atthefollowing expression forthepotential outside thespheroid: ii.=§[Q0(coshoi)-Q2(coshon(cosoi- Similarly, wecaneasily ﬁndthepotential inside thespheroid. Finally, using (7.9.l), wecanexpress thepotentials 4.»,andii»,interms ofelementary func- tions. 8.9. The Dirichlet Problem foraHyperboloid ofRevolution Theability toseparate variables inLaplace’s equation written inspheroidal coordinates alsoallows ustosolve boundary value problems forthedomain bounded byahyperboloid ofrevolution. Ifoi,(5,cparethespheroidal coor- dinates described by(8.6.l), then thesurface [3=(50corresponds toahyper- boloid ofrevolution (seeFigure 29).TheDirichlet problem forthecase of axially symmetric (i.e., <p-independent) boundary conditions canbestated as follows: Find thefunction u=u(ot,(3)such that1)uisharmonic inthedomain Os(3<B0andcontinuous intheclosed domain 0<(3<Bo,and2)usatisﬁes theboundary condition u[B=B0 =f(a) andthecondition atinﬁnity u|,,_,,,, —>0 uniformly in[5,where f(a) iscontinuous intheinterval 0<on<00and f<<»>i..,.. »0.Aswenow show, under certain conditions, thesolution ofthisproblem isgiven byasuperposition ofthefollowing particular solutions ofLaplace’s equation: u=u,=M,P_i/2,“ (cosh oi)P_i/2+“ (cos(5), 1'20. (8.9.l) Infact, setting it=0,v=-—}+itin(8.6.6—7), weobtain B=CP—V2+i1: (cos "l'DP—%+i1('_cOS F3), A=MP_i/H" (cosh oi)+NP_,,+,, (—cosh 0!), andthenthecondition thatthesolutions bebounded ontheaxisofthehyper- boloid, where either ororBvanishes, implies D=N=0.Moreover, accord- ingto(7.6.3), wehave 1)__l/2+5-1; (cosh a)|a_,,, —>0,andhence u,|,,_,., —>0,as required. Thepossibility ofmaking asuperposition ofsolutions (8.9.l) which SEC.8.10 SPHERICAL HARMONICS2 APPLICATIONS 221 satisﬁes theboundary conditions isbased ontheMehler-Fock theorem, which states that“ f(x)=fem1'tanh1"c'rP_%+,,(X)d'rJIDf(Z)P_%+,,(§)dE, 1<x<00(s.9.2) atevery continuity point off(x),provided that 1.Thefunction f(x), deﬁned intheinﬁnite interval (1,oo),ispiecewise continuous and ofbounded variation inevery ﬁnite subinterval [x1,x2],where l<x1<x2<oo; 2.Theintegrals fl|f(X)l(x—1)"°"‘*dx, f|f(x)|>t"’21<>s><dx areﬁnite, forevery a>1. Thus, iftheboundary functionf(<z) satisﬁes appropriate conditions,“ we canwrite f(a)=fa”F(-r)P_i/2+" (cosh1)dr, 0<0t<oo, (89.3) where 0 F(r) =1-tanh Tc-rJ:°f(a)P_ 1/,H,(cosh (Z)sinhatdoc. Then theintegral u=lowF(T) P_.,,,,,(cosh0!)at (29.4) gives thesolution ofourproblem, atleast formally. Forfurther details, including thesolution ofaproblem ofelectrostatics, werefer thereader else- where.“ 8.|0. Solution ofLaplace’s Equation inToroidal Coordinates Inaddition tospherical and spheroidal coordinates, there areother coordinate systems whose useisintimately connected withLegendre functions. 2“SeeN.N.Lebedev’s dissertation (cited onp.131), andV.A.Fock, Ontherepre- sentation ofanarbitrary function byanintegral involving Legendre’s functions with a complex index, Doklady Akad. Nauk SSSR, 39,253(1943). Atdiscontinuity points, the integral intheright-hand side of(8.9.2) equals %lf(X +0)+f(x-0)]- “E.g., iff(cx) iscontinuous in[0,A]forevery ﬁnite A,andiff(ot) falls offlike e“1/2*‘>°‘, e>0asix->co. 25SeeN.N.Lebedev, Solution oftheDirichlet problem forhyperboloids ofrevolution (inRussian), Prikl. Mat. Mekh., 11,251(1947). 222 SPHERICAL HARMONICS: APPLICATIONS CHAP. 8 First weconsider toroidal coordinates oi,B,<p,related totherectangular coordinates x,y,zbytheformulas x_ cSi1'1hotCOS0isascale factor.” This coordinate system isuseful forsolving boundary value problems involving thedomain bounded byatorus, orthe domain bounded bytwointersecting spheres.” Ifapoint hascylindrical coordinates r,<pandz,then csinha csinﬁF=is Z=iii-—s cosh at—cos(3 cosh on—~cosB ormore concisely, z+ir=iccoth%5- The corresponding triply orthogonal system ofsurfaces consists ofthe toroidal surfaces at=const, described bytheequation 2 (r-Ccothone+Z2= , (8.10.2) thespheres (3=const, described bytheequation 2 2 C 2(Z~ccotl3) +r= . (8.l0.3) andtheplanes rp=const (seeFigure 31).Itshould benoted that allthe spheres (8.l0.3) intersect inthecircle r=c,z=0. Itfollows from (8.l0.l) thatthesquare oftheelement ofarclength is 2 use= (d¢2 +use+Si1‘lh2otd<p2), (8.10.4) corresponding tothemetric coefﬁcients c csinhat h“:h°= ’ h“: ' 26Itisclear from (8.l0.l) that x,yand zareperiodic in(5andcp,with period 21:. Therefore wecanchoose B1<[3<Bi+21:,cpl<q;<cpl+2r:instead of—n:<[5str, -1:<<psnr(which corresponds totheparticular choice [5,=<p1=—1r), anditis sometimes convenient todoso(seeSec. 8.12). 2’Later on,inSec.8.13, wewillconsider aclosely related coordinate system, i.e., bipolar coordinates. sec.8.10 SPHERICAL I-IARMONICSI APPLICATIONS 223 / d=const .0.testé’B=const >1 FIGURE 31 Therefore Laplace’s equation intoroidal coordinates hastheform 8 sinhat 8u 8 sinhat éiu amlcosh on—cos(5ﬁat)+?(5lcosh at—cos(56(5) I 52“ (8.105) +o 6?=°- Unlike thecases considered previously, wecannot separate variables inthis equation. However, ifweintroduce anewunknown function vbymaking thesubstitution u=\/2coshot— 2cos(5v, (8.l0.6) then (8.10.5) goes intotheequation 8212 62v 80 l 1820b?+8?’2+COthM$(+ZU+ é?=0, which belongs totheclass ofequations permitting separation ofvariables. Infact, setting v=A(a)B((3)<D(<p), (8.l0.8) weﬁndthat ‘n1-12 l@+lﬂ3+;Otha%+1-_l.L2(D- 2 S‘“Ado? Bd(%2 Add4‘ dq>=_“’ 224 SPHERICAL I-IARMONICSZ APPLICATIONS CHAP. 8 where (L2isaconstant. This implies d2<DW"L“'26)20’ (80).1.9 iantc0thocdA 1(.2 1d2B _i .___.__i __N-___..i—22+ + 2— 2”)’,Adoc A doc 4 sinh at Bd(3 where v2isanother constant, which leads totheequations d2B W "l"V2B =O, (8.10.10) 1d. dA 1 2$]—m(%t(s1nh EX.E)_(ya~Z+5-H-1l%—o‘)A =0.(8.10.11) Thus Laplace’s equation intoroidal coordinates hasinﬁnitely many solutions oftheform ll=\/2coshM-2costaA(<z)B((5)(<p), (8.l0.12) where A,Band(Darethesolutions oftheordinary diﬁerential equations (8.10.9—11). Bysuperposition ofthese solutions, wecansolve various boun- dary value problems ofmathematical physics forthedomains mentioned at thebeginning ofthissection. Asusual, thecase ofrotational symmetry, where thefunction uisindependent ofthecoordinate cp,corresponds toset- ting(J.=0and(D=1.Inthiscase, solving equations (8.10.10—l1), weﬁnd that u=\/2cosh or—2cos(3[APv_i/2 (cosh fl)+BQ,,_i/,(cosh ot)] (81013) ><[Ccosv(5+Dsinv(3]. '' 8.|l.The Dirichlet Problem foraTorus Toillustrate theapplication oftoroidal coordinates, wenow solve both theinterior andexterior Dirichlet problems forthedomain bounded bythe toroidal surface or=oto.Tokeep things simple, weconsider thecaseofrota- tional symmetry, corresponding to(1.=0,<1)=1.Wealsohave thecontinuity conditions u|,=_,, =u|,,-,,, = (8.l1.1) which areequivalent tothephysical requirement that thesolutions be periodic inthe“cyclic” coordinate (3.Theconditions (8.11.l) arepossible only iftheparameter visaninteger, which, without lossofgenerality, wecan assume tobenonnegative, i.e.,v=n(n=0,1,2,...). Fortheinterior problem, weneed solutions bounded inthedomain sec.8.11 SPHERICAL I-IARMONICSI APPLICATIONS 225 oto<at<oo.Therefore, because ofthe behavior ofP,,_1/2 (cosh at), Q,,_1/,(cosh oi)forlarge oz,given byformulas (7.10.l, 8),wemust setA=0. Ontheother hand, fortheexterior problem, which corresponds tothedo- main 0<or<<10,wehave toconsider thebehavior ofP,,_1/2 (cosh Ot), Q,,_i/2 (cosh at)asot—>O, andthen, according to(7.3.13, 23),wemust set B=Oifthesolutions aretoremain bounded. Thus thesolutions ofLaplace’s equation suitable forsolving theinterior Dirichlet problem foratorus are u=u,,=\/2cosha -—2cos('l [Mncosn(5+N"sinn(5]Q,,_i/2 (cosh oz), n=0,1,2,...,(8.l1.2) while those suitable forsolving theexterior problem are u=u,,=V2cosh ot—2cos(5 [M,,cosn(3+N,,sinn(3]P,,_y, (cosh oi), n=0,1,2,... (8.l1.3) Forthisreason, P,,_i/, (cosh ix)andQ,,_1/2 (cosh oi)areoften called toroidal functions. Example. Find theelectrostatic ﬁeld duetoacharged toroidal conductor atpotential V. This problem reduces tosolving the exterior Dirichlet problem with theboun- dary condition i(.(,=,,O =V, (8.ll.4) where Illistheelectrostatic potential. According to(8.l0.2), therelation between thequantities c,oneand thegeometric 2! parameters oi,lofthetorus (seeFigure 32) z isgiven by CCOtl'l<Z0=l, =0, g andhencelr I C=(/12 _a2’ cosh mo:Z FIGURE 32 Asshown above, weshould look forasolution intheform ofaseries (8.1l.2), where, because ofthesymmetry oftheproblem withrespect totheplane z=0, wemust setN,,=0,obtaining u=\/2cosh oz—2cos(3 2M,,P,,_i/2 (cosh oi)cosn(5. (8.11.5) 226 SPHERICAL HARMONICS2 APPLICATIONS CHAP. 8 Theboundary condition (8.l1.4) willbesatisﬁed ifwedetermine thecoeﬂ‘i- cients Mnfrom therelation TDV G3 i__. = M"Pn_I2 h ) , — < Q . \/2cosh <10-2cosB Z0 /(cos docosng it n 3 (8.11.6) Expanding theleft-hand sideof(8.ll.6) inaFourier series intheinterval [—n, TC]andusing (7.10.10), weﬁndthat cosn(3—-i————d(3\/Zcoshao —2cos(3 =%/Q,,_i),(coshao), n=l,2,...,M,,P,,_;/, (coshoto)=2%’In0 VMop-1/2 (cosh °‘o)=;Q-1/2 (Cosh <10)» which leads tothefollowing formal solution foralt: V ——-i--i— P_ hill=;\/2coshot —2cos(5 [ Q_1,2 (cosh oto) °°P,,_g(cosh oz)+2"Z1 -Q3 Q,,_16(cosh oco)cosn-ti(8.11.7) Byusing theasymptotic representations ofSec.7.11, itcanbeshown thatthe series (8.11.6) converges andactually gives thesolution ofourproblem. Finally, wenotethatthecharge density onthetoroidal surface isgiven bythe formula [cf.(6.6.10)] l8u 1 8u0'-——(‘Y/1°‘5La=ao -—%(cosho<0 —cos(3)aa=a0 Remark. Itiseasytoseethatinthecasewhere uisafunction ofallthree coordinates (X,(3andcp,theappropriate solutions ofLaplace’s equation are u=u,,,,,=V2cosh on—2cos(3[M,,,,, cosn(3+N,,,,,sinn(3] 8.11.8)(XQ;,"_./,(cosha) m,n=0,1,2,... fortheinterior problem, and u=u,,,,,=\/2cosh U.—cos(3[MM cosn(5+N,,,,,sinn(5] (8.11.9) ><P,’{‘_i/,(cosh (Z) m,n =0,1,2,... fortheexterior problem. SEC. 8.12 SPHERICAL HARMONICS2 APPLICATIONS 8.l2. TheDirichlet Problem foraDomain Bounded byTwo Intersecting Spheres Toroidal coordinates canalsobeused tosolve boundary value problems involving adomain bounded bytwospheres S1andS2which intersect ina circle Y.Letx,y,zbeasystem ofrectangular coordinates with it origin atthecenter ofY,and B letthez-axis pass through the 1 center ofthe spheres (see , Figure 33).Letoi,(5,q>beasys- ‘ tem oftoroidal coordinates related tox,y,zbythefor- mulas (8.l0.l), andchoose the 0 >1 constant cequal totheradius ofY.Finally, let(5,,betheangle between theplane z=0and thetangent plane tothesphere Sp(p=1,2),drawn through anypoint ofthecircle Y,where 0<(51<(52<21:.Then itfol- lows from (8.l0.3) that the equation ofthesphere S,intoroidal coordinates is(5=(5,,.Moreover, of thetwodomains bounded bythespheres, theinterior domain D,corre- sponds totheinterval (51<(5<(52,while theexterior domain Decorresponds totheinterval (52<(5<(51+2n.Inboth D,and De,thevariable 0!. ranges over theinterval O<0!.<oo,where points onthez-axis corre- spond toat=0andpoints ontheedge Ycorrespond toon=co.“ Wenowconsider theDirichlet problem forthedomains D,andDe,con- ﬁning ourselves totherotationally symmetric case. Justasbefore, westart from thesolutions (8.l0.l3), butunlike Sec.8.11, there isnolonger anyneed torestrict vtobeanonnegative integer. Infact, asweshall soon see,the solution ofourproblem canbeconstructed bysuperposition ofsolutions of theform u=u,=V2cosh at—2cos(5[M, cosh -r(5+N,sinh-r(5] ><P_i/,+,, (cosh oi), 1'>0,(8.l2.1) obtained from (8.10.l3) bychoosing v=itandsetting B=0.29Webegin withtheinterior problem, andassume thatthefunctions f,,=f,,(ot) appearing intheboundary conditions u|Y,=Bp =f,,, p=1,2 (8.12.2)FIGURE 33 2“Notealsothatx =y=0,2-> iooif<x= 0,(5->2"rri. 29Thisisnecessary forthesolution tobebounded onthez-axis. 228 SPHERICAL HARMONICSI APPLICATIONS CHAP. 8 aresuch thatthefunctions <1)(Pp(G) = : P=I,2 \/2cosha -2cos (5,, canberepresented asintegrals oftheform <p,,(ot)=foo,,(¢)P_./,... (cosh8)dr, 0<8<oo, (8.i2.3)0 where theexpansion coefﬁcients lI>,,(-r) areindependent of(X.According to theMehler-Fock theorem (8.9.2), such arepresentation ispossible, andthe functions ,,(¢) canbecalculated from theformula <I),,(-r) =1tanh tttInq>,,(ot)P_ i/2H,(cosh ot)sinh atdot, (3.12.4) ifthefunctions f,,(<z) arecontinuous andofbounded variation in[0,A]for every ﬁnite A,andiftheintegrals low8|f,,(8.)| dot, p=i,2 (812.5) areﬁnite. Thelastcondition presupposes thatthef,,(a) approach zero sufﬁciently rapidly asoi~>oo,i.e.,asthecircular edge Yisapproached. Ontheother hand, limfp=fi»(°°) =Hy, where u,isthevalue taken bythesolution uontheedge Y,andthisvalue is usually notzero.3° However, inmost cases ofpractical importance, the modiﬁed functions f?(<*) =fp(<1) ~ﬁ»(°O), P=1,2 falloffsuﬂiciently rapidly asat—>oo,andhence there exists anexpansion 83(8)=V3%% =(Om(D:('L')P_%+i1 (cosh<1)dr,(812.6) where <I):('r) isgiven by <1>;:(.)=1tanhTIT(0q>:(a)P_.,,,.. (cosh<1)sinhatdot.(8.12.?) 3°Here weassume that theboundary function iscontinuous, butallourconsidera- tions caneasily beextended tothecaseofpiecewise continuity, where limflmaynot equallim/2. °‘"°°or-vm SEC.8.12 SPHERICAL HARMONICSZ APPLICATIONS 229 Moreover, itisnothard toshow that“ _-—-_~i-1._.__ =F P_,,,,. (cosh8)d¢(8.i2.8)\/2cosh at—2cos(5,, 0 C0511 TIT forO<(5,,<21:.Multiplying (8.l2.8) byf,,(oo) andadding theresult to (8.l2.6), weobtain anexpansion for<p,,(<x) oftherequired form (8.l2.3), where ,,(¢)=;:(.)+%:°T:T cosh(1.-(a,,)¢. (812.9) Now consider theintegral u=\/2cosh oi—2cos(5low(D2sinh(B—Si§g)&3:_q)g1s)i:h (B2TBk ><P_i/2+), (cosh ot)dr, (8.l2.l0) made upofparticular solutions oftheform (8.l2.1). Weseeatonce that (8.l2.l0) satisﬁes theboundary conditions (8.l2.2) andhence gives thesolu- tionoftheinterior Dirichlet problem. Similarly, thesolution oftheexterior Dirichlet problem canbewritten intheform --~iif8‘B1.25%.:i%?'lhi§§?.*ti—><P_.,,,,. (cosh8).11,(8.12.11) where f1(°l') =ulB=B1+21I9 f2(°‘) ="li1=e2’ andtherestofthenotation isthesame asbefore. Example. Consider the“spherical bowl” orzone obtained bysetting (51=(52=(50inFigure 33.Find theelectrostatic ﬁeld duetoathincharged conductor ofthisshape atpotential V. This isjusttheexterior Dirichlet problem fordetermining theelectro- static potential kl),inthespecial casewhere P1:P2=Po» f1(°‘) =f2(°‘) =V- “Combining theformulas =ZJ\mc0§x1dTJm d6 \/2coshx+2c0shot YTo 0\/2cosh6+2coshot [Cf.(6.5.3—4)] and 2 “° 9P_1/2+1‘ (C0511 1) =;C0$h TCTJZJ [Cf.(7.4.6)], W6ﬁnd that 1 °°cosx-r_i__.__ = P_ if ll d, \/2coshx +Zcoshoi -lo°°5l1T" I/2+(cos cl)T which gives (8.l2.8) after setting x=i(i-c—(5,). 230 SPHERICAL I-IARMONICS2 APPLICATIONS CHAP. 8 Thefunctions (D1and(D2canberead oﬁatonce from (8.l2.8): _ _cosh (1:—(50)-: %_%*V_R$F— Then formula (8.12.l1) becomes ii w h_ il;= I/\/2COShO(—-2COSl5J‘O cosh(n+B0—p): ><P_1/2+" (cosh oi)d-:, (50<(5<(50+21:. (8.12.12) Substituting 2 e 'eP._1/2+" (COSII 0!.)=;COtI1 RTL [cf.(7.4.7)] into(8.12.12), andintegrating ﬁrstwith reslpect to~:andthenwith respect to6,weﬁndafter some manipulation thatthesolution canbeex- pressed inclosed form interms ofelementary functions:32 A \/2cos§ ¢=ZP+/_;2m:1§£_]+KFmw 22 C0311 <1—C05(230—5) T‘ \/cosh oicos(5 -2150-13A \/2cosi——- _/_£@1;@£_2mm_____;L__cosh °‘*C05(230*l3) K/cosh oi—cos(2(50—(5)ll(812.13) The fact that (8.l2.l3) satisﬁes theboundary conditions isimmediately apparent. 8.l3. Solution ofLaplace’s Equation inBipolar Coordinates There isstillanother coordinate system which leads tosolutions of Laplace’s equation involving Legendre functions, i.e.,three-dimensional bipolar coordinates oi,B,ip,related totherectangular coordinates x,y,zby theformulas x_ csinoicos<p _csinoisin<p z_ csinh(5 _coshﬁ —cos0!, y—cosh(5 —cosOi’ Tcosh(5 —cosoi (8.l3.l) 3’Inintegrating with respect to':,usetheformula . TU‘ °°c0Shp'r 11: Smh?~[FS1nrTdT=Z ! 0<p<q. °S1q cosh— +cos-q ll sec.8.13 SPHERICAL I-IARMONICSI APPLICATIONS 231 where Og <1:, —oo<(5<oo, —-r:<<p<i-:, R andc>0isascale factor. This system isclosely related tothetoroidal coordinates studied inSecs. 8.l0—l2, andissuitable forsolving boundary value problems forthedomain bounded bytwononintersecting spheres. Ifa point hascylindrical coordinates r,ipandz,then csinoi csinh(5r=-—--——-—i, z=—-—-—-is cosh (5—cosoi cosh (5—cosoi ormore concisely z+ir= iccot%ll3~ Ar a=const 7:}ifs?'i8"l\‘%i' ”"Q‘~ FIGURE 34 The corresponding triply orthogonal family ofsurfaces consists ofthe “spindle-shaped” surfaces oi=const, described bytheequation (r_8cot0t)2+Z2= (8.i3.2) thespheres (5=const, described bytheequation _ 22:L2, (z ccoth (5)+r (sinh 6) (8.l3.3) andtheplanes cp=const (seeFigure 34).Thepoints r=0,z=iccor- respond tothevalues (5=oo,while r=0,z~> iooifoi=0,(5—>0i. Itfollows from (8.l3.l) thatthesquare oftheelement ofarclength is 2 G182=(c0i—_-ffa)2 (doe+d(52+sinz8482), (8.l3.4) 232 SPHERICAL I-IARMONICSZ APPLICATIONS CHAP. 8 corresponding tothemetric coefﬁcients c csinoih.,=h0=—i, h0,=———-Acosh (5—cosoi cosh (5—cosoi Therefore Laplace’s equation inbipolar coordinates hastheform 8( sinoi 8n)+8( sinoi 8u) 8oicosh (5—cosoi8oi 8(5cosh (5—cosoi8(5 1 82u(8.13.5) . i =O. +(cosh (5—cosoi)sinoi8<p2 Justasinthecaseoftoroidal coordinates, wecanseparate variables, provided weﬁrstintroduce anewunknown anewfunction vbymaking thesubstitution u=\/2808118 -280$8U, (813.6) which transforms (8.13.5) intotheequation 82v 82v 8v l 182vﬁ+8T52'l-COtOt-é;‘—zU'l'SiTaé'?—0. Tosolve (8.l3.7), weset v=A(oi)B((5)€I>(<p). (8.13.8) This gives thefollowing equations fordetermining thefactors A,Bandfl): d2(I) TP2 -l'(J.2(D =0, (8.139) d2B id.dA 2gi—I1—o‘E(S1n 8%)+[v(v+1)-§1iiﬁJA =0. (8.13.l1) Theﬁrsttwoequations canbesolved interms ofelementary functions, and thethird interms ofLegendre functions. Inparticular, fortherotationally symmetric case, where thesolution uisindependent ofip,weﬁndthat u=\/2cosh (5—2cosoi[AP,, (cosoi)+BQV (cosoi)] ><[Ccosh (v+-1r)(5+Dsinh(v+2)(5]. (8.13.12) Inproblems involving thedomain bounded bytwo nonintersecting spheres (5=(51and(5=(52,thevariable oiranges over theclosed interval [0,1:],andhence toobtain solutions which areﬁnite onthez-axis wemust set B=0andv=n(n=O,1,2,...),asinSec.8.3.Thus, forthisclass ofprob- lems, theappropriate particular solutions ofLaplace’s equation are u=u,,=\/2cosh(5 —2cosoi[M,, cosh (n+%)(3+N,,sinh(n+%)(5] ><P,,(cosoi), n=0,1,2,... (8.l3.l3) SEC.8.13 SPHERICAL HARMONICS: APPLICATIONS 233 Ontheother hand, inproblems involving thedomain bounded bythesurface oi=oi0,theappropriate particular solutions areobtained bychoosing v=-1}+i'T.'(-r20),andareoftheform u=u,=V2cosh(5 —2cosoi[M,cos -:(5+N,sin-i-B]8.13.14 XP-%+i1(i COS“), ‘F209 ( ) where theplus signcorresponds totheexterior problem (0<oi<oi0)and theminus signtotheinterior problem (oi0<oi<1:). Example. Find theelectrostatic ﬁeld between twospherical conductors of radius a,whose centers areadistance 2lapart, ifoneconductor isatpotential —Vandtheother isatpotential +V. Thespheres have equations (5=1(50inbipolar coordinates, ifwechoose thequantities c,(50such that ccoth (50=I, Fm) =a, i.e., c=\/l2—a2, cosh(50=£ r W=-V \l'=+V 0 >2 21 FicuRE35 (seeFigure 35).Then theproblem reduces toﬁnding afunction Ll)(where kl) istheelectrostatic potential) which isharmonic inthedomain —(50<(5<(50 andsatisﬁes theboundary conditions '~l’lo=—o<> ="Vt ‘l’lo=no ZV- Using (8.l3.13) andnoting thatuisanoddfunction of(5,welook forasolu- tionoftheform il.»=\/2cosh(5 ~2cosoiEM,P,, (cosoi)sinh(n+%)(5. (8.l3.l5) 234 SPHERICAL I-IARMONICS2 APPLICATIONS CHAP. 8 Theconstants M,canbedetermined from thecondition RV on .? Cosh B0_2cosat=2:0M,,P,, (cosoi)sinh(n+-})(50. 0<<TE. Using (4.2.3) toexpand theleft-hand sideinaseries ofLegendre polynomials, weobtain V _ Ve“‘lo/2 \/2cosh $0-2cos oiT\/l—2e“”<> cosoi+e'“<> U) =V2e""*‘/=)°oP,, (cosoi), n=O which implies M,sinh(n+-})(50 =Ve“""‘/*’°<>. Thus theformal solution oftheproblem isgiven bytheseries llJ=V\/2 cosh (5—2cosoi"E0e“”*‘/M0 P, (cosoi).(8.13.l6) The factthat (8.13.16) converges andsatisﬁes theboundary conditions is easily veriﬁed. 8.14. Solution ofHelmholtz’s Equation inSpherical Coordinates Inmathematical physics, Legendre functions arise notonly when dealing with Laplace’s equation, butalsowith other equations, among which Helm- holtz’s equation Vzu+kzu=0 (8.l4.1) isofparticular importance. Tosolve (8.l4.l) inspherical coordinates, we look forparticular solutions oftheform H=R(r)@(9)‘1>(), justasinSec. 8.2.Thevariables separate immediately, andweobtain the following differential equations fordetermining thefactors R,(E)and<1): d2<l>W +(1.24) =0, (8.l4.2) 1d .d® 2 (%(r2 +[kzrz ——v(v+1)]R =0. (8.14.4) SEC.8.14 SPHERICAL 1-IARMONICSI APPLICATIONS 235 Here (1.andvarearbitrary realorcomplex parameters, butwithout lossof generality wecanassume thatRe(1.20,Rev2-2(cf.footnote 2,p.206). Equations (8.l4.2—3) coincide with equations (8.2.5—6), andcanbesolved Z interms ofelementary functions inthe ﬁrst case, and interms ofLegendre functions inthesecond case. Under the substitution R=r‘1'2v, equation (8.14.4) goes into /1 1/ + 2 v+70 +[k2—(—iv r2%)]v=0. (814.5) This isBessel’s equation ofargument z=kr,whose general solution canbe expressed interms ofcylinder func- FIGURE 36 tions. Inparticular, intherotationally symmetric case, where uisindependent ofthecoordinate ip,wehave u=r‘1'2[A.I,,i/,(kr) +BHf,%Z 1/.,(kr)][CP,, (cos 6)+DQ, (cos 6)], (8.14.6) where J,(z) istheBessel function oftheﬁrstkind andH531/2 isthesecond Hankel function.” Inproblems where 0varies over theinterval [0,1:],the boundedness requirement compels ustosetD=0andv=n(n=0,1,2,...). Bysuperposition oftheparticular solutions (8.l4.6), wecansolve many problems ofmathematical physics, including theimportant problem ofdif- fraction ofelectromagnetic waves bytheearth’s surface.“ PROBLEMS 1.Letthesurface ofasphere ofradius abedivided intotworegions S1and S2asshown inFigure 36.Find thestationary distribution oftemperature u inthesphere ifS1isheld attemperature u0,while S2isheld attemperature zero. Ans. °° 1|. u(r,6)=539{l—cosoi—2[P,1,1 (cosoi)—P,1_1 (cosoi)] P1(cos 6)}- Tl1 33This form ofthesolution isconvenient forproblems involving steady-state oscilla- tions, when thetimedependence isdescribed bythefactor e“°'.Ifthetimedependence is described bye"‘°', wereplace H551"/,(kr) byH,$5,’y,(kr). 3‘G.A.Grinberg, op.cit.,Chap. 23. 236 SPHERICAL HARMONICS: APPLICATIONS CHAP. 8 2.Find thepotential il»oftheelectromagnetic ﬁeldinside asphere ofradius aifonehemisphere (corresponding to0s6<1:/2)isheldatpotential V, while theother hemisphere (corresponding tor:/2<6<1:)isheld atpoten- tialzero (cf.footnote 17,p.160). Ans. V °°4n+3 r2"“=l»(r.9)=;ll+"ZO2—nT2P...(0) (5)P2...(cosol- 3.Find thestationary distribution oftemperature Llinaprolate spheroid if half ofitssurface (corresponding toz>0)isheld attemperature u0,while theother half(corresponding toz<0)isheld attemperature zero. 4.Calculate thegravitational potentials il»,,(la,(seeSec.8.8)ofahomogeneous solid oblate spheroid. Introducing spherical coordinates r,0andqa,derive an asymptotic representation ofil»,forsmall c,where cisthedistance from the origin tothefocus andverify thatKlle—>m/rasc—>0,theresult tobeexpected. Derive thecorresponding asymptotic formula fortheprolate spheroid. Hint. Note that 2 2 COS1'ld=L[//I +Ecos0+%+A/1 —£ccos6+C—2]»2c r r r r 2 c2 2c c2cos(5=§rZ_[A/1 +Tccos6+?—A/1—-;cos6+F]- Ans. Fortheprolate spheroid, 1 2 il/..l2_.1, zm[7+£5P2(cos6)]- 5.Find theelectrostatic potential (lainside ahollow prolate spheroid with semiaxes aandb,which hasapoint charge qatitscenter andwhose surface isheldatpotential zero. Hint. Write kl!asthesum ofthepotential 4:0duetothesource andthe potential uduetotheinduced charges. Usetheformula 35 1 Pgn(X') , ——-—iz =21>,0 -1'1..l—1\/sinh2 oi+x2X 2()Q2n lcos M) Ans. 1 *"‘°"B)=1/zoqﬁo -§0(4n +1)P2,,(0) 1l%3 P2,,(cosh0t)Pg,1(cos(3)1. where tanh oi0=b/a. 55SeeProblem 16,formula (ii),p.202. sec.8.14 SPHERICAL I-IARMONICSZ APPLICATIONS 237 6.Calculate thesurface charge density 0onaconducting disk ofradius adue toapoint charge qadistance Ifrom thediskalong itsaxisofsymmetry (see Figure 37). Hint. Note thatthedisk isalimiting case ofanoblate Z spheroid. Usetheformula“ a 1 Pzémd —21>0 '"h l_1V x 1o..()Qo..(1S1n <1)- 1 TAns. _~L E2 azsinzp -3/2 "o41i2ll»ll1+—i- l2i °°_,,(4n+1)nl ,+lV;cosBnZo( 1)-——,(n+,)Qo..(a)P2..(<=<>SB)l where (5=arcsin(r/a)andristhedistance from thecenter FIGURE 37 ofthedisk toanarbitrary point onitssurface. 7.Suppose aconstant electric ﬁeld E0acts along theaxisofsymmetry ofa grounded conducting torus. What istheelectrostatic potential il»along this axis? Hint. Useformula (7.l0.10), after integrating byparts. Ans. ____._5 _ Q1/2(cosh oi0) l)l’=° TTE°z +215°‘/I2 Tasin ,.n_T1/2(cosh oio)Sin"B, l\)'@3 21\/18“U-3 where coshoi0=l/a,andthenotation isthesame asinSec.8.11. 8.Solve thepreceding problem, assuming instead that theexternal ﬁeld is duetoapoint charge qatthecenter ofthetorus. 9.Find theelectrostatic potential il»outside aconductor atpotential V,which hastheform ofthe“spindle-shaped” surface mentioned onp.231inconnec- tionwithbipolar coordinates. Hint. Cf.(8.13.14) and(7.4.6). Ans. il»(oi,(5)=V\/2 cosh(5 —Zcosoi °°cos(5:P_1/,.i1,(—cos oi0)in 1 4.Xlocosh -r:-:P_1/2,1, (cosoi0) "’+“(cos Oi)T asSeeProblem 16,formula (i),p.202. HYPERGEOMETRIC FUNCTIONS 9.l. The Hypergeometric Series andltsAnalytic Continuation Bythehypergeometric series (already introduced inSec.7.2)ismeant the power series 20 Zk, (9.1.l) where zisacomplex variable, oi,(5andYareparameters which cantake arbitrary realorcomplex values (provided thatYaé0,—l,—2,...),andthe symbol (>1),denotes thequantity (>i)0=l, (A)2= =%(k+l)---(k+k—l), k=1,2, Ifeither oior(5iszero oranegative integer, theseries terminates after aﬁnite number ofterms, anditssumisthen apolynomial inz.Except forthiscase, theradius ofconvergence ofthehypergeometric series is1,asiseasily seenby using theratio test.‘ Thesumoftheseries (9.l.l), i.e.,thefunction °°(¢)i.((5)F(°¢>l3§Y§Z) =Z(TM. lzl<1. (9-1-2)1c=0 - 1Writing _(<1)CB) wehave ( ii..._oi+/t)(l-5+Ii)uk'(Y+k>(1+Ii)’“lzlask->oo,sothatthehypergeometric series converges forlzl<1anddiverges for zl>1. 238 SEC.9.1 HYPERGEOMETRIC FUNCTIONS 239 iscalled thehypergeometricfunction, butthisdeﬁnition isonly suitable when 2liesinside theunitcircle. Wenowshow thatthere exists acomplex function which isanalytic inthez-plane cutalong thesegment [1,oo]andcoincides with F(oi,(5;Y;z)for|z|<1.This function istheanalytic continuation of F(oi,(5;Y;z)intothecutplane, andwillbedenoted bythesame symbol. To carry outthisanalytic continuation, weﬁrstassume that ReY>Re(5>0 andusetheintegral representation Q: F(Y) 1-1+1: __ Y—-1 =Mk F-M1,(Y_B)(0tB (11)“ dt, ko,1,2,..., (9.13) implied bytheformulas ofSec.1.5.Substitution of(9.l.3) into(9.l.2) gives F(oi, (5;Y; z)= k£o%°Zr:f01tB-1+i<(1__t)v-is-i dr F(Y) (1_ - °°(4)1.=-——i 1°11—tY °‘1dt — t“,P(l5)F(Y—B).‘l .2.,/ii(Z) where, asusual, reversing theorder ofsummation andintegration isjustiﬁed byanabsolute convergence argument? According tothebinomial expansion (cf.footnote 17,p.121), Ni ,§o(%)!"(zt)" =(1—tz)'°‘, 0<<1,lzl<l, andhence F(oi,(5;Y;z)hastherepresentation F(0t, = JT01tBT1(1*l)vTDT1(1—lZ)Tadl, ReY >Re(5 >0,lzl<1. (9.l.4) Thenextstepistoshow thattheintegral in(9.l.4) hasmeaning andrepre- sents ananalytic function ofzintheplane cutalong [1,oo].Ifzbelongs to theclosed domain p<lz—ll<R, |arg(1—z)létr—8, (9.1.5) where R>0isarbitrarily large andp>0,8>0arearbitrarily small, and if0<t<1,then theintegrand 1°-1(1-t)Y-°*1(1 —iz)-°‘ iscontinuous intforevery zandanalytic inzforevery t,andweneed only 2Infact,ifRey >Re(5 >0andlzl<1,then G} 1 in _ 2KL?'¢l(z|xf |tB—1+k(1_t)'V-B-1|dt< 2§_lll;l1~|Z|i¢J_1ti:oa-1+;¢(1_,)no-i~nou-id, ho k. <1 W0 k. 0 = F(|4l.R¢B;R@Y;l1l)- 240 HYPERGEOMETRIC FUNCTIONS CHAP. 9 show thattheintegral isuniformly convergent intheindicated region.“ But thisfollows atonce from theestimate ltB—1(1__t)v—B—1(1_tz)—oc|< MtRe B—1(1 _t)ReY—Re B-1 where Misthemaximum value ofthecontinuous function [(1—tz)|‘°‘fort in[0,1]andzinthedomain (9.1.5), andfrom thefactthattheintegral 1 MJ‘ tReB—1(1_t)Rev—Re B—1dt 0 converges forRev>Re>B>0.Therefore thecondition |z[<1canbe dropped in(9.l.4), andthedesired analytic continuation ofthehypergeo- metric function isgiven bytheformula P(Y) J‘_ __F,;;= t“11—”°11—'°‘d,(“@Y” F@NY—@o ‘’)‘w’ Rey >Rep >0,|arg(l—z)|<1:.(9.l.6) Inthegeneral casewhere theparameters have arbitrary values, theanaly- ticcontinuation ofF(ot,(3;Y;z)intotheplane cutalong [1,oo]canbewritten asacontour integral obtained byusing residue theory tosum theseries (9.l.2).‘* Amore elementary method ofcarrying outtheanalytic continuation, which, however, does notleadtoageneral analytic expression forthehyper- geometric function inexplicit form, involves theuseoftherecurrence rela- tion5 to+1)F(°‘>l3§Y§Z)= v(v-at+1>F<@.@ +1§Y+2;» +an~(Y—@>z1F<<1+1.@+1;Y +2;z>-‘H7’ Byrepeated application ofthis identity, wecanrepresent thefunction F(ot,B;Y;z)with arbitrary parameters (yaé0,—l,—2,...)asasum P F(<=<,B;Y;z) =2r1sp(<»,B;Y;Z)F(<=< +8,8+P§Y+21>;z), (9-1-3)s=0 where pisapositive integer andtheas,,(ot, B;Y;z)arepolynomials inz.Ifwe 3E.C.Titchmarsh, op.cit.,pp.99—100. 4E.T.Whittaker and G.N.Watson, op.cit.,p.288. 5Toverify (9.l.7), wesubstitute from (9.1.2), noting that thecoefﬁcient ofz"inthe right-hand side of(9.1.7) becomes (MB+1),. (+l)t(l3+1)». (<1+1)~_1(l5 +1)t-1 Y<Y'°‘+1)?Y+2)./a+“YQ(Y+2):J<! 'a(Y_B) _(<1)t.(l5)». B+k +kB+k (y+k+l)k )’€( )k K‘ K = (Y+k)(Y+k+1)EY(Y+1) > SEC. 9.2 HYPERGEOMETRIC FUNCTIONS I choose psolarge that Reﬁ >-p, Re(Y—B)>-p, then wecanuse formula (9.l.6) tomake theanalytic continuation ofeach ofthefunctions F(<z+s,Q+p;Y+2p;z)appearing intheright-hand sideof(9.l.8). Sub- stituting thecorresponding expressions into (9.l.8), weobtain thedesired analytic continuation ofF(o<,B;Y;z),since theresulting function isanalytic intheplane cutalong [1,00]andcoincides with (9.1.2) for|2|<1. The hypergeometric function F(0£,[5;Y;z) plays animportant role in mathematical analysis anditsapplications. Introduction ofthisfunction allows ustosolve many interesting problems, such asconformal mapping of triangular domains bounded bylinesegments orcircular arcs, various prob- lems ofquantum mechanics, etc.Moreover, aswillbeseen inSec. 9.8,a number ofspecial functions canbeexpressed interms ofthehypergeometric function, sothatthetheory ofthese functions canberegarded asaspecial caseofthegeneral theory developed inthischapter (cf.footnote 20,p.176). 9.2. Elementary Properties oftheHypergeometric Function Inthissection weconsider some properties ofthehypergeometric function which areimmediate consequences ofitsdeﬁnition bytheseries (9.1.2).6 First ofall,observing thattheterms oftheseries donotchange iftheparameters at and[3arepermuted, weobtain thesymmetry property F(oi,B;Y;Z)=F(B,<1;Y;Z)- (9-2-1) Next, differentiating (9.2.1) with respect toz,weﬁndthat d ,__w (°¢)r¢(i3)k k_1_co(°‘):¢+1(l3)k+1 k 2zF‘“>‘*>Y’Z>".-Zi.<Y—>;<T1>@Z = a$k%0(a l)kzk = Z), andhence” 4 . “F3EF(ot,B,Y;Z)=-;F(ot+1,B+1;Y+l;Z). (9.2.2) Repeated application of(9.2.2) leads totheformula i:F(a,§;Y;z)= F(a+m,B+m;Y+m;z), m=1,2,... dZ’" (Y)... (9.23) 6Itfollows from theprinciple ofanalytic continuation thatalltheformulas proved here, under theassumption that |z|<1,remain valid inthewhole domain ofdeﬁnition ofF(@,l3§Y;Z)-7Cf.formula (7.12.25), p.197. HYPERGEOMETRIC FUNCTIONS CHAP. 9 From now on,tosimplify thenotation, wewrite F(¢,B;Y;Z) EF, F(¢i-1,B;Y;Z)E F(<1i1), F0116 i1;Y;Z) EF(Bi1), F(<1,B;Y i1;Z)E F(Yi1)- Then thefunctions F(ot11),F(Bi1)andF(Yi1)aresaidtobecontiguous toF.Thefunction Fandanytwofunctions contiguous toFareconnected byrecurrence relations whose coefﬁcients arelinear functions ofthevariable 2.8Among therelations ofthistypewecitetheformulas (Y_1._t5)F+0(.(I_Z)F(0t+1)-(Y-(a)F((a-1)=0,(9.2.4) (Y_V.-1)F+0tF(0t+1)-(Y-l)F(Y_1)=0,(9.25) Y(1-z)F-YF(¢-1)+(Y-{3)zF(Y +1)=0,(9.2.6) which canbeveriﬁed bydirect substitution oftheseries (9.l.2). Forexample, substituting (9.l.2) into(9.2.4), weobtain (Y-<1—B)F+ <1(1— Z)F(¢+ 1)—(Y-l3)F((5 —1) _°° (1)1415) (1+1)t((5);. r.Z.l<Y-at-@>rYm“"°‘"W<1—__ __ (°‘)k(15 _l)k_“((1 +1)k-1(13)1¢-1 1, ‘YB)(mt! <Y>..-.</< —1)!1’ =kZ1§“§i“i’l‘;*1<Y-¢-@>(@+k-1>+<<»+k><@+k-1)= Y)t¢k! _ —(Y—l5)(l5— 1)—(Y+/<—1)k]z"E0, andsimilarly for(9.2.5—6). Three other formulas areanimmediate conse- quence of(9.2.4—6) andthesymmetry condition (9.2.l): (Y—<1—B)F+B(1—Z)F(l1 +1)—(Y—°<)F(°< —1)=0, (9-2-7) (Y—B—1)F+BF(B+1)—(Y—1)F(Y -1)=0, (9-2-8) Y(1-z)F—YF(B —1)+(Y—ot)ZF(Y +1)=0. (9.2.9) The restoftherecurrence relations canbeobtained from (9.2.4—9) by eliminating acommon contiguous function from anappropriate pair of formulas. Forexample, combining (9.2.5) and(9.2.8), or(9.2.6) and(9.2.9), weobtain (ot—t3)F—otF(ot +1)+BF(t3 +1)=0,(9.2.l0) (<1—B)(1~Z)F+(Y—<1)F(¢ ~1)—(Y—l5)F(l5 —1)=0,(9-2-11) andsoon.” °Obviously, thetotal number ofsuch relations is G)=15. 9Thelistofallﬁfteen recurrence relations involving Fanditscontiguous functions isgiven intheBateman Manuscript Project, Higher Transcendental Functions, Vol. 1, p.103. sac.9.3 HYPERGEOMETRIC FUNCTIONS 243 Besides therecurrence relations justgiven, there exist similar relations between thefunction F(Ot,B;Y;z)andanypairoffunctions oftheform F(o1+l,(5+m;Y+n;z),where l,mandnarearbitrary integers. Some simple relations ofthistype are‘° F(=1,B;Y;z) —F(=1,l5;Y —1;Z) =— F(a +1,(.1+ 1;Y+1;z), (9.2.12) F(<><,B+1;Y;Z)— F(<1,B;Y;z) =%F(0t-l-I,B-l-1;Y-l-I;Z), (92.13) F(°¢>(5'1‘1§Y‘1'1§Z)—F(°l»l5§Y§Z) = F(¢ +1,(s+1;Y +2;z), (9.214) F(<1—1,l1+1;Y;z)— F(<»,l$;Y;z) ——z=(1-Q7-ll F(0t,(s+1;Y +l;z). (9.215) Formulas (9.2.l2—l5) areproved bydirect substitution of(9.l.2), orbyre- peated useoftherelations between F(<x,(5;Y;z)anditscontiguous functions. Finally, werecall from Sec. 7.2that thehypergeometric function u=F(0t,(5;Y;z)isasolution ofthehypergeometric equation z(1—z)u”+[Y-—(Ot+B+l)z]u’ —mﬁu=0, (9.2.16) which isanalytic inaneighborhood ofthepoint z=O. 9.3. Evaluation oflimF(o<,(3;Y;z)forRe(Y—oz—(3)>O Z—>l — Indeveloping thetheory ofthehypergeometric function, itisimportant toknow thelimit asz—>l—ofthefunction (9.l.2), where theparameters satisfy thecondition Re(Y—at—(3)>0.“Suppose thatbesides thiscondi- tion, ReY>Re(3>0aswell. Then thedesired result canbeobtained by passing tothelimit behind theintegral signin(9.l.6), which gives limF(0t,B;Y;z) =-i—J-1:“-1(1—z)Y"°‘-B-1 dt,z»1- 1"(B)1"(Y —B)0 1°Formula (9.1.7) isalso arelation ofthistype. 11Itcanbeshown thatifthiscondition isnotsatisﬁed, then, with certain exceptions, thesumofthehypergeometric series becomes inﬁnite asz->1-. 244 HYPERGEOMETRIC FUNCTIONS CHAP. 9 or,inview of(l.5.2, 6), 211151F(oi,(1;Y;Z)= (9.31) where, forthetime being, weassume that Re(Y—oz—(3)>0, ReY >Re(3 >0. (9.3.2) Tojustify thepassage tothelimit, itissufficient toprove thattheconditions (9.3.2) imply thattheintegral (9.l.6) isuniformly convergent forO<z<1. Tothisend, wenote that 1—t<|l—tzl<1 for0 <<1,0<<1,andhence [t°‘1(l Tt)Y‘“'1(1 —tz)'°‘| <tR°°‘1(l —t)’"‘1, (93.3) whereN ~ )\_ Re(Y—ot—B) ifReo1>0, _Re(Y—(3) ifRe<z<0. Theestimate (9.3.3) shows thattheintegral (9.l.6) isuniformly convergent for0<z<1,since theintegral J1tB.ea-1(l _1);.-1 dt, 0 which majorizes (9.l.6), isconvergent iftheconditions (9.3.2) hold. Wenow show thatthesecond oftheconditions (9.3.2) isnotessential. Suppose thatinstead of(9.3.2), theparameters ofthehypergeometric func- tions satisfy theweaker inequalities Re(Y—-ot— ti)>0, Re(Y—-[5)> —l, ReB> —l. Then therestrictions under which weproved (9.3.1) aresatisﬁed byeach of thehypergeometric functions intheright-hand sideoftherecurrence relation (9.l.7). Itfollows that . __ _ — +lI‘(+2)F(—-ot—(3+l) 3l‘I‘_F(°"l*’*’Z)‘YYi1 F(:’"°1+Y2)F(Y—l3+1) +atF(Y+2)P(Y—°‘—§)Y(Y+1)1“(Y—¢+1)1"(Y—l1+l) EF(Y)F(Y—°‘—l3),Po—are—s) which isjusttheprevious result. Repeating thisargument, wecanprove by induction that - .._F(Y)F(Y-at-(1),zlPP_ F(as B1Y9Z)“T _ ___ sec.9.4 HYPERGEOMETRIC FUNCTIONS 245 provided onlythatRe(Y—or—(5)>O.Formula (9.3.4) plays animportant roleinthederivation ofvarious relations satisﬁed bythehypergeometric function. 9.4F(Ot,(3;Y;z) asaFunction ofitsParameters Inthissection weshow thatthefunction l f(a9BsY>z):?Y)F(“1i5>Y9Z) isanentire function of0t,[5andY,forﬁxed z.Iflzl<1,theproof isanim- mediate consequence oftheexpansion fM@wﬂiiﬁ%%%% |4<L (Mm obtained bysubstituting (9.l.2) into(9.4.l). Infact, since theterms ofthe series (9.4.2) areentire functions ofoc,(3,Y,andsince theseries isuniformly convergent intheregion lot]<A,|B]<B,|Y|<C(where A,BandCare arbitrarily large),12 itfollows thatf(ot,(3;Y;2)isanentire function ofitspara- meters. Now letzbeanarbitrary point inthecomplex plane cutalong [1,oo],and consider theformulas 1 1 .M%wﬂ=———~—t“W—W”W—®“Mnwn-oi gm)ReY >Ret) >0,]arg(l—z)l<1-c, f(¢,l3§'Y§Z) =Y(Y—91+1)f(¢,B +1;Y+_2;z) +m-n-@mu+tw»w+aa 6“) which aretheanalogues of(9.l.6) and(9.l.7). Since theintegrand inthe right-hand sideof(9.4.3) isanentire function oftheparameters EX,(5,Yforany tin(0,1),andsince theintegral isuniformly convergent intheregion |a|<A, 8<Re(5<B, 8<Re(Y—(3)<C, 1’Usethecriterion given infootnote 4,p.102,noting thatif u= (°‘)1¢(B)t¢ Z), "I‘(Y+k)k! ’ then 10.11 _(<X+k)(B +k) (A+k)(B+/<) at"(Y+k)(1+/<)’ ‘(k-c)(1+/<)l’l<"<1 for]z|<1andsufﬁciently large k. 246 HYPERGEOMETRIC FUNCTIONS CHAP. 9 where 8>0isarbitrarily small, itfollows thatf(ot,(5;Y;z)isananalytic function ofitsparameters intheregion lot]<oo, ReB>0, Re(Y—(5) >0. Byrepeated application oftherecurrence relation (9.4.4), wecanrepresent thefunction f(ot,(3;Y;z)asasum to.aw)=ibsp(°1>l5§Y§Z)f(°‘ +8,8+p;Y+2/1.1).(94.5)s=0 where theb,,,(a, (5;Y;z)arepolynomials in0t,B,Yandz,andpisapositive integer. Asjustshown, each term ofthissumisananalytic function inthe region |o<|<oo,Ret) >—p,Re(Y—(3)>-p,andhencef(a, (5;Y;z)isan entire function ofitsparameters. Itfollows thatforﬁxed zintheplane cut along [1,oo],thehypergeometric function F(0t,(3;Y;z)isanentire function ofatand(3,andameromorphic function ofY,with simple poles atthepoints Y=O,—l, -2,... 9.5. Linear Transformations oftheHypergeometric Function Consider theclass ofallfractional linear transformations z,_az+b _cz+d carrying thepoints z=0,l,oointo thepoints z’=O,1,oochosen inany order. Itiseasy toseethatbesides theidentity transformation z’=z,this class consists ofthefollowing ﬁvetransformations: z 1 1 z-1z'=—i> z'=1—z, z’=—i1 z'=—» z’=i-z—-1 1-2 z z Wenowderive various linear relations connecting thehypergeometric func- tions with variables zandz’.Relations ofthiskind areamong themost im- portant inthetheory ofthehypergeometric function, andareknown aslinear transformations ofthehypergeometric function. Inparticular, these formulas enable ustomake theanalytic continuation ofF(U.,(3;Y;z)intoanypartof theplane cutalong [1,00].” Webegin byderiving arelation which isuseful inthecase where one requires theanalytic continuation ofthehypergeometric function into the half-plane Rez<1.Suppose zbelongs totheplane cutalong [1,oo],and assume forthetime being that ReY>Re(5 >0.Then, using theintegral 13Thetheoretical possibility ofsuch ananalytic continuation hasalready been proved inSec. 9.1. sec.9.5 HYPERGEOMETRIC FUNCTIONS 247 representation (9.l.6), and introducing thenew variable ofintegration s=1—t,weﬁndthat F(oi,B;Y;Z)= L1SY'°“(1 —S)°‘1(1 —Z+SZ)‘°‘dS =(1'Z)‘“‘r(@'>*F((»i)— 11')‘w(l'"Y_H11 _"1"“where l3'=Y_l3, /=5 andourassumptions imply thatReY>ReB’>0,while z’belongs tothe plane cutalong [1,00].“ According to(9.l.6), theexpression ontheright is just (1-Z)-“F(oi, 1-1';Y;/>.andhence F(0t,('1;Y;Z)=(1" z)-9tF(ot,Y —t3;Y;:-€—1), |arg(1-z)|<1:. (9.5.1) Formula (9.5.l) was proved under thetemporary assumption that ReY>Re(3>0,but,asweknow from Sec.9.4,after dividing byF(Y), both sides become entire functions of(3andY.15Therefore, bytheprinciple of analytic continuation, (9.5.1) remains valid forarbitrary BandY,with the exception ofthevalues Y=O,—1,—2,...forwhich F(0t,(5;Y;z)isnotde- ﬁned. Moreover, ifRez<-1,then Z ?:l<I, andthehypergeometric function intheright-hand sideof(9.5.1) canbere- placed bythesumofthehypergeometric series, i.e.,(9.5.1) gives theanalytic continuation ofF(CX.,(3;Y;z)intothehalf-plane Re2<-1;. Permuting atand(5in(9.5.1), and~using thesymmetry property (9.2.1), wearrive attherelation F(Y.aw)=<1-Z)-“F(Y -Y.13§Y§Z%T)’ |=1rg(1-z)l<W. (9.52) which canalsobeused tomake theanalytic continuation ofthehypergeo- metric function intothehalf-plane Rez<-1.Toobtain another important 11Note that under thetransformation z’=z/(1 —z),theplane cutalong [1,oo] goes into itself. 15The expression F[f(o1, B,Y,...), g(ot, B,Y,...),...] isanentire function of oz,(3,Y,...ifF,f, g,...areentire functions oftheir arguments. 248 HYPERGEOMETRIC FUNCTIONS CHAP. 9 result, weperform thetransformations (9.5.1) and(9.5.2) consecutively, ob- taining F(Y.@;Y;z>=<1—z>-~(1- ,—§—Y)-<1-WY -Y.Y-@;Y;z>. [arg(1—z)|<1:, or F(<1,t1;Y;Z)=(1— Z)*‘°“‘1F(Y —<1.Y—11;Y;z), |arg(1—z)|<1:.(9.53) Toderive arelation between thehypergeometric function with variable z andthehypergeometric function with variable 2’=l—z,weuseageneral method from thetheory oflinear differential equations. First wenote that thegeneral solution ofthehypergeometric equation z(1—z)u”+[Y—(oz+B+l)z]u’ —aﬁu=0 (9.5.4) canbewritten intheform 16 u: +A2Z1_YF(1— Y'1'“:1 _Y+ —Y;Z)9 ]arg(1—z)|<rt,|arg2|<11:,Yaé0,i1,i2,... (9.5.5) Under the transformation z’=l—z,the domain |arg(1—z)|<rc, |argzl<TCgoes intothedomain |arg(1 —z’)|<1:,|argz’|<rc,andequa- tion(9.5.4) goes intothehypergeometric equation with parameters ot’=oi, B’=(5,Y’=1+oz+(3—Y.Therefore theexpression u=B1F(<».B;1+ 1»+B—Y;1—Z)+B2(1— Z)*““‘° ><F(Y—¢,Y~l1;1—9<—l1+Y;1—Z),(9-5-6) |arg(1— z)|<1:,|argz| <rc,at+(3—-Yaé0,1-1,12,... isalsoageneral solution ofequation (9.5.4). Inparticular, thisimplies the existence ofalinear relation oftheform F(@<.t1;Y;z) =C1F(<»,B;1+ <1+B—Y;1— Z) + _Z)Y-a_BF(Y '—“Q-Y '_ _a_F3 °‘+l5—Y¢0, i1,i2,-~ Todetermine theconstants CYand C2,weassume temporarily that Re(oi+(3)<ReY<1,andthen take thelimit ofthelastequality, ﬁrstas z->1-andthen asz—>0+. Using (9.3.4), weobtain C=1-l(Y)P(Y-'°‘_l5), 1F(Y-Y>P<Y-11>P(l+<x+(3—)F(l—) I‘(l—oc—B+)l"(l—)_ C1P<1+Y—Y>P<(+@—i>+C* P(1—Y>P<1Y—@>Y“' 1°SeeSec.7.2,noting thatbytheprinciple ofanalytic continuation, formula (7.2.6) remains valid inthewhole domain |arg(1—z)|<rr,|argz|<-rc. SEC.9.5 HYPERGEOMETRIC ruucrrous 249 Itfollows that CZ1"(Y)P(Y+11-Y),2 1“(<1)1"(B) after some simple calculations involving theidentity (l.2.2). Therefore the required formula is .. __F(Y)P(Y—°1—l5) . .F(¢,5,Y,Z)— F(¢,3,1 -1"<1-1-15- Y,1—Z) _ P(Y)F(°@+1-Y) '1“(1 z)l’ B (9.5.7) '_'Z)1 |argz| <11, |arg(1 —z)] <11, a+(3—Yaé0, il,i2,... Togetridofthesuperfluous restrictions imposed ontheparameters 0t,(3 andY,wenote thatafter multiplication bysin1'c(Y—at-(5)/F(Y), both sides of(9.5.7) areentire functions oftheparameters." Therefore, according tothe principle ofanalytic continuation, therelation (9.5.7) isvalid forallvalues of theparameters except those forwhich at+(3—Y=0,1-1,i2,...For- mula (9.5.7) gives theanalytic continuation ofthehypergeometric function intothedomain |z-—1|<1,|arg(1—z)|<1:. The remaining relations between thehypergeometric functions with variables zandz’canbeobtained bycombining theformulas justderived. Forexample, consecutive application of(9.5.1) and(9.5.7) leads totherela- tion18 F(°‘>l3§Y§Z) =<1—z>~“ F(Y.Y -t;1+ Y—11.1%,) +(1-Z)-B ;F(Y -ot,(5;1— 11+e;1—i-2_) [arg(—z)] <1:,|arg(1— z)|<1:,at-(5aé0,11,352,...(9.5.8) which enables ustomake theanalytic continuation ofF(tl,(5;Y;z)intothe domain |z—1|>1,[arg(1—z)l<1:.Then, combining (9.5.8) with (9.5.1—2), weobtain F(Y.s;Y;z>=(-Z)-" F(¢.1+01-Y;1+ Y-1.1) _F()F(°1—15) _ ,1+(—Z) ° F(@,1+5—Y,1+$—¢,;)’ |arg(—z)| <rt,|arg(1— z)|<rt,or—(3¢0,il,12,...,(9.5.9) 11Here weagain make useof(1.2.2). 1°Note that under thetransformation z’=z/(z—1),thedomain |arg(—z)| <rt, |arg(1—z)|<rtgoes into thedomain |argz’]<1r,|arg(1—z’)]<rt,which guaran- teesthat(9.5.1) and(9.5.7) canbeapplied consecutively. _ 250 HYPERGEOMETRIC FUNCTIONS CHAP. 9 which gives theanalytic continuation ofF(Ot,(3;Y;z)intothedomain |z|>1, |arg(1—z)|<1:.Finally, consecutive application of(9.5.7) and (9.5.1) gives m@WFfW¢WWr~w+Pw+HMw—— .u_.+rwuw-n+Z“Z’° umm ><F(Y—ot,l——ot;1+Y—ot—($;:g—1),F(Y)P(Y —at— F( 2;1) |argz|<-rc, |arg(1—z)|<rc, ot+(5-—Y#O,il,i2,..., (9.5.l0) which canbeused tomake theanalytic continuation ofF(oi,(3;Y;z)intothe domain Rez>1;,|arg(1—z)l<TC. Theproblem oftheanalytic continuation ofthehypergeometric function intothez-plane cutalong [1,oo]issolved byusing formulas (9.5.1-3) and (9.5.7—10). Some exceptional cases, where these formulas arenotapplicable, willbeconsidered inSec.9.7. 9.6. Quadratic Transformations oftheHypergeometric Function Therelations between hypergeometric functions derived inthepreceding section arevalid forarbitrary values oftheparameters oc,B,Y(apart from certain exceptional values). Onecanalsoconsider relations where thepara- meters satisfy certain constraints; although lessgeneral, relations ofthistype arealsouseful inmaking various transformations andcarrying outanalytic continuation. Among such relations, themost interesting involve hypergeo- metric functions with twoarbitrary parameters. Aswillbeseen below, they alsocontain expressions like 1-l-\/I-—Z I—\/I-—Z -42 2 1+\/1-2 (1—z)1 andhence arecalled quadratic transformations ofthehypergeometricfunction. Asanexample ofaformula belonging tothisclass, consider therelation F(¢.t1;¢ +15+ %;z)=F(2¢.21%;“+t1+ %;l—_—\g—1;z)’ (9_6_1) ]arg(1—z)[ <11, o1+(5+%¢0,—1,—2,..., which canbeproved asfollows: Theleft-hand sideisasolution ofthehyper- geometric equation (9.5.4) with parameter Y=oi+(3+-1;,which isanalytic inthedomain |arg(1—z)|<rt.Under thesubstitution” Z’=Y(1-1/1-Z), 19By\/1—zismeant thebranch which ispositive forrealzintheinterval (0,1). SEC.9.6 HYPERGEOMETRIC FUNCTIONS 251 thisequation goes intoanequation ofthesame form with parameters cx":2a1l3,=2B1Y/=°1+§+'%a andthedomain |arg(1—2)]<1-:goes intothedomain Rez’<1,which is partofthedomain [arg(1—z’)|<11:.Butaccording to(7.2.6), thehypergeo- metric equation cannot have twolinearly independent solutions which are analytic inaneighborhood ofthepoint z=0,andhence there must exist a relation oftheform F(o1,(3;a +B+-};z) =AF(2ot,2[3;0t +(3+- Z), where Aisaconstant. Setting z=0,weﬁndthatA=1,thereby proving (9.6.1). Alarge number ofother relations ofthesame type canbededuced by applying thelinear transformations ofSec.9.5toformula (9.6.1)andchanging theindependent variable ortheparameters. Forexample, using (9.5.3) and (9.5.1) totransform theright-hand sideof(9.6.1), weﬁndthat F(<1,i5;=1+ B+%;Z) _l+\/l——z‘2°‘ L 1_\/I—Z—l -(+1 )F(Y»;1+~*+t+r>T~Y—1-..1)1[arg(l —z)l<rc,or+(3+-{Yaé0,—l,—2,..., (9.6.2) F(°11B;°1+F3+%;Z) =(———1+‘§1'Z)%_HF(Y—@+9.11-Y+%;<»<+@+%;%l5), larg(1— z)l<11,on+(3+1;aé0,-1,—2,... (9.6.3) Using (9.5.1) totransform theleft-hand sides of(9.6.1) and(9.6.2), andthen making thesubstitution z%1'_)Z1 weobtain twoother useful relations: F(Y.Y+-1.Y;z>=<1- Z)-~F(2<». 2Y-20¢-1;Y;§- larg(1—z)l<Tc,(9.6.4) f -20: _ f F(ot,ot+%;Y;Z)= F(2ot,2ot—Y+ l;Y; ), larg(1—z)|<1:.(9.6.5) 252 HYPERGEOMETRIC FUNCTIONS CHAP. 9 Finally, using (9.5.3) totransform theleft-hand sides of(9.6.1) and(9.6.2), andthen making thesubstitution ¢—>¢ -1, 13—>B -1, wearrive attherelations F(@=,l%;<» +B—%;Z) =ﬁF(2oc -1,2(1-1;“ +(1- -51%-Z), |arg(l—z)|<1-c, ot+B~~};éO,—1,—~2,..., (9.6.6) F(¢,l$;¢+l5—%;z) =1(1+v1—-?)1-*1 \/YT? 2 ><F(2a-1,61-(1+1;<Y+(1-1;‘/M‘-1),\/1—z+1 |arg(1— z)|<11:,or+(-3—-1;¢0,-1,—2,,.. (9.6.7) Itisinteresting tonote thatformulas (9.6.2, 5,7)continue thecorresponding hypergeometric functions intotheplane cutalong [1,oo].Infact, ,1-1/1-Zkill. <I 1+1/1-Z if|arg(l—z)|<1:,andhence thehypergeometric function intheright-hand sideofeach ofthese formulas canbereplaced bythesumofthecorresponding hypergeometric series. Further results canbeobtained bytaking inverses oftheformulas just derived. Forexample, inversion of(9.6.l—3) gives“ F{¢,l5;%(9< +B+1};Z)=F{%¢,%i5;%(¢ +(1+1);4Z(1- 1)}, Rez<1,161+(1+1)as0,-1,-2,..., (9.6.8) F(oc,(5;ot— (5+l;Z) -01 . . 42F{%‘M!%(G+1)—B,¢~$+1,_ }s |z|<1,11-(1+1110,-1,-2,..., (9.69) F(9<,1 —<1;Y;Z) =(1—Z)*"F{=1z(Y —°‘)’%(Y +1—1);Y;4Z(1 —1)}, Rez<1.(9.6.10) 2°Inparticular, (9.6.8) isobtained from (9.6.1) bymaking thesubstitution _\/ _ 2ot—>ot, 2{3—>ﬁ, 1——2;-—z-—>z. SEC.9.6 HYPERGEOMETRIC FUNCTIONS 253 Moreover, combining these formulas withthelinear transformations given in Sec.9.5,wecanobtain stillanother group offormulas. Forexample, apply- ingthetransformation (9.5.7) totheright-hand sideof(9.6.8) andmaking the substitution <z—>2ot, B—>2B, z—>1—2-Z1 weﬁndthat“ F(2<1<, 2(1;<1+(5+%;L;-1) _Y(1+11+%)1“(11) O,.1. _F(Y+1)1“(B+1)F1’B’2’Z2) (9.6.11) +Z F(a+%,@+%;%;zz)’ P(<»>P<@> |arg(1 iz)l<rt,o(+(3+1}aé0,—-1, —2,... Formula (9.6.11) plays animportant roleinthetheory ofspherical harmonics. Forexample, therelation (7.6.9) isanimmediate consequence of(9.6.ll). Weconclude thissection byderiving afewformulas ofamore compli- cated nature. Theﬁrstresult is F(M,l1;2t1;Z) 1+v1—_Z-2~ __1_~/(T22=(2)F{“»“-(+1911-(1.Y—1_.)}’|arg(1—z)|<1-c, 2(5;é—l,—3,—5,..., (9.6.12) which isproved inthesame wayas(9.6.1), bynoting thatunder thechange ofvariables z,_(l—\/l—z)2 u_(1+\/1-2)-“U 1+\/1—z 2 ’ equation (9.5.4) goes intothehypergeometric equation with thenewpara- meters dz“: l5’=a_B+%9 Yl:13+%' Since theveriﬁcation ofthisfactisquite tedious, wesupply some intermediate 21Inthecourse ofthederivation, itisconvenient toassume temporarily that |arg(1—z)l<rr,Rez >0.The result can then beextended the whole domain |arg(11-z)l<1':byusing theprinciple ofanalytic continuation. HYPERGEOMETRIC FUNCTIONS CHAP. 9 steps which willserve tokeep thereader ontheright track during thecourse ofthecalculation: Z_1_(V?-1)2 dz’__z'(\/7 +1)8 _ \/?+1’ dz‘ 2(\/?-1)’ u=(Y?+1)2°‘v, (9.6.13) dd’d X/_'+l2°‘*2 -,-, ddz=didz“,=_(2(Z\/;_)1) [ow+\/Z(\/Z +1)6T:,]. (9.6.14) d2 z(l—z)ﬁ =Yaw? +1)2Y{[..+1- f[(1.1)+\/?(\/? +1) +1/Y(1/Z +1)[(u +1+ 5TL—?)% +1/?(\/? +1)%]}- (9.6.15) After using (9.6.13—15) towrite thehypergeometric equation satisﬁed byu, wemultiply theresult by 1_V?, V?(1+Y?) obtaining *- a+1--1 av+\/Z'(\/Z'+1)-,llgl iﬁilll ‘eZ51 +\/;(l-1/?)[(1 +1+ -3;”,+\/?(\/? +1)$] + [(1—(at+(1+ 1) ]lav +\/?(\/F +_1)%,] _ot(5(1-\/Z’) = Y/?(1+1/7)” 0’ which cannowbereduced quite easily tothehypergeometric equation I /d2 rd z(l-@5111<(1+1>-(2<Y-11+1>z1;,§-to-(1+1>v=o. satisﬁed byv.Making thesubstitution 1-1/1-Z l+\/1—z—)Z SEC.9.6 HYPERGEOMETRIC FUNCTIONS 255 in(9.6.12), weobtain theformula (1;2(1;%} =(1+l)w“.Y—1+1.1+1;z2>. lzl<1, 2B1*—l,-3,-5,... (9.6.16) Ourﬁnal result is z'°‘ z 2 F(Y.(1;211»)=(1-5)F{1<»1(Y +111+1; |arg(1——z)|<1-c,2Baé-1,—~3,-5,..., (9.6.l7) which canbederived asfollows: Applying thetransformation (9.5.1) tothe right-hand sideof(9.6.9) andreplacing (3byat—(3+1,weobtain _, 4F(<1,<1-1+1.1+1;z>=<1+ 2)F{1w1<¢ +1>;1+ |z|<1,2?)aé—1,—3,—5,... (9.6.18) Then, comparing (9.6.l3) and(9.6.15), weﬁndthat __4z _(1+z)2°‘ I _ 1_ 422 Flt"1’11’(“W1~ F(1°"1<°‘11111+1"<1W>1}’ andthedesired result isobtained bymaking thesubstitution 42(1-_-‘_7)5—>z, which implies l+z2__>2—z 422 \ z 2_ (1+z)2 2’ (1+z2)2'l2-Z) Thetheory ofquadratic transformations ofthehypergeometric function wasdeveloped byGauss, Kummer andGoursat, andalsofrom amore general point ofview inRiemann’s investigations ofaclass ofdifferential equations including thehypergeometric equation asaspecial case.” Werefer the reader tothese sources foramore detailed treatment ofthesubject.” 22SeeE.Goursat, Surl‘e'quation dtﬂérentielle linéaire, quiadmet pour intégrale la série hypergéometrique, Ann. Sci.Ecole Norm. Sup. (2),10,3(1881). Therelevant references byGauss, Kummer andRiemann aregiven onp.296ofthebook byWhittaker andWatson (op.cit.). 2“See also theBateman Manuscript Project, Higher Transcendental Functions, Vol.1,p.110ff.,foranextensive listofquadratic transformations ofthehypergeometric function. 256 HYPERGEOMETRIC FUNCTIONS CHAP. 9 9.7. Formulas forAnalytic Continuation ofF(o<,Q;Y;z)in Exceptional Cases Theformulas derived inSec.9.5allow ustoobtain theanalytic con- tinuation ofthehypergeometric function into anypart ofthez-plane cut along [1,oo].However, some ofthese formulas arenolonger meaningful for certain values oftheparameters, andmust therefore bemodiﬁed inawaywe nowindicate. Thegeneral approach istostart from theformulas ofSec.9.5 andthen carry outappropriate passages tothelimit. Forexample, suppose wewant toﬁndtheanalytic continuation ofthe function F(ot,{5;y;z) into thedomain |z—l|<1,[arg(1—z)l<1:.If at+[3—Yaé0,1-1,1-2,...,wecanuse(9.5.7), butthisformula isnot applicable ify=on+Bin(n=0,1,2,...).Toderive aformula allowing ustocarry outtheanalytic continuation inthelatter case, wereplace the hypergeometric functions intheright-hand sideof(9.5.7) bythecorrespond- ingseries, anduse(l.2.2) totransform theresult, obtaining 1 W F(°1,F5;Y;Z) _ Tr 1 0° _ ;¢ 'sinwe—1-elm-com—an;Pu+1+B*kY+k)/<!(1Z) _ 1 E (Y_°‘)r¢(Y '“BM (1-Z)kH—u_B] P(oz)F([3)k=oF(l —o¢—l3—l-Y+k) kl = W_@ (81-g2)- (9-7-1) Itiseasily veriﬁed that - _ - _ 1 w(°‘'l'")k(l5 ‘l'")k _ TL i»1l‘P@+,.g‘ "i~llﬁ+ng2 "r(¢)1*(@) Z, (n+k)!k! (1Z)“’ and hence theright-hand side of(9.7.l) becomes indeterminate forY= on+[5+n.Using L’Hospital’s ruletoeliminate thisindeterminacy, wehave 1 831 382 ]——€- F; ;=-1"— -— -F(°l-+l5'l'n) (“,3 lX+ B+n Z) ( )l8Y V=oc+D+1|. 5Yv=a+l3+n (9.7.2) After some calculations resembling those made inSec.5.5,weﬁndthat“ Fg _ 1 <—1>"~*<n -k-l)l(U')lc(l3)k _ﬁt ‘Po+n>1“<@+ n)20 kl (1z)l +13(oi+nus+mlP(ot)P(f3) no (n+k)!k! ><[*1/(k+1)—=l(¢+'1)—W5+")](1—Z)"*", (9-7-3) 2"Indifferentiating gz,weusetheformula d 3(7%=(7\)kl\l’(7\ +k)"‘l)(7~)l- From nowon,weassume thatcc,l3950,—1,—2,... 55¢,9_7 HYPERGEOMETRIC FUNCTIONS 257 =130»+"ma+mtav P<<»>P(@>,,=, (n+k)!/:1><[tl»(<z+n+k)—il/(ot+n)+cl/([3+n+k) —'~l-(B+n)—il/(1+n+k)+log(1—z)](1 —z)""", (9.7.4) where tl1(z)=F’(z)/1"(z) isthelogarithmic derivative ofthegamma function. Substituting (9.7.3-4) into(9.7.2), weobtain F(<x,l3;o: +L5+n;z) _F(°<+B+H)"'1(—1)"(" —/<—1)!(@=)t(B)t k _F(<»+n)F({5+n)Z, k! (1_Z) +(' +")go(°‘JEn”f(£)!L”)" [Mk+1)+¢(n+k+1) -i,b(ot+n+k)—gb({3+n+k)—log(l —z)](l—z)""’°, |z—l] <1,]arg(1 —z)l <1c, n=0, 1,2,..., <x,i3;é0, —1,—2,... (9.7.5) Asusual, themeaningless sum 1 20 which appears when n=0,issetequal tozero. Formula (9.7.5) isnolonger applicable ifonorBequals 0,—1,—2,...,but thenF(ot,(5;at+B+n;z)reduces toapolynomial, andthere isnoneed for analytic continuation. Moreover, thecase Y=ot+B—nreduces tothat justconsidered byusing thetransformation (9.5.3), which becomes F(oz, (3;on+B—n;2)=(1—z)'"F(oc’, B’;oz’+B’+n;z) (9.7.6) ifoc'—-oc—n,l3'=l3—n. Similar considerations apply totheother formulas ofSecs. 9.5—6. Togive another example, wederive aformula suitable formaking theanalytic con- tinuation ofF(ot,[3;Y; z)into thedomain |z|>1,|arg(—z)] <TCinthe case where on—{3=O,il,12,... Here wehave topass tothelimit t3—>on1n(n=0,1,2,...)in(9.5.9). Acalculation likethat given above leads tothefollowing formula (forthecase(5=on+n):25 F(ot,ot +n;Y;z) =P<Y><-Z)-~ "2101-k-1>!(¢>t<1—Y+cot(_z)_,,F(Y —ot)F(oc +n)k=o kl +P(Y)<—z>-" 3(Q1+~>k<1+Q—Y+rmF(ot)F(Y —oc—fl),_,=o (n+k)lkl ><[i.l»(k+l)+\lJ(n+k+1)——\]1(o<+n+k) —\l’(Y—<1—H—k)+1<>g(—Z)]Z“""‘, |2|>l,]arg(1 —z)| <Tc,n=O,l,2, ..., on79O,—l, —2,..., Y—on#O,il, i2,..., Y75O,—l, —2,... (9.7.7) 25Inthelaststepofthecalculation, useformula (l.3.4). HYPERGEOMETRIC FUNCTIONS CHAP. 9 Wenow examine thecases where formula (9.7.7) isnotapplicable. If on=0,—1,—2,...,thefunction F(a,on+n;Y;z)reduces toapolynomial, andthere isnoneed foranalytic continuation. According to(9.5.3), F(ot,at+n;Y; z)=(1—z)"‘2°“"F(Y —oz,Y—at—n;Y;z),(9.7.8) andtherefore F(ot,at+n;Y;z)reduces toanalgebraic function ifY—on=O, —1,—2,...orY—on=1,2,...,n,andanalytic continuation isagain un- necessary. IfY—oz=n+1,n+2,... andonaé0,il,i2,..., then the hypergeometric function intheright-hand sideof(9.7.8) satisﬁes thecondi- tions allowing ittobecontinued byusing formula (9.7.7). IfY—oc=n+1, n+2,...and<1=1,2,...,thehypergeometric function canberepresented byanintegral ofthetype(9.l.6) witharational integrand, i.e.,F(ot,on+n;Y;z) canbeexpressed inﬁnite form interms ofrational functions. Finally, we note thatthecase[5=oz—nreduces tothatjustconsidered ifweagain use thetransformation (9.5.3). 9.8. Representation ofVarious Functions inTerms ofthe Hypergeometric Function Aswenowshow, various familiar functions ofmathematical analysis are special cases ofthehypergeometric function F(oc,_15; Y;z),corresponding to suitable choices oftheparameters ot,[3,Yandthevariable 2:26 1.Elementary functions. The hypergeometric function F(oc,13;Y;z) re- duces toapolynomial ifon=0,-1,—2,...orB=O,-1,—2,. .. Forexample, F(a0'Y'z)=1 F(cx —2'Y’z)=1—25z+@z2, J S J 9 3 3 ! Y + andsoon.Thetransformation F(<»,l5§Y;Z) =(1—z)*'°"°F(Y —ow—B;Y;z), |arg(1 —z)l<Tr [cf.(9.5.3)] shows thatF(a,B;Y;z)reduces toanalgebraic function if Y—at=0,-1,—2,... orY —[3=0,—1,—2,... Inparticular, F(oi,B;l$;z)=(1—Z)“, |arg(1—z)l<W (9-8-1) foranyvalue of13,and (1—Z)v=F(_v11;1;z)> (1—z)_l/2=F(%11;1;z)9 (9.s.2)z"=F(—n,1;l;l —Z), n=O,1,2, 2“Further examples aregiven intheBateman Manuscript Project, Higher Trans- cendental Functions, Val.I,pp.89,101. SEC. 9.8 HYPERGEOMETRIC FUNCTIONS 259 Other representations ofthistypecanbederived from theformulas of Sec.9.6.Thus, setting [3=oz+iin(9.6.2) and(9.6.7), weobtain —;- -2°. F(a,a+};2m+1;z)=( ) ,|arg(1—z)|<1r, _ 1-11+\/1 2“F(oz,a+-};2ot;z) = ,larg(1 -z)l<TE. (9.8.3) Bystarting from theseries expansion °° k+1 °° 1 1 lOg(1—Z)=—kZ0ki-ﬁ=—ZkZ0 Zk, lZl<1 ofthelogarithm, weﬁndthat log(1—z)=——zF(1, 1;2;z), |arg(1—z)|<-rc. (9.8.4) Similarly, wededuce thefollowing formulas fortheinverse trigono- metric functions: arctanz =zF(-13 l;%; —z2), |arg(1izi)|<-rc, arcsinz=zF(-1, -1;%;zz), |arg(1iz)|<1:. (9.85) Elliptic integrals. Thecomplete elliptic integrals 1:/2 1:/2 K(z)=lo(1-Z2S1I12(p)_1/zdfp, E(z)=fa(1-Z2Sll'12<P)1l2d<p oftheﬁrst andsecond kinds [cf.(7.10.11)], where zisacomplex variable belonging tothedomain |arg(1iz)|<1:,canalsoberepre- sented interms ofthehypergeometric function. Assuming temporarily that|z[<1andusing thebinomial expansion, weﬁndthat which implies K(z)=gm,-g;1;z2), |arg(1iz)|<-=. (9.s.6) Similarly, wehave thefollowing representation oftheelliptic integral of thesecond kind: Eu)=§F<--1,1;me). |arg(11~z)l<R.(9-81> Starting from these formulas, onecandevelop thetheory ofelliptic integrals, regarded asfunctions ofthemodulus z. 260 HYPERGEOMETRIC FUNCTIONS CHAP. 9 3.Spherical harmonics. One ofthemost important classes offunctions which canbeexpressed interms ofthehypergeometric function consists ofthespherical harmonics studied inChapter 7.Infact, formulas (7.12.27) and(7.12.29) immediately imply thefollowing representations oftheassociated Legendre functions: m_F(v+m+ 1)(Z2-1)'~/2 P42)“ F(V-—-WI +1)2’"F(m+ 1) ><F(m—v,m+v+1;m+l;%E), [arg(zi1)]<-rc, m=0,1,2, ..., (9.8.8) ere=‘i———',l1'§§(f‘j§”,§’,ff.T.‘) <12—1)" m+ +2m+ +1 31 |arg2]<rc, ]arg(zil)|<TE, m=0,1,2,...(9.8.9) Inparticular, theLegendre polynomials (seeSec.4.2)aregiven bythe formula P,(z)=F(-n,n+1;1;l-;_z), n=0,l,2,... (9.s.10) Byregarding (9.8.8—l0) asdeﬁnitions andusing thegeneral theory of thehypergeometric function, itisasimple matter todevelop thetheory ofspherical harmonics. This approach isespecially convenient for deriving therelations ofSec.7.6andtheir generalizations tothecase ofarbitrary m. 9.9The Confluent Hypergeometric Function Besides thehypergeometric function F(oc,B;Y;z),animportant roleis played inthetheory ofspecial functions byarelated function <1>(-)-3%? ||<oo -£0-1-2 (991) G,Y,Z —k=0(Y)kk!> Z ,Y , , ,..., .. known astheconﬂuent hypergeometric function. Here zisacomplex variable, onandYareparameters which cantakearbitrary realorcomplex values (except thatYaé0,—1,—2,...),and, asalways, . F01+k)(x)0=1, (x),,=_1,-(T=x(x+1)---(x+-k-1), k=1,2,... SEC.9.9 HYPERGEOMETRIC ruucrrous 261 Asindicated, theseries (9.9.1) converges forallﬁnite 2,”andtherefore repre- sents anentire function ofz. Ifweset <P(°%Y;Z)=%)<1><Y.Y;Z)= (9-9-2) then<p(<x,Y;z)isanentire function ofatandY,forﬁxed z.Infact, theterms of theseries (9.9.2) areentire functions ofonandY,andtheseries isuniformly convergent intheregion |a|< A,lY|<C(where AandCarearbitrarily large)?“ Therefore, forﬁxed z,lI>(ot,Y;z)isanentire function ofonanda meromorphic function ofY,with simple poles atthepoints Y=0,—l, —2,... Acomparison of(9.l.2) and(9.1.3) shows atonce that <1>(Y.Y;z)-8193 s;Y; (9-9-3) Thefunction (<z,Y;z)isvery frequently encountered inanalysis, mainly because ofthefactthatalarge number ofspecial functions canbeobtained from d)(a,Y;z)bymaking suitable choices oftheparameters oz,Yandthe variable z(seeSec.9.13). Thismakes itpossible todevelop thegeneral theory ofthese functions inasimple andcompact form. The deﬁnition oftheconﬂuent hypergeometric function immediately implies theidentities dE(I)(0t,Y; Z)=g(I>(ot+1,Y+1;Z), (99.4) %",;<D(oc,Y;z)=€o%'(ot+m,Y+m;Z), m=l,2,..., (9.9.5) Ym 2”Usetheratio test,noting thatif k ,,,=QL,(Y)k kl then “L2 =I___l_'l_i__z _,O uk (Y+k)(1+k) ’ ask—>oo. 2"Usethecriterion given infootnote 4,p.102,noting thatif IcUK= (‘x)x L, F(Y+k)k! then Um-1_ 0!-l'k A-l-k Y,._l(Y+k)(1+/()2<(k—C)(1+k)lzlsq<1’ forsutﬁciently large k. 262 HYPERGEOMETRIC FUNCTIONS CHAP. 9 andtherecurrence relations (Y—at—1)+a<I)(a +1)—(Y—1)d>(Y —1)=0, (9.9.6) Y(1)~Y<l>(a —1)—z(Y +1)=0, (9.9.7) (at—1+Z)<I) +(Y—ot)(I)(ot —1)-—(Y——l)(D(Y —1)=0, (9.9.8) Y(1+Z)<1>—¢Y<1>(<== +1)—(Y—¢)Z<P(Y +1)=0,(9-9-9) (Y—ot)<D(ot ——1)+(2ot —Y-1-Z)(D —ot<D(ot -1-1)=O, (9.9.l0) Y(Y—1)¢’(Y ~1)-Y(Y—1+Z)‘1>+(Y~<1)Z<1>(Y +1)=0,(9-9-11) connecting thefunction (I)E(I)(ot,Y;z)with any two contiguous func- tions <D(o1i1)-2(D(oci1,Y;z)and(I>(Y-31)2(o1,Yi1;z).Formulas (9.9.6—7) canbeveriﬁed bydirect substitution oftheseries (9.9.1), andthen theother recurrence relations canbeobtained bysimple transformations of (9.9.6—7). Besides therecurrence relations justgiven, there exist similar relations between thefunction (oc,Y;z) and anypair offunctions oftheform <I)(<x+m,Y+n;z),where mandnarearbitrary integers. Two simple rela- tions ofthiskind are” <I)(ot,Y;z) =(D(ot+1,Y;z) -€-(I)(0t+1,Y +1;z), (99.12) <I)(o:,Y;z) =$<I)(ot,Y +1;z)+$q>(¢ +1,Y +1;z), (99.13) ascanbeveriﬁed bydirect substitution of(9.9.l), orbyrepeated useofthe relations between (D(a,Y;z)anditscontiguous functions. 9.10. TheDifferential Equation fortheConfluent Hypergeometric Function andltsSolutions. TheConfluent Hypergeometric Function oftheSecond Kind Itiseasytoseethattheconﬂuent hypergeometric function isaparticular solution ofthelinear differential equation zu”+(Y—z)u’—au=0, (9.10.1) where Y¢O,——1,—2,...Infact, denoting theleft-hand sideofthisequa- tionbyl(u),andsetting u=ul=<D(a,Y;z),wehave _°°k(/<—1)(<1)Y. ..-, _°°(Qt ,,_,_ °°(“)1,. l(u1) —22 Z +(Y Z)E1 Z akgo Z _ (L),_] °°(oz),,z"[koc+k <x+k_k__ ]=0 Y(Y)1 oi+YZ1(Y)k/<1 Y+k+YY+k oi_' ,_i,a- 29Note thesimilarity between formulas (9.9.6—13) andformulas (9.2.4—l5). SEC.9.10 HYPERGEOMETRIC FUNCTIONS 263 Toobtain asecond linearly independent solution of(9.l0.1), weassume that |argz|<7!andmake thesubstitution u=z1'Yv. Then equation (9.10.1) goes intoanequation ofthesame form, i.e., zv”+(Y'—z)l)’—-oc'v=O, withnewparameters a’=1+at-Y,Y’=2—Y.Itfollows thatthefunction u=u2=z1'Y(1+ a——Y,2~—Y;z) isalsoasolution of(9.10.1) ifYaé2,3,....Thus, ifYaé0,i1,i2,..., both solutions ul,L12aremeaningful andarelinearly independent ofeach other,“ sothatthegeneral solution of(9.10.1) canbewritten intheform u=A(I>(a, Y;z)+Bz“Y<D(1 +01—Y;2—Y;z), |argz| <Tc,Y950,;I_-1, i-2,... (9.102) With aview toobtaining anexpression forthegeneral solution of(9.10.l) which issuitable forarbitrary Yaé0,—1,—2,...[see(9.10.1l) below], we introduce anewfunction ‘F(Y.Y;Z)=F(%;%Y) (Y.Y;Z)+lg-)1-)z1~~(1+ Y-Y.2—Yiz), |argz| <1:,Y¢0,il,12,..., (9.l0.3) called theconﬂuent hypergeometric function ofthesecond kind. Formula (9.l0.3) deﬁnes thefunction ‘F(oi, Y;z)forarbitrary nonintegral Y,andmore- over, aswenow show, theright-hand sideof(9.l0.3) approaches adeﬁnite limit asY—>n+1(n=0,1,2,...).Replacing the<1)functions in(9.l0.3) by theappropriate series, and using formula (l.2.2) from thetheory ofthe gamma function, weobtain ._W 1 °°(1)1 Z_" W"Y’Z)*sin1;Ylr(1+on-Y),2,F(Y+k)kl1no(4 )“H (9.10.4)at—Y z _ -rc _r(a),,Z, T‘(2-Y+k)/<1l_S1I1TEY(g1 _gzl Since - _ 1 ac (°‘)1¢ ik_ 1 O0 (°‘)1¢ ff Yyrlll}-lgl _F(ot —n),2:0 I‘(k +fl+1)klTF(oc —n),2,(n+k)lkl’ 1°° (ex-—n),, z"'"1" =_ i ___ ,_‘,f‘I,g2 1“(¢),Z,1“(/< -fl+1)k! ___ 1ﬁ:(°‘_n)n+n Zk _ 1 E (“lie it _1"(¢),,=, r(/<+1)(71+/<)1_F(O(._n),,=,(n +k)!/<1’ 3°Note thatulEllzifY=1. 264 HYPERGEOMETRIC FUNCTIONS CHAP. 9 theright-hand side of(9.10.4) becomes indeterminate asY->n +1,and approaches alimit whose value canbefound byusing L’Hospital’s rule, i.e., ‘l"(ot,n +1;z)= lim\1*(YY,Y;z)=(-1)"+1[92 _951’Y"71+1 8'YY=n+1 8TY=n+1 |argz|<1:,n=0,1, 2,...(9.l0.S) Calculations likethose made inSec.5.5show that“ 31 _ 1 w (1)11: la -1%_n)k§0(———-n +,j),k,1Y<Y -Y)-¢(n+k+1)1. % Z13(ot),,z" aY,.,,,, F(0t_n),,=,(n +k)!k! ><[t)(l +k)-1.l:(ot+k) +1l1(ot—n)—l0gz] 1"'1(—1)"""(n -k-1)l(ot-Y),,,_,,+r(@Y),,Z, k! Z’ which leads tothefollowing series expansion: ‘F(<z, n+1;z) _<—1)"+1 °°(ow _ _ ,, _M_nnzogﬁ [(1(¢+k)¢(1+k)¢(n+1+ k)+1<>.,z] (9.l0.6) |argz| <rc, n=0, 1,2,..., (1750, ~—1,—2,... Here 19(2)=F’(z)/F(z) isthelogarithmic derivative ofthegamma function, andthemeaningless sum -1 go which appears when n=0,issetequal tozero. Ifa=—m(m=0,1,2,...),passage tothelimit Y—>n +1(n=0,1, 2,...)in(9.l0.3) leads totheexpression” ‘1’(—m;n +1;z)=<-1>""’%")’ @(—m,n +1»)."' (9.l0.7) m=0,1,2,..., n=0,1,2,... 31Indifferentiating gg,weusetheformula d30)..=onllo +k>—1»<1)1. From now on,weassume that oz¢0,—1,—2,... “ZHere weagain useformula (l.2.2). sac.9.10 HYPERGEOMETRIC FUNCTIONS 265 Moreover, itisanimmediate consequence of(9. 10.3) thattheconﬂuent hyper- geometric function ofthesecond kindsatisﬁes therelation ‘F(oi, Y;z) =z1'Y‘I"(l +on-—Y,2—Y;z), [argz|<Tc.(9.10.8) Using thisformula, wecandeﬁne thefunction ‘F(oi, Y;z)forY=O,—1,—2, ...,obtaining ‘1’(9<,1 —n;z)=111111‘1’(<1,Y;Z) =Z"‘1’(<X +'1,"+1;Z),Y“‘" 9.10.9|argz|<rc, n=l,2,... ( ) Thus weseethat‘F(oi, Y;z)ismeaningful forarbitrary values ofthepara- meters ctandY.Itfollows from thedeﬁnition (9.l0.3) andtheproperties of (a,Y;z)that‘F(oi, Y;z)isananalytic function ofzintheplane cutalong [—oo,0],andanentire function ofozandY. Next weshow that ‘F(oi, Y;z)isasolution ofthedifferential equation (9.10.1). For Yaé0,il,i2,...,this isanimmediate consequence of (9.l0.3), andforintegral Y,theresult follows from theprinciple ofanalytic continuation (cf.footnote 12,p.167). Foronaé0,—1,—2,...,thesolutions <I)(ot,Y;z)and‘F(oc, Y;z)arelinearly independent, ascaneasily beveriﬁed by calculating theWronskian 33 W@aYaW@wm=—§@r%1°‘) (9.10.10) |argz| <1:,Yaé0,—1,—2,..., andthen thegeneral solution of(9.l0.1) canbewritten intheform =A ,;+B‘I’ ,;.u (“YZ) (“YZ) (9.10.11)|argz| <1:,oc,Y750,-1,—2,... Thefunction ‘F(oi, Y;z)hasanumber ofproperties analogous tothose of (a,Y;z).Forexample, wehave thedifferentiation formulas %W@wa=-wn+LY+nadm (9.lO.l2) (§‘P'(ot,Y;z)=(—1)’"(oc),,,‘1"(ot+m,Y+m;z), m=1,2,..., therecurrence relations ‘F—a‘F(a +1)—‘F(Y—1)=0, (9.l0.l3) (Y—a)‘I’+‘P'(a—1)—z‘P'(Y +1)=0, (9.10.14) 3“Equation (9.10.1) implies W{, ‘~P‘}=Cz"'e’. Comparing both sides ofthisidentity asz—>0,weﬁnd that I‘() °=u%' 266 HYPERGEOMETRIC FUNCTIONS CI-IAP. 9 (OL-1+z)‘I"—‘F(ot—1)+(oz—Y+l)‘F(Y —1)=0, (9.l0.15) (ot+z)‘P'+oc(Y—at—1)‘I/‘(oz +1)—z‘P'(Y +1)=0, (9.l0.16) ‘I’(a—1)—(20;—Y+z)‘F+ot(ot—Y+l)‘F(<x +1)=0, (9.l0.17) (Y—ot—1)‘IJ'(Y— 1)~—(Y—1+z)‘I’+z‘I’(Y+l)=0. (9.10.18) ‘YE‘F(=1,Y;Z), ‘Y(1i1)E‘Y(1i1;Y§Z)» ‘F(Yi1)E‘I’(¢,Y i1;Z) andsoon,whose validity follows from thedeﬁnition ofthe‘I’function and thecorresponding properties ofthe<1)function. 9.1l.Integral Representations oftheConﬂuent Hypergeometric Functions Thefunctions (<z,Y;z)and‘F(oi, Y;z)have simple integral representa- tions which playanimportant roleinthetheory andapplications ofconﬂuent hypergeometric functions. Here weconsider onlythebasic representations in terms ofintegrals evaluated along aninterval oftherealaxis, referring the reader elsewhere formore general representations interms ofcontour integrals.“ Thesimplest integral representation ofthefunction <D(a,Y;2)canbeob- tained bysumming theseries (9.9.1) with thehelp offormula (9.1.2): (L)k_ F(Y) 1tZ—1+lC _ Y—0t—1 (Y).'P(Y)P(Y—Y)’(1’)d”ReY >Rea >O,k=0,1,2,... This gives (ot,Y;z) = kE:0%TJ;1;“-1+n(1_t)Y-a-1 dt _ ) 1at-1 Y'Ot—1 an(t)k "mnli-al.’ “"’>"’,.Z..%’OT q) ._ F(Y) 1ztoc—1 _ Y—t!-1 R R 0(a,Y,z) -F(a)F(Y _cc)0et(1 t) dt, eY>ea>, (9.l1.1) where reversing theorder ofintegration andsummation isjustiﬁed bythe usual absolute convergence argument (cf.footnote 2,p.239). 34SeetheBateman Manuscript Project, Higher Transcendental Functions, Vol. 1, pp.256,27111‘. sec.9.11 HYPERGEOMETRIC FUNCTIONS 267 Wecanusetheintegral representation (9.11.1) todeduce animportant relation satisﬁed bythefunction (a,Y;z). Assuming temporarily that ReY>Reat>0,wemake thechange ofvariable t=1—s.Then (9.ll.l) becomes (D(ot,Y;z)= —;) ezL1e‘“s"‘°‘“1(1 -—s)°“1 ds, which implies ‘P01,Y;Z)=@"‘I>(Y —<1,Y;Z), (9-11-2) since ReY>Re(Y—at).Therelation (9.11.2)wasproved under theassump- tionthat ReY >Re02>0,butafter dividing byF(Y), both sides become entire functions of01andY.Therefore, according totheprinciple ofanalytic continuation, (9.11.2) remains valid forarbitrary 01.andY,provided that Y;éO, —l, —2,... Toobtain anintegral representation of‘F(oi, Y;z),weﬁrstnote thatthe function u,deﬁned by u=ﬁfom e'2‘t°“1(1+t)Y'°"1dt, Rea>0,Rez>0,(911.3) isasolution ofthedifferential equation (9.10.1). Infact, denoting theleft- hand sideof(9.11.3) byl(u),wehave” <1) 1 —zt (1-1 Y—u—1 2_ __ _atl(u)=wfOe t(1+t) [Zl(Yz)l ]dr 1 O0d -2 oz -02 _ I —z on Y"-Gt=a)..__=-ﬁfe an't(l-1-t)” ]dt-—F(;Ye 't(1+t) t=0=0. According to(9.l0.2), thesolution ucanbewritten intheform 11=/1<P(9<, Y;Z)+BZ"*<1>(1+ <1—Y,2—Y;Z),(9.11.4)|argz| <1:,Yaé0,1-1,12,... Assuming temporarily that0<ReY<1andz>0,wetake thelimit of (9.ll.3) asz->0+. This gives ._ ‘ _ 1 co(Y-1 —o1—1 _ F(1_Y)A-211131 u-—-1_,(a)_L t(1+1)” dt-7-il,(1+ at_Y), where wehave used formulas (l.5.3) and(1.5.6) from thetheory ofthe gamma function, andthepassage tothelimit behind theintegral signiseasily 35With ourrestrictions onozandz,thedifferentiation behind theintegral signis justiﬁed. 268 HYPERGEOMETRIC FUNCTIONS CHAP. 9 justiﬁed. Moreover, differentiating (9.11.4) with respect toz,multiplying by 2*andthentaking thelimitasz—>0+,weobtain _ 1 ‘ ’_ 1 1 ' no —ztoc —-11-1B-l_Y2l_1Y)rfrzYu-Y__1I,(a)zl1rgYz"J‘o et(1+t)* dt _ 1 ‘ so —s -0:-1 '0-0u0i$l.“““+” “ _1”_Ha _W—0‘0—0u0ke‘ “"ho‘ Itfollows that *Y _ PU ) .u- Pu +a_Y)(I)(ot,Y,Z) P_ +£;)z1'Y<D(1+ on—Y,2—Y;z)E‘F(ot,Y; z). (9.11.5) F01) Since both sides areentire functions oftheparameter Yandanalytic functions ofthevariable zinthehalf-plane Rez>0(seeSec.9.10), thetemporary restrictions imposed onYandzcanbedropped, andwearrive attheintegral representation ‘Y(1, Y;z)=T-—(l“—)J~: e“""t°“1(l +t)Y'°“1 dt, Rea >0,Rez >0. (9.1l.6) Some other integral representations ofthefunctions (a,Y;z) and ‘F(oi, Y;z)aregiven inProblems 11-13, p.278. 9.12. Asymptotic Representations ofthe Confluent Hypergeometric Functions forLarge lzl Webegin byderiving theasymptotic representation of‘F(oi, Y;z)forlarge lz|,which turns outtobesimpler than thecorresponding representation of <D(a,Y;z).Suppose that Rea>0, |argz|<g—8, where 8>0isarbitrarily small. According to(5.11.2), (1.=9<_;1>:<1,;;Y-Y»t" icO + t"*1lo1 (1-s)"(1+sr)”'°“""Zds. SEC.9.12 HYPERGEOMETRIC FUNCTIONS 269 Substituting thisexpansion intotheintegral representation (9.11.6)andinte- grating termbyterm, weobtain 36 _u"-1"1+— _%,w,=Z [2 k_|_rn(z)]’0R‘ where _(_1)n+1(l +“_Y)"Za w-2 non 1 n —o¢—n-Fn(Z)— O2 tl+ dt 0(1—-S) (1+St)Y 2dS. Estimating |r,,(z)] weﬁndthat |n.(z)| < 2°‘J:e""""‘“°t""P“°°‘dt ><flu-s)"(l+st)R°‘”‘°‘>‘"‘2ds.0 Ifwechoose nsolarge thatRe(Y—oz)—n—2<0,then (1+st)R°“"°”'"‘2 <1, andhence” (1+ _ )7‘P(n +Re +1 Rea 1r|Im a| _n_ |r"<z>|<(H+°j),F(1) (|2|Q“mllilnf =0<|z|1). Itfollows that q/'a,Y;Z) =2-0: g:0 Z—k +0(|Z|—n—1)],/'\ r-iia- R¢<»>0, |argz|<g-8, n>Re(Y—oc)—2 (9.12.1) forlarge Wenow show that theconditions under which this formula has been proved canbeconsiderably weakened. First wenote that even if Re(Y—ot)—n—2>0,aninteger m>ncanalways befound such that Re(Y—oz)——m—2<0.Since theexpansion (9.12.1) certainly holds with nreplaced bym,wehave III Tl kZ0...+0(lZ|—m-1)=kZ ...+ = ..._|_O(]Z|—m—1) O R‘ ii\/15b-' 7|. =Z---+0<lz|-"-1) 3“According to(l.5.l), T,-<15]: e"‘t°"""1 dz=(oc),.z‘°‘“". Rea>0,Rez>0,k=o,1,2,... 5”Forcomplex aandbwehave |ah| =|a|Re be-Irn h-arr a<|a|Re berlllm bl HYPERGEOMETRIC FUNCTIONS CHAP. 9 which again gives (9.l2.l). Therefore thecondition imposed onncanbe dropped, and(9.l2.l) isvalid forarbitrary n. Next wegetridoftherestriction imposed ontheparameter <1.Suppose at satisﬁes theweaker condition Rea >—l.Then Re(oz+1)>0,and formula (9.l2.l) canbeapplied toeach ofthehypergeometric functions inthe right-hand sideoftheidentity ‘P'(ot,Y;z) =z‘F(<z +l,Y+l;z) +(1+on—Y)‘I’(o: +1,Y;z), (9.12.2) obtained byreplacing onby0!.+1in(9.l0.l4). Carrying outthenecessary calculations, weagain arrive attheasymptotic representation (9.12.1),butthis time with thecondition Reoz>—l.Repeating thisargument, weseethat (9.l2.1) holds forarbitrary values ofot.Moreover, byslightly m’odifying the method used toprove (9.l2.1), wecanreplace thecondition |argz|<-ht—8 bytheweaker condition |arg2|<1:-—8.38Thus, ﬁnally, wearrive atthe following asymptotic representation of‘I"(o<, Y;z)forlarge |2|: \p(a,Y;Z) =2-012 Z—k _|,_0(|Z|—n—1)], p@4<n-a any Thecorresponding asymptotic representation ofthefunction (I>(o<,Y;z) canbededuced from (9.12.3) andtherelation I‘ . F <@=,Y;Z)= e*~""~I’(@<,Y;z> +em-*>"‘e“1’<Y —<»,Y;-Z), |arg2|<1:,—z=ze*"‘, Yaé0,—l,—2, ..., (9.l2.4) which istheinverse of(9.l0.3), where theplussignischosen ifImz>0and theminus signifImz<0.Toprove (9.l2.4), weassume thatYaé0,i1, 12,...anduse(9.l0.3): MY;Z)= @(“>Y§Z) +P(%‘)z1-~<1><1 +<><-Y,2—Y;Z)- aux Replacing onbyY—<1andzby—z=ze*"‘, weobtain I‘l— ez‘F(Y '—(X9 —Z) : q)(as Z) _F(Y _' _ M ,ﬁ7:52YwY¢u+¢-$2-%@,@um 3°Instead of(9.l1.6), usetheintegral representation @4319 ‘P'(oc,Y;z) =Féji e‘=’t°“1(l +t)Y'°"1dt, Rec: >O, where MP1—if—(rc—8)<argz< —(E—8), e= 2 —— ifg—3€argz<T:—8.NF! sac.9.13 HYPERGEOMETRIC runcrrous 271 where wehave used (9.l1.2). Eliminating (l+on—Y,2—Y;z) from (9.l2.5—6), wearrive at(9.l2.4) after some simple calculations, where the validity oftheresult forpositive integral values ofYfollows from theprin- ciple ofanalytic continuation. Substituting (9.l2.3) into(9.12.4), weﬁndthe desired asymptotic representation of(<z,Y;z)forlarge |z|: ‘F(oi,Y;Z) F() taxi_an(—1)"(<X)k(1 +<1—)k:_,, _,,_,=W{—-oT)e z[’go z +O(|z| )] + ezz-(Y-a>[iO Z—k +0(|Z|—1»—1)], Pi‘ |arg1|<TC-s,Yat0,-1,-2,...(9.12.?) Asbefore, theplus sign corresponds toImz >0andtheminus sign to Imz<0.If|argz|<1}1r—8,theﬁrstterm issmall compared tothesecond, and(9.12.7) takes theform (D(oc,Y;z) =%e*z“Y‘“> so kz‘k +O(|z|‘"‘1)], ‘W |argz| <g-s,ot,Yas0,-1,-2,... (9.12.s) 9.l3. Representation ofVarious Functions inTerms ofthe Confluent Hypergeometric Functions Aswenowshow, various familiar functions ofmathematical analysis are special cases oftheconﬂuent hypergeometric functions (ot,Y;z)and ‘P'(ot, Y;z),corresponding tosuitable choices oftheparameters ot,Yandthe variable z.Particular attention willbedevoted tothespecial functions intro- duced inChapters 2-5. l.Elementary functions. Some typical relations involving elementary functions are <I)(<x, ot;z) =ZH=ez, PF O °° 2" ez—lc1>1,2; =i_=i,(Z),Z0(k+1)! Z (I>(—2, l;z) =1—22+%z2. 272 HYPERGEOMETRIC FUNCTIONS CHAP. 9 2.Error functions. Itfollows from (2.l.5) and(2.l.2) thattheerror func- tionhastheexpansion lc2lc+ 1 °° 1 (__ —1) ()1. 2)"Erfz= —————(2kz_|_1)=zkZ:0-(;—),€———](z! i K‘ ,,1\/18E/\ andhence Erfz =2(I> '-22) (9.l3.1) Ml-*..Nbl(— . Similarly, thecomplementary error function (2.l.6) canbewritten in theforrn co co -225 Erfcz =|z e"2dt=~}2e"’2 LR-/€i?gds, ifwesett=z\/1 +s.Then, according totheintegral representa- tion(9.ll.6),°9 Erfc z=%2e"2‘P'(l, %;22), or Erfcz={,8-z2\r(g,.1,; Z2), |arg2|< (9.112) where wehave used (9.l0.8). 3.Thefunction F(z). Next weconsider thefunction F(z), related tothe probability integral ofimaginary argument (seeSec.2.3). Itfollows from (2.3.4) that °° (_1)k2kz2k+1 Z°°(1)k(_z2)k’ F i -ii Z (Z) 120 1'3’ ''(2k +1) ego ki(%)k andhence F(z) =2<D(1, %;-22). (9.l3.3) 4.Fresnel integrals. Combining (2.4.6), (2.l.5) and(9.l3.1), weﬁndthat z 13_rci22 13_ 1-cizz 6(2)=2|“’|2’2,7)+“’|2’2’" 2 l3rcizz l3 1:122 S“)=2-|“’|2’5;7|‘‘bl?2;“ 5.Theexponential integral. Bydeﬁnition,(913.4) co —t Ei(-—z) =—f €Tdt, |arg2|<1: 3”Inthederivation weassume that 2>0,and then useanalytic continuation to extend (9.l3.2) into thedomain |arg2|<1:/2. SEC.9.13 HYPERGEOMETRIC FUNCTIONS 273 [cf.(3.l.2)], andhence, setting t=z(l+s)andusing theintegral representation (9.ll.6), wehave 00 —zs€ Ei(—z) =—€_z-L mdS =—€_z\F(1,1;Z), or Ei(z) =—e"Y(l, 1;-2), |arg(—z)| <TC. (9.l3.5) 6.Thesineandcosine integrals. Combining (3.3.6) and(9.l3.5), weﬁnd that Ci(2) =—%e“Z ‘Y(1, 1;2e"”2) —%e‘“I’(1, 1;ze"“’2), |arg2|<g, Si(z) =g+%e""F(1,1;2e"”2) —%;e‘z‘I"(1,1;ze"‘”2), |arg2|< (9.13.6) 7.Thelogarithmic integral. Itisanimmediate consequence of(3.4.3) and (9.l3.5) that li(z)=—2(D(l, 1;—log2), |arg2|<1:, |arg(1—z)|<1:. (9.131) 8.Hermite polynomials. According to(4.9.2), theeven Hermite poly- nomials canbewritten intheform H...<z)=§0<-1>"m‘;,35’_’—’;,5,<2z>*"-2" =<-1>"<2n>! ..<2>1"(-M2)“ .<2>1"->..2* since (Zk)!=22"(%)kk!, andtherefore H2,,(2) =(—1)" (-—n, %;22). (9.l3.8) FortheoddHermite polynomials wehave theanalogous formula H.....<z) =<—1)" 2z<1><-~. %;Z2). (9.13-9) 9.Laguerre polynomials. Itfollows from (4.l7.2) that ,_"1“<n+i»+ 1)<—z)"_<<»+1>.. "<~n>z"L4’)‘,2,F(k+0t+1)k!(n-k)!_n!,Z,<..+1k),k!’ andhence ,1 (oz+1),, _Ln(Z) = T (I)(—n, 1+ 1,Z). 274 HYPERGEOMETRIC FUNCTIONS CHAP. 9 10.Cylinder functions. Assuming temporarily that Rev >—{¢, weset s=%(1+t)intheintegral representation (5.10.3), obtaining _ 22v( —iz 12izs v—1¢ v— 2J,(z)_ |oe S/(1-5‘) Vds. Therefore, according to(9.ll.l), Jv(Z)= <1>(»+1,2)+1;212), OI‘ J,,(z)= e"*fI>(v +1,2v+1;212), |arg2|<TE,(9.13.11) where wehave used theduplication formula (l.2.3) forthegamma function. Then weusetheprinciple ofanalytic continuation toshow that(9.l3.ll) holds forarbitrary v. Similar representations canbeobtained fortheother cylinder func- tions. Forexample, itfollows from (5.6.4), (9.l3.l 1)and(9.l0.3) that“ H\‘,1’(2) =—%_e“’""")(22)"‘P"(v +i,2v+1;2ze"‘”2), TC -T‘2-<argz<TE,(9.13.12) H§2’(z) =%_e"“Z‘”")(22)"‘P'(v +1,2v+1;22¢‘/2),T: -—TC<argz<(9.13.13) Then, using (5.7.6), weobtain thefollowing representations ofthe Bessel functions ofimaginary argument: 2V I,,(z) =%|T)T) e"(I>(v +§,2v+1;22), |arg2|<1:,(9.l3.14) K,,(2)=\/?=(2z)ve-“F(Y +1,,29+1;22), |arg2|<n.(9.13.15) ll.Whittaker functions. Aclass offunctions related totheconﬂuent hypergeometric functions, andoften encountered intheapplications, consists oftheWhittaker functions, deﬁned bytheformulas“ M1..11(Z) =Z“*‘/’e‘Z’2<P(% —k+11,21»+l;z), |argZl<W,, , (913.16)W,,_,,(z) =2“/1e“/2‘I"(% —k+|L,2111+1;2), |arg2|<TE. 4°Wealsouseformulas (9.11.2) and(l.2.2—3). ‘*1E.T.Whittaker andG.N.Watson, op.cit.,Chap. 16. sac.9.14 HYPERGEOMETRIC FUNCTIONS 275 9.|4. Generalized Hypergeometric Functions Consider thepower series co IF? (|ZT)lc Z 00 r=1 _k: (°¢1)1¢"'(°‘z>)i¢ik, Ill(Y)k/Yr.2.,(Y1).-'<Y.)./<1 6'14"‘) where pandqarenonnegative integers (p,q=0,1,2,...)satisfying the condition p<q+1,2isacomplex variable, oz,andY,arearbitrary para- meters (except that Y,aéO,—l,—2,...),and (1),.=F(7Y+k)/l‘(7.),‘*2 Using theratio test,weseeatonce thattheradius ofconvergence oftheseries (9.14.l) equals ooifp<qand1ifp=q+1.Thesumofthe series (9.14.1) iscalled thegeneralized hypergeometricfunction, andisdenoted bythesymbol 11 ...1'Z Y1! '''9Y(1 ormore concisely, by,,F,,(u,; Y,;2),i.e., P oo ]_i,[ (“role zk ,,F,,(a,; Y,;Z)= (9.14.2) k—o 1—I(Y8)lc s=1 Clearly, ,,F,,(<x,; Y8;2)isanentire function of2ifp<q.The function ,,+1F,,(a,; Y8;2)isoriginally deﬁned only inthedisk |2|<1,butcanbeex- tended outside thisdiskbyusing analytic continuation. Thefollowing arethesimplest generalized hypergeometric functions: 9°ic oFo(°‘r§'Ys§ Z)=Z5=8*,k=0kl 1FO(°‘r;Ys; Z)=2 zk = _Z)—a1a k=0 - .F.<Y.;Y.-Z)=ii=1"<Y.>z~<~1-1>/21. _.<2z1/2).’ ;¢=0(Y1)1¢k! 1 w k 1F1(ar; Ys; Z):go :(I)(a19 Z); 2F1(°‘r§Ys§Z) =Z §,;o%Pc =F(°‘1, °<2§Y1§Z)- Ic=0 ' 4’Asusual, themeaningless products 0 0 r=1 s=1 which appear when p=0orq=0,aresetequal to1. HYPERGEOMETRIC FUNCTIONS CHAP. 9 Thelasttwoexamples show thatthehypergeometric functions considered in thischapter arespecial cases ofthemore general function (9.l4.2). Some features ofthetheory ofordinary hypergeometric functions canbe carried over tothecase ofgeneralized hypergeometric functions. Forex- ample, itiseasily seen that thefunction u=,,F,,(a,; Y5;2)isaparticular solution ofthelinear differential equation |sﬁ|(s +Y,-1)-21ff|(s +oc,)|u=0 (9.14.3) oforder q+1,where 8denotes theoperator z(d/d2).‘*3 This equation reduces to(9.l0.l) ifp=q—1,andtothehypergeometric equation (9.2.16) of p=2,q=1.There isawell-developed theory ofgeneralized hypergeometric functions, with appropriate recurrence relations, integral representations, etc.“ PROBLEMS 1.Starting from theintegral representation (9.l.6), prove that F(°¢,i5;Y;X +i0)—F(<1,ii;Y;x —1'0) _ 2rciF(Y) __B ___ ______ — ¢ 1)”°‘F(Y <1,Y F311+Y *1F311 X), x>1, Y#0, -1,—2,... Hint. During theproof, assume that Reat<1,ReY >Re(5 >0,and then useanalytic continuation. Comment. This formula shows why thecut[1,oo]isnecessary indeﬁning F(ot,|3;'Y;Z)fOI‘ ot,(3:/:0,—l, —2,... 2.Derive theformulas d dFZ(z°‘F) =¢»Z°"‘F(¢ +1),E(z”'1F)= (Y—l)z*‘2F(Y —1), where thenotation isthesame asinSec.9.2. 3.Prove thefollowing identities: F(2¢,2a;¢+a+1;%)= . 0t+@+%"¢0,-l,—2,..., F(a,B;l +m__ §;_1)=2—a , r(1~11+§)r|5+5) l+ot—-|3;£0,—l,—2,... 4°Note that applying 8toucorresponds tomultiplying ubyk. 4"Forasummary ofthetheory andreferences forfurther reading, seetheBateman Manuscript Project, Higher Transcendental Functions, Vol.I,Chap. 4.Some newresults aregiven byN.E.Norlund, Surlesfonctions hypergéométriques d’ordre supérieur, Mat.- Fys.Skr.Danske Vid.Selsk., 1,no.2(1956). PROBLEMS HYPERGEOMETRIC FUNCTIONS 277 4.Show thatthehypergeometric polynomials F(—n, B;Y;2)(n=0,1,2,..., Y;’=0,—1,—2,...)canbedeﬁned astheexpansion coeﬂicients ofthe generating function w(z,t)=(1—t)°'*(1—— t+zt)'” =ﬁ:%F("", 15;Y;Z)”, |t|<rnin{1,|z—1|'1}. 5.Derive theintegral representation T‘(<=<)T‘(i3) _,_1°"‘°°F01+s)T(i5 +S)T(—S) , F(a9 B>Y1z) '_Znl-J;_iw _|_S) (_z)ds1 Reoc >0,Re|3 >O,|arg(—z)| <1:,Y960,—l,—2,..., where min{ReOt,ReB}<c<0. Hint. Complete thecontour ofintegration ontheright with thearcofa circle ofradius R,=n+%(n->oo),andthenuseresidue theory. Comment. Therestrictions imposed ontheparameters canbeeliminated bysuitably deforming thecontour ofintegration.“ 6.Using term-by-term integration, verify thefollowing formulas: F(Y.l3§Y§Z)= 1“-10-11*-C-1F<<==.@;c;z:>d1. ReY>Rec>O, |arg(1—z)|<rc, 1 F(<x,|3;Y +1;2)= YJO F(oi, B;Y;2t)tY'1dt, ReY >0,|arg(1— z)|<1:. 7.Byanalogy with Sec.9.10, thehypergeometric function ofthesecond kind G(ot,(5;Y;2)canbedeﬁned as G(°‘, 15;Y;Z)= F(¢, F3;Y;Z) 1-‘(Y_1)-Y . .+g;)—Z‘ F(1+<1—Y.1+i5—Y,2—Y,Z), |argz| <1:,|arg(1— z)|<1:,Y¢0,:1, 12,... Prove thatG(a, B;Y;2)satisﬁes therelation G(<1,B;Y;z) =z1'”G(<>< —Y+1.B— Y+1;2 —Y;z)- 8.Repeating theconsiderations ofSec. 9.10, show that G(<x, (-3;Y;2)isan entire function ofoi,B,Y,andderive theformula , , (-1)"+1 “O()1<(i5) ,, G(°"B’"+1")=I‘(oc-n)F(B-n),,Z,,(na+ 1<)i1<1’ ><[¢(YY+k)+<|»(|1+k)—¢(1+k)—¢(n+1+k)+log2] 1 ”'1(-1)"(" ~k-1)!(°‘ -'l)1Y(i3— "hi _,, +r(a)r(|a),,;, /<1 zk’ |arg2|<r:, |2|<1, n=0,l,2,..., oc,|5;éO,—l,—2,... *5E.T.Whittaker andG.N.Watson, op.cit.,p.286. 278 HYPERGEOMETRIC FUNCTIONS CHAP. 9 9._Prove that thefunctions F(0€, B;Y;2)andG(oc, B;Y;2)areapair ofsolu- tions ofthehypergeometric equation (9.2.16) with Wronskian T‘ W{F(<1.13;)/3Z).Go.l3;Y§Z)} =-$2-Y(1 —z)*-~-B-1. |arg(1— z)|<1:,|argz| <1:,Y;¢0,-1,—2,... Comment. Itfollows that thetwo solutions arelinearly independent if oc,|3#0, —l,—2,... 10.Find differentiation formulas andrecurrence relations forthefunction G(@=,B;Y;Z)- Hint. Usethecorresponding relations forthefunction F(ll, B;Y;2). 11.Derive theintegral representation r<> __1~'e1"< +)1"<-) , D(as Y,2) _'2?"-J~c_£m °‘1w(Ys+ S) S(_Z) dss Rea >0,—Reo1 <c<0,Y¢0,-1, —2,... |arg(—-2)| < Hint. Useresidue theory. 12.Derive theintegral representation 1‘ 9,_a -@(<X, Y;Z)= )e‘z‘1'”)’2|o e"t/2”“) J|_1(2\/zt) dt. Re(Y —OC)>0,|argz| <1-c,Y¢0,-1,—2,... Hint. Expand theBessel function inpower series, andthen integrate term byterm. 13.Derive theintegral representation 2(1-1)/2 ,0__l _ ‘Y(1, Y;Z)= |0 6‘l‘°‘/¢(l*")KY-1(2)/Zt) dt, Reot >0,Re(a —Y)>-1, |argz| <1:, where K,,(z) isMacdonald’s function. 14.Prove theformulas (<».Y; Z)= _|01l““(1 —t)”"°“(<1,c;zt)dt, ReY >Rec >0, 1 <D(a,Y +1;2)= YJO (a,Y;2t)tY“1dt, ReY >0. 15.Show thattheLaplace transform of(a, Y;x)is 501.Y;X)=;1,F|<1, 1§Y;%)' HYPERGEOMETRIC FUNCTIONS 279 PROBLEMS 16.Verify that theWhittaker functions M,,,,,(2) and W,,_,,(2) areapair of solutions ofWhittaker’s equation . 1/<l—1-2ll+(—Z+;+i?—)M=O, with Wronskian F(2*’"+1). 2|1+1.=0 -1-2 W{Mk.u(Z), Wt.u(Z)} =*Wjm (913.16). 17.Derive theintegral representation ‘*6 Zke—z/2 <10 _t “k_l/ tu+k—§§ Wk_u(Z) — o eI“ 1+Z dt, Re(p.—k+§)>0, |argz|<1-:. blem, prove theasymptoHint. Usethedeﬁnitions ticformula 18.Using theresult ofthepreceding pro 2" |2|—>0O, |argz|<r:-8. W1....(z) z6'2’Z. 19.Using theresults ofSec. 9.13, derive thefollowing representations of various special functions interms ofW,,_,,(2): Erfc2=5%;-Z e“=2/2 W_1/2.‘/X22). |arg2|<g’ . 1E1(2)=—1 e”'2W- 1/,_O(—z), |arg(—z)| <rt, \/-2 02), |arg2|<1:,|arg(1—z)|<Yr, li(z) =—A/__Z___ W_1/,,Y,(—l g —logz K.<z>=A/iW...(2z). 20.Prove that 17 (1"£,1rFq(°‘r§Ys§z) =1¢;TPFq(°‘r +1§Ys ‘|'1§Z)~ Yss=1|arg2|<Tr. 21.Prove that F+1(°‘r; Ya; Z) 1 °‘-1F(<1;Y.;Zl)dt.0+1 <1 J t°‘r+1“1(1-—l)Yq+1“ P+1 PqT >0—F(°¢p+1)F(Yq+1 _°‘p+1 0 ReYq+1 >Re12,,“ where|arg(1— 2)|<nifp =q+1. 22.Derive theformula 20¢213,on+B‘2‘ ' 2:, , ’ .1F(Y.9.Y +9+1.2)] .F.(a+B+Y,2“+2,) citp.340. 46E.T.Whittaker andG.N.Watson, op. ., 280 HYPERGEOMETRIC FUNCTIONS CHAP. 9 Hint. Find athird-order linear diﬂerential equation satisﬁed bythesquare ofthefunction F(d,B;on+|3+-};2),“andshow thatthefunction F 2ot,2B;oc+|3;2) 82a+|=1+=1.;21Y+2|3 isthesolution ofthisequation which isanalytic inaneighborhood ofthe point 2=O. 47E.T.Whittaker andG.N.Watson, op.cit.,Problems 10-11, p.298. PARABOLKICYLDHHHKFUNCTHJNS l0.l. Separation ofVariables inLaplace’s Equation in Parabolic Coordinates Tosolve theboundary value problems ofpotential theory foradomain whose surface isaninﬁnite parabolic cylinder, itisappropriate tousea coordinate system such thatthecylinder corresponds toaconstant value of oneofthecoordinates. Thus, letx,yand2beasystem ofrectangular co- ordinates with the2-axis parallel tothegenerator ofthecylinder andthex- axisalong theaxisofsymmetry ofanyoneoftheparabolas inwhich the planes perpendicular tothe2-axis intersect thecylinder. Choosing theorigin atthefocus ofthisparabola, weintroduce athree-dimensional system ofpara- bolic coordinates a,B,2,related totherectangular coordinates x,y,2bythe formulas x=%ﬁ—W) y=maz=2 amp where —oo<a<oo, 0<|5<oo, —oo<2<oo, andc>0isascale factor. Thecorresponding triply orthogonal system of surfaces consists oftheparabolic cylinders a=const with fociattheorigin,‘ described bytheequation 2 y2=—2ca2|x —1;)» (10.1.2) 1Thesurface or=const >0,isthehalfoftheparabolic cylinder (10.1.2) with y>0, andthesurface oz=—const istheother half, asindicated inFigure 38. 281 282 PARABOLIC CYLINDER FUNCTIONS CHAP. 10 theparabolic cylinders B=const with fociattheorigin, described bythe equation yz=2c|52|x +g-$5)’ (10.1.3) andtheplanes 2=const (seeFigure 38).Inparticular, given aparabolic cylinder with equation y2=2p|x+5;) (10.1.4) instandard form,2 suppose wechoose theproduct c|3§equal top.Then the cylinder (10.1.4) hasequation B=[30inthecoordinates a,3,2,andthedo- main inside thecylinder tothevalues 0<B<B0,while thedomain outside thecylinder corresponds tothevalues Bo<(3<oo. Al’ at=const a/>0 =const a=O B=O>-X a<O FIGURE 38 Itisanimmediate consequence of(10.1.1) thatthesquare oftheelement ofarclength inthecoordinates a,|:l,2is dsz=c2(a2+B2)(dotz +dliz) +dzz. (10.1.5) Therefore themetric coeﬁicients are h,,=h|,=c\/a2+|52, h,=1, 2Here pisthedistance from thefocus (attheorigin) tothedirectrix. SEC. 10.2 PARABOLIC CYLINDER FUNCTIONS andLaplace’s equation takes theform [cf.(8.l.3)] 1 32 82 82V214 =c 'l'ii -l-C2(Oi.2 -l"$2) = Now suppose welook forsolutions of(10.1.6) oftheform u=A(a)B(|5)Z(2). (10.1.7) Then thevariables separate, andweobtain 1|l€Zé+L@| __ls12_Z_,2C2(ot? +B2) Adotz Bd|32 — Zd22 T ’ where 7.isanarbitrary constant. Itfollows that d2Z F +X22 =O, 1d2A 1d2B 222 2 (10.1.8) K21?‘-l"E?d—|5-§—)\C(0t +|5)=0. Thelastequation, inturn, canhold only if “A% +((1—7Y2c2a2)A =0, (10.1.9) 2 6%; ——(|1.+).2c2|32)B =0, (10.1.10) where |I-isagain aconstant. Thus Laplace’s equation hasinﬁnitely many solutions oftheform (10.1.7), depending ontwoarbitrary parameters Aand|l.. Inmost physical problems, theparameter Aisapositive realnumber (cf.Sec.9.10). Then, introducing newvariables g=\/12..., Y|=\/X25, —oo<E,<oo,0<~r,<oo, andanewparameter vrelated to|1.bytheformula |J.=7tC(2v +l), wereduce equations (10.l.9—10) totheform 2 % +(2v+1-—§2)A =0, (10.1.11) QB5277?-(2Y+1+'I|2)B=0. (10.1.12) l0.2. Hermite Functions Wenow investigate equations (1.l0.11—l2), which, asjustshown, arise when separating Laplace’s equation inparabolic coordinates. Clearly, the problem reduces tostudying thelinear dilferential equation u”+(2.+1-z2)u=0 (10.2.1) 284 PARABOLIC CYLINDER FUNCTIONS CHAP. 10 forarbitrary realorcomplex zandv.Ifwemake thesubstitution u=e'Z2'2v, (10.2.2) (10.2.1) goes intotheequation v”—2zv'+2vv=0, (10.2.3) which fornonnegative integral v=n(n=0,l,2,...)isjustthedifferential equation (4.l0.4) fortheHermite polynomials studied inChapter 4.There- fore, inthecase where theparameter visarbitrary, itisnatural tocallthe solutions of(10.2.3) Hermite functions, while thecorresponding solutions of (10.2.1) arecalled parabolic cylinder functions.“ TheHermite functions canbeexpressed interms oftheconﬂuent hyper- geometric function (oc,Y;z).Infact, ifwechoose t=22asanewinde- pendent variable, equation (10.2.3) goes into dzv 1 dv vtZF+(§—t)E+iv=0, (10.2.4) which isthespecial caseofequation (9.l0.l) corresponding totheparameter values v la Z Z 5, ‘Y Z in Therefore, according to(9.l0.2), thegeneral solution ofthedifferential equation (10.2.4) is v=A<1>(_ --1+Bx/?q>(-_1 5“,g;1), (10.2.5)l\J__<|\,)>—l or l l—3u=A(- 5,5;22)+Bz (——-fl, 5;Z2), (10.2.6) after returning totheoriginal variable z.Inparticular, choosing theeon- stants AandBtobe A=l2VP(%) ,B=i2vF(_%)’ (10.2.7) rel) P<~1>2 2 3Thedeﬁnition given here differs somewhat from that prevalent intheliterature (see theBateman Manuscript Project, Higher Transcendental Functions, Vol. 2,Chap. 8), where theterm parabolic cylinder function refers toasolution oftheequation 2 u”+{v+-15-3-)u=0, which reduces to(10.2.1) ifwemake thesubstitution z=\/21. One ofthesolutions of thisequation isthefunction D,,(z), related toourfunction H,(z) [see(10.2.8)] bythe formula Dy =2—v/2 —=2/4Hv(_z;).(Z) e ‘/2 SEC.10.2 PARABOLIC CYLINDER FUNCTIONS 285 wearrive atthesolution 2"I"1 l 2"F -—l l— 3U=Hv.(Z) =%% (I)(— 51 Z2) + Z(D(i2;)9 Z2)» Fl?) Fl‘5) which wecalltheHermite function (ofdegree v).4Itfollows from (10.2.8) and theknown properties ofthegamma function andtheconﬂuent hypergeo- metric function thatHV(z) isanentire function both ofthevariable zandthe parameter v. Ifv=n(n=0,1,2,...),oneoftheterms in(10.2.8) vanishes andthe other reduces toapolynomial inz.Using formulas (1.2.l—3) from thetheory ofthegamma function, weﬁndafter some simple calculations that H2..<z>=<-1>'"%'Zl <-m.a;Z2).' (10.2.9) H2...1(z> =<-0'" 2z(~m. %;Z2). Comparing these formulas with (9.l3.8—9), weseethatifv=n,thefunction Hv(z) reduces totheHermite polynomial ofdegree n. Ifvaé0,1,2,...,thegeneral solution ofequation (10.2.3) canbeex- pressed interms ofHermite functions. Infact, since equation (10.2.3) does notchange ifwereplace zby——z,thefunction v2=Hv(—z), aswellasthe function v1=Hv(z), isasolution of(10.2.3). Bytheusual method (cf. Sec.5.9),itiseasily shown thatthepairofsolutions vi,v2hasaWronskian oftheform W{v1, v2}=C622, where Cisaconstant. Setting z=0andtaking account oftheformulas H,(0)= 115(0)= (10.2.10)—v wﬁ iswhich areimmediate consequences of(10.2.8), weﬁndthat emwaeﬂaﬁfgﬁﬁi 1“____1*__ 4Itshould benoted that according to(9.l0.3), theHermite function Hv(z) bears the following simple relation totheconﬂuent hypergeometric function ofthesecond kind: H\,(z)=2v\r(_ é;z=)- 286 PARABOLIC CYLINDER FUNCTIONS CHAP. 10 where inthelaststepwehave used formulas (l.2.2-3) from thetheory ofthe gamma function. Itfollows that W{H,,(z), Hv(—z)} = c e22. (10.2.11) Therefore, ifv¢0,1,2,...,thesolutions H,,(z) andHv(—z) arelinearly independent andthegeneral solution of(10.2.3) canbewritten intheform v=MHv(z) +NHv(—z). (10.2.12) However, suppose v=n(n=0,1,2,...),sothat WE0.Then Hv(z) and Hv(—z) arelinearly dependent, andinfact, H,,(—z) =(—l)"H,,(z). (10.2.13) Therefore theright-hand sideof(10.2.12) isnolonger thegeneral solution of (10.2.3). Toobtain anexpression forthegeneral solution of(10.2.3) which is suitable forarbitrary values oftheparameter v,weﬁrstobserve thatthesub- stitution v=ezzw, C=iz transforms (10.2.3) intotheequation w”—2Zw’ —2(v+l)w=0, (10.2.14) which isthesame as(10.2.3) except thatvhasbeen replaced by—~v—1.It follows thatthefunctions vs=e*2H_\,_1(iz), v4=eZ2H_v_1(—iz) (10.2.15) arealsosolutions ofequation (10.2.3). Calculating theWronskians W1H.(z>. @Z*H_._,.(iz>1 = (M16) W{Hv(Z), @’2H_v-1(—iZ)} =622*1/*‘“””’", weﬁndthateach ofthesolutions (10.2.15) islinearly independent ofHv(z). Therefore, forarbitrary v,thegeneral solution of(10.2.3) canbewritten in either ofthefollowing equivalent forms: v=MHv(z) +Ne22H_,,_1(iz) =PHv(z) +Qez2H_v_1(—iz). (10.2.17) Finally, comparing (10.2.17) and(10.2.2), weﬁndthefollowing expres- sions forthegeneral parabolic cylinder function: =M -22/ZHV N22/2H_v_ - u e (Z)+e 102) (10.2.18)=Pe‘*2/2H,,(z) +QeZ2’2H_(,_1(—iz). SEC.10.3 PARABOLIC CYLINDER FUNCTIONS 287 l0.3. Some Relations Satisﬁed bytheHermite Functions Inthepreceding section, itwasshown thateach ofthefunctions U1=Hv(Z): U3=ez2H—v—1(iz)a U2=Hv(—Z): U4:eZ2H—v—1(—iZ)(10.3.1) isasolution ofequation (10.2.3). Since asecond-order linear differential equation cannot have three linearly independent solutions, itmust bepossible towrite each ofthefunctions (10.3.1) asalinear combination ofanytwo others. Inparticular, ifvaé—1,—2,...,5there must exist arelation ofthe form Hv(z) =Mez2H_,,_1(iz) +Ne*2H_.,_1(—iz). (10.3.2) Todetermine theconstants MandN,weusetheconditions (10.2.10), obtain- ingthesystem ofequations 1+v22v+1F(1 +X) 22v+1F(___) M+1v=-i_i M-1v=l_2_~1-—v ’ _ v P<—.-> ii»)Transforming theright-hand sides ofthese equations byusing formulas (1.2.2—3) from thetheory ofthegamma function, weﬁndthat _2v+1F(v +1) vr: _2"+1F(v +1)_.vrrM+N-———T/;_————cos 2, M N- V; lSl1'l?' (10.3.3) Solving thesystem (10.3.3) andsubstituting theresulting values ofMand Ninto(10.3.2), wearrive attherelation Hv(z)Z e”2[e‘”"’2H_.,_1(iz) +e“”‘”2H_.,_1(—iz)]. (10.3.4)T: Formula (10.3.4) remains valid fornegative integral vifwetaketheright-hand side tomean itslimit asv—>-n(n=1,2,...). Replacing zby-2in (10.3.4), weobtain therelation Hv(—Z) = e‘*'2[e""“2H_.,_1(—iz) +e‘“"”2H_v_1(iz)]. (10.3.5)TC 5Ifv¢ -1, —-2,...,then . '\/#W{e”H-V-1(iz), e=2H_v-1(-—zz)} = 3 ezz950. 288 PARABOLIC CYLINDER FUNCTIONS CHAP. 10 Further relations canbededuced from (10.3.4—5) bypurely algebraic operations. Forexample, wehave HV(z) =e‘”"H,,(—z) + r e"2*‘/1‘”*‘”"H_.,_1(——iz), (10.3.6) H.(z>=e~“"H.(-Z) + ‘*“‘”""*""*H_._.(iz>. (10-3-1) andsoon. l0.4. Recurrence Relations fortheHermite Functions The Hermite function H.(z) satisﬁes simple recurrence relations which generalize thecorresponding formulas forHermite polynomials (seeSec.4.10) tothecase where thedegree visanarbitrary complex number. Toderive these recurrence relations, weﬁrst make apreliminary transformation of (10.2.8), which leads toasimple power series representation ofH.,(z). Replac- ingthehypergeometric functions in(10.2.8) bytheir explicit series representa- tions [cf.(9.9.1)], andusing theformulas r(- =2v+1vEr(-v), (10.4.1) 1 13 12(2)="flﬁlflil ="4implied by(l.2.2-3), wehave v 1——v 3/E (_5) °°Flk+2)Hv(z) =T) = (+5) "k€o z2k+1 X/_ as : Tc 2 Z2k_Z 2 Z2k+ 1 _ 0 2“”’=(L£“‘)P(?‘5“—2) P(L5"’)P(%3)R‘l\/18OW‘,1'1 P?‘P?N[OPi‘ R‘ ol\/18"1 _so(_)»1pH =21§(/fv)"; F( 2'". (10.4.2) Since, according to(l.2.3), 2'"r(’l§l)r(”%2) =\/Er(m +1)=V;/11!, sec.10.4 PARABOLIC CYLINDER FUNCTIONS 289 formula (10.4.2) canbesimpliﬁed toe 1...(-1)mr('l’-2-3) Hv(z) =ﬁmgoﬁi (2z)”', |z|<00. (10.4.3) This expansion, which isofindependent interest, allows ustogive avery simple derivation oftherequired recurrence relations. Differentiating theseries (10.4.3) andintroducing thenew summation index n=m-—1,weﬁndthat 1...,2(-1)mr(%) Hi“)Z2r(-om; (m-1)!(zzym ,02(_1)»p('$') 1 2 H =_2r(_v),.Z., nl (22) zr(1-v)=""TF7 Hv—1(z) =2\'Hv-1(Z)- Thus theHermite function H(,(z) satisﬁes therecurrence relation H.j(z) =2vH\,_1(z), (10.4.4) which generalizes formula (4.l0.2). Next wedifferentiate (10.4.4), obtaining H$’(Z) =2vH6_1(-Z), which, together with thedifferential equation (10.2.3) written intheform H(,’(z) —2zH;(z) +2vH.,(z) =0, implies 2vH(_,(z) =2zH§(z) -—2vHv(z). (10.4.5) Using (10.4.4) toeliminate H(_1(z) andH§(z) from (10.4.5), weobtain Hv(z) —2zHv_1(z) +2(v—1)H\,_2(z) =0. (10.4.6) Finally, replacing vbyv+1in(10.4.6) leads toanother recurrence relation Hv,.1(z) —2zH,(z) +2vHV_,(z) =0, (10.4.7) which agrees with ourprevious formula (4.l0.l) when visapositive integer. 5Because oftheintervention oftheduplication formula (10.4.1), theseries (10.4.3) canbeused fornonnegative integral v=nonly ifweagree that theindeterminate ratio F(—1) F(—2) isformally equal to-4[thevalue consistent with (10.4.1)], andallother indeterminate expressions areevaluated withthisinmind. 290 PARABOLIC CYLINDER FUNCTIONS CHAP. 10 l0.5. Integral Representations oftheHermite Functions Various integral representations oftheHermite functions H.,(z) involving contour integrals ordeﬁnite integrals canbederived bysumming theseries deﬁning Hv(z). Thesimplest such representation isobtained from (10.4.3) byassuming thatRev<0andreplacing 1“[-}(m —v)]byanintegral ofthe type (1.1.1). This gives 1°°(—1)”'(2Z)"‘ °° = —s1/2( —v-H(,(z) 2P(_v)mZ0 ml Les"'’ids _1=0_,_,2,_ °°(—1)'"(2z\/§)"‘$10 eS/1d3mZo__"ﬂ? (10.5.1) 1 0°—s—2z~/§ -1/v—1-2F(_v) Le s=ds, where reversing theorder ofsummation andintegration isjustiﬁed byan absolute convergence argument. Introducing thenewvariable ofintegration t=\/s,wecanwrite (10.5.1) intheform _ 1 0° -£2-2tz -v—1H.,(z)_F(_V)f0 e 1dz, Rev<0. (10.5.2) This formula resembles theintegral representations ofSec. 4.11, derived earlier fortheHermite polynomials. Inparticular, itfollows from (10.5.2) that theHermite functions ofnegative integral degree canbeexpressed in closed form interms ofthecomplementary error function (2.l.6). Infact, setting v=~1in(10.5.2), weobtain H_1(Z) =fooe"2'2‘Z dt=ezzJ“e‘“""2 dt=ea”J“e"z ds, 0 0 z i.e., H_1(z) =ezzErfc z, (10.5.3) andingeneral _l 71. dn 2 H_,,_1(z) = E;(e’ Erfc z), n=0,1, 2,...(10.5.4) Another important integral representation ofHv(z) canbededuced from (10.3.4) byreplacing theHermite functions intheright-hand sidebyintegrals oftheform (10.5.2). Under theassumption thatRev>—1,thisgives Zvezz "2w 221t '2O0 :22Hv(Z) =T/= e‘”“’ e“‘'2!"dt+e"”“’ e‘"mt"dt>TC 0 o OI‘ v1z? rt) H.,(z)=Fifi e“2t"cos(221-E)dz, Rev>-1.(10.5.5)\/Tt 0 2 sec.10.6 PARABOLIC CYLINDER FUNCTIONS 291 Formula (10.5.5) isthegeneralization oftheintegral representations (4.11.2—3) oftheHermite polynomials, towhich itreduces when v=n (n=0,1,2,...). Some other integral representations oftheHermite functions aregiven in Problems 1-4attheendofthischapter. 10.6. Asymptotic Representations oftheHermite Functions for Large lzl Toderive asymptotic representations oftheHermite functions Hv(z) for large |2|andﬁxed |v|,weﬁrstassume that Rev <0,|argz|<rt/2. Then, using (10.5.2) torepresent H,,(z), wereplace e“'2byitsTaylor series expansion with remainder, i.e., '1(_1)1¢t21¢ 6-1“=2—k,— +<.>,,(1), (10.6.1),= .O where t2n+ 2 l¢°n(Y)l < Integrating term byterm andnoting that low6-mi“-V-1 dr= /<=0,1,2,...(10.6.2) ifRez>0,Rev<0[cf.(1.5.1)], weﬁndthat Hv(z)=(2z)‘[ (igfit (22)-Zr +r,,(z)], (10.6.3)R‘ O where r,,(z)= ff6,(1)@-“=1-V-1 dz and (-0..=1.(-0...= =(—»><—» +1>---1-v+2/<-1) (k=1,2,...).Now suppose that |arg2|<g—8, where 8>0isarbitrarily small. Then itiseasily seen that 2 -Re v1/g1r|Im v| lrn(Z)l s I000 e—2t|z| sin6t2n+1-Re vdt=0(|Zl—2n—2) 292 PARABOLIC CYLINDER FUNCTIONS CHAP. 10 (cf.footnote 37,p.269), andhence (10.6.3) canbewritten intheform H(,(z)=(zzylki (2z)‘2k +O([z|'2"‘2)]- (10.6.4) Next weshow that(10.6.5) remains valid forarbitrary v.Infact, letthe condition Rev <0bereplaced bytheweaker condition Rev<1.Then, using therecurrence relation (10.4.7), werepresent Hv(z) intheform HV(z) =2zH.,_1(z) —2(v—1)Hv_2(z), (10.6.5) where therealpart ofthedegree ofeach Hermite function ontheright is negative. Applying (10.6.4) toeach ofthese functions, andmaking some simple calculations, weobtain anexpansion ofthesame form as(10.6.4), thereby extending (10.6.4) tothecaseRev<1.Repeating thisargument as often asnecessary, weﬁndthat(10.6.4) isvalid foranyvalue ofv.Moreover, byslightly modifying themethod used toprove (10.6.4),7 wecanextend the result tothelarger sector |argz] <%—-8. Thus, ﬁnally, wearrive atthefollowing asymptotic representation ofH.,(z) forlarge zandﬁxed v: 11.12)=<2z>"[§0 (ikllkt-»)..(2z>-2* +0<|z|"*""2>]» |argz|<Z1‘~8- 1 (10.6.6) Asymptotic representations ofHV(z) which arevalid inother sectors ofthe complex plane canbederived from (10.6.6) byusing therelations (10.3.6—7). Forexample, if Tc 51':Z<argz <T then |arg(—z)| =|argz —1-:|<3%, |arg(—iz)| =argz —g < Therefore, applying (10.6.6) toeach Hermite function intheright-hand side of(10.3.6), weﬁndthat H.<z)=(2z)"[ (-v>..<2z)-2* +0<|z|"2"-2)] V; ‘mi 22 —v—1 n(V+1)2l¢ -216 —2n-2—ﬁe z [kZ:o——k-!—— (22) +O(|z| )], 53+s<argz<f_s.(10.6.7) 7Instead of(10.5.2), usetheintegral representation 1 wig” —t2-2tz —v—1Hv(Z) —ifv) J0 8 I dl, where |0|<1:/4andtheintegration isalong therayargt=0. sec.10.7 PARABOLIC CYLINDER FUNCTIONS 293 Similarly, itfollows from (10.3.7) and(10.6.6) that H.(z)=<22)" (-v)2..(2z>~2* +01121 \/;e_v\;1t1ez2 -v~1[kﬁo (2Z)—2k +0([Zl—2n—2):|, —r(- Z = . -(5-3) <argz< -(5+s)- (1068) 4 \ \ 4 II Together, formulas (10.6.6-8) give acomplete description ofthebehavior ofthefunction H(,(z) forlarge These formulas donotcontradict each other intheir common regions ofapplicability, since thesecond terms of (10.6.7—8) aresmall compared totheﬁrstterms if _E<M2<_E E<a.<ﬁ 4 g 4’4 g 4’ andcantherefore beincluded intheterm 0(|2|"2"‘2). Finally, wenote that(10.6.4) isanimmediate consequence oftheasymp- totic representation (9.12.3) fortheconﬂuent hypergeometric function ofthe second kind andthefactthat H.,(z)=2v\r(--.-l\)<\\,)>—~Nto \¢/ (cf.footnote 4,p.285). |0.7. The Dirichlet Problem foraParabolic Cylinder Thespecial functions studied inthischapter allow ustosolve theboundary value problems ofpotential theory forthecase ofadomain bounded bya parabolic cylinder. Toﬁnd theappropriate setofsolutions ofLaplace’s equation, weintroduce theparabolic coordinates (10.1.1) andlook forsolu- tions intheform oftheproduct (10.1.7), thereby arriving atequations (10.l.8-10). Ifwerequire thatthesolutions bebounded inthewhole domain, inparticular atinﬁnity, itmust beassumed thattheparameter Aisreal.” Then thecorresponding solution of(10.1.8) is Z=Ccos >12+Dsin 7.2, A20, (10.7.1) which isbounded for—oo <z<oo. Introducing thenewparameter vrelated toitbytheformula it=7\c(2v +1), °Without lossofgenerality, wecanassume thatXisnonnegative, since changing the sign of71does notaffect theseparation constant 7?. PARABOLIC CYLINDER FUNCTIONS CHAP. 10 andusing (10.2.18), weﬁndthatthegeneral solution of(10.1.9) canbewritten intheform A=M6-M“/2H,(\/E6) +New/2H_,_,(i\/E6). (10.7.2) According totheasymptotic formulas ofSec.10.6, H.(\/166.) z(2\/mt)", 6_>66, H_,_.(i\/E6) z6-‘/1<"+1>"'(2\/E6)-V-1, 6._>66, andhence wemust setN=0ifthesolutions aretobebounded. Moreover, forv aé0,1,2,..., wehave \/i Acct? \/_' -v—1Hv( 7.eot)zf(—_-T)e ()\C|0i.|) , 0i.—>— (I) andtherefore wemust also setM=0.Itfollows that unless visanon- negative integer, there arenosolutions which arebounded asot—> ioo (except thetrivial solution identically equal tozero). Forintegral v=n(n=0,1,2,...),theHermite functions reduce to Hermite polynomials, andthesolution ofequation (10.1.9) bounded inthe interval (—oo,oo)is A=M6-W“/2H,(\/E6), n=0,1,2,... (10.7.3) Substituting thecorresponding value p.=7.c(2n +1)into(10.1.10), wecan write thegeneral solution ofthisequation as B=Pew“/*H,(i\/E0) +Q6-M’/2H_,_.(\/E11) (10.7.4) [cf.(10.2.18)]. Combining (10.7.1) and (10.7.3,4), weseethat Laplace’s equation hasinﬁnitely many solutions oftheform u=um,=e""‘°‘2'2H,,(\/ﬁat)1P,_,,e"°°2'2H,,(i\/E13) - 7.+Qi...e‘“°”’H_..-1(\/Ml5)l:$,5» )12O, n=0,1, 2,...,(10.7.5) which arebounded for—oo <0t<oo,—oo <z<oo.For theexterior problem, (5varies over theinterval [50<B<oo,where thesurface ofthe parabolic cylinder corresponds to[5=(30,andhence wehave tosetPM=0, inview oftheasymptotic formulas H,,(i\/E0) zi"(2\/E0)", (3_>66, H_,,_,(\/M13) z(21/16(1) "1,(3_>66. Wenow show that QM,must besetequal to0ifthesolutions (10.7.5) sec.10.7 PARABOLIC CYLINDER FUNCTIONS 295 aretobeharmonic inthecase oftheinterior problem, where 0<[5<[30. Here thedecisive consideration isthebehavior ofgradunearthesingular curve ofthetransformation (10.1.1), i.e.,thelineat=[5=0onwhich the Jacobian 8(x,y,z)/8(a, (3,z)vanishes. Itisanimmediate consequence of (10.1.5) that 1 8a2 ﬁu2 Eiu2 ‘g'*“‘">’= llal1“fall+la)'Since thedenominator intheright-hand sidevanishes onthecurve 6t=(5=0, anecessary condition forgrad utobeﬁnite isthattheexpression inbrackets should alsovanish forOt=13=0,i.e.,that QM =0,since (10.7.5) impliess 8u2 Eiu2 — cos7.22 1(6)+(6)-(“~'Q~"6. 6)~°-Moreover, thiscondition isalso suﬂicient, since itiseasily veriﬁed that if QM=0,then theexpression let+1-">280: (81-3 ] isdivisible by612+(32,sothatgrad uiswell-behaved onthelineat=13=0.1° Thus theappropriate particular solutions ofLaplace’s equation are u=6,,=P,_,,e-<40/20*‘-l*’>H,(\/XE6)H,,(i\/X213) 7120,n=0,1,2,... (10.7.6) fortheinterior problem, and - - 1-4=u...=Q1.n@“*°”"°‘2*“"H..(\/M41)H-6-1(\/M15) jff,2» 7.20,n=0,1,2,... (10.7.7) fortheexterior problem. Boundary value problems involving parabolic cylinders aresolved by superposition oftheparticular solutions (10.7.6—7). Forexample, consider theinterior Dirichlet problem, assuming, forsimplicity, that thefunction f=f(ot,z)appearing intheboundary condition u|B=(,° =f (10.7.8) 9Inthecourse ofthecalculations, weusetheformulas H,,(0)H,{(0) =0,H,,(0)HL,.-1(0) =-666 H;(0)H_,._.(0) =sin n=o,1,2,..., which follow from (10.2.10). 1°Cf.theanalogous treatment foranoblate spheroid onp.217. PARABOLIC CYLINDER FUNCTIONS CHAP. 10 isaneven function ofz,which implies thatthesame istrueofthesolution u=u(oc,15,2).“ Suppose thatfcanbeexpanded inaFourier integral f=fwfA(a) cosAzdx, —oo <z<oo, (10.7.9)0 where ft=3Jmfcos )\zdz, (10.7.10)T50 andmoreover suppose thatthesolution ucanalsoberepresented asaFourier integral u=foou,\(oc, B)cosAzdx, —oo <z<oo. (10.7.11)0 Then, according to(10.7.6), wecanlook foru)(oc, (5)intheform ofaseries um.s)=ZPme-<“’2><“’~*>’>H.<~/i?==@>H.(i~/iii»,n.=0 —oo <on<oo,0<B<B0,(10.7.12) andwehave thecondition rm=ZPt..e-<*~2><~”-**%>H.<@=~/i?>H..W%s<,>.n=0 —oo <at<oo (10.7.13) fordetermining thecoefﬁcients PM. Assuming thatf(a) satisﬁes thecondi- tions ofTheorem 2,p.71,weﬁndthat PAehcﬁg/2Hn(i,\/')\—C. go)2% Jun e—ma2/2fA(a)Hn(\/Q“) dot, '" 2"n!\/ -to (10.7.14) andhence theexpansion coefficient uA(ot, (5)isgiven bythesum go _ 2_ 2 2 '— M,(5)= '3*“°’-I;I"(—l—:———H,,(\/7\ O!)u e X/_H,,(i\/1030) C (10115) Ac co —7tcoc? 2 '—><27?; Lne/fA(ot)H,,(\/71¢-oz) du.fl. Substituting (10.7.15) into (10.7.11), weobtain theformal solution ofour problem. 11Thecase wherefis anoddfunction ofzishandled inthesame way. Then thesolu- tion inthegeneral case isrepresented asthesum ofthesolutions ofthetwo simpler problems withthefollowing even andoddboundary conditions: fl =%If(a> Z) +.f(a: —Z)]a f2 =%If(qa Z)—-f(a) "—z)]" SEC. 10.8 PARABOLIC CYLINDER FUNCTIONS l0.8. Application toQuantum Mechanics TheSchrodinger equation foralinear harmonic oscillator ofmass m, angular frequency tooandtotal energy Ehastheform 2 22 d4’+(2';l”2E ”‘h"°°)¢ —0, (10.8.1)?0?___'T' where LI»isthewave function andhisPlanck’s constant.” Inquantum mech- anics, itisrequired toﬁndthevalues ofEforwhich (10.8.1) hasbounded solutions intheinterval —00<x<oo.Ifweset 2mE mm 1*=F’ ‘C=*7?’ equation (10.8.1) coincides with equation (10.1.9). Itfollows from the results ofSec. 10.7thatthesolutions of(10.8.1) arebounded in(—oo, oo) onlyif u=7\c(2n+l), n=0,l,2,..., i.e.,onlyif 2Elg?=(2n+1)"’—",;"’, n=o,1,2,..., which implies E=5,,=(n+%)h(o0, n=0,1,2,... (10.8.2) Thecorresponding wave functions canbeexpressed interms ofHermite polynomials. PROBLEMS 1.Derive thefollowing integral representations oftheHermite functions: 2v+1 w-:2 —v2 2v/2 T‘H(,(z)=i;— 0et(t+z) dt, Rev<1, |argz|<5, P(‘%)V+1 00 H(,(z) =LT zf e"2t“"1(t2 +z2)<"‘1>'2 dt, Rev <0,|argz|< Pi‘1)°Hint. Useformulas (9.l0.3) and(9.11.6), andtherepresentation ofHV(z) in terms of‘F(1x, Y;z),theconﬂuent hypergeometric function ofthesecond kind (seefootnote 4,p.285). 1”SeeD.Bohm, Quantum Theory, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1963), p.296. 298 PARABOLIC CYLINDER FUNCTIONS CHAP. 10 2.Derive thefollowing integral representation oftheproduct oftwoHermite functions: Hu(Z)Hv(Z) 1"_ _ n/2 _ _ =ii-‘i)— Hu+\,[z (cosq> +sinq>)]cos‘“‘1 q>s1n“’“1 q>dq>,P<—1L>P<-0L Rep <0,Rev <0. Hint. Use (10.5.1) and transform topolar coordinates inthedouble integral. 3.Prove theintegral representation _ _.2 1 no-12-22: -—v—1 ‘L, v,%tH,.(z)Hv(z) =i e t" 2F2 it+v’1—y.—vdt, ° 2 21“(—1L— -———— where ZFZisageneralized hypergeometric function (seeSec.9.14). Hint. Use(10.5.1) torepresent theleft-hand sideasadouble integral over thesquare 0<s<oo,0<t<oo,andthen transform tothenew variables u=s+t,v=t/s. 4.Prove theformulas 1 ==° 12[H\,(z)]2 =Wye e"2‘2"t_2""1<D(—v, —v+ dt, Rev <0, 1 °° 1:2Hv(Z)Hv+1(Z) = e"2'2”t'2"‘2 q)<—V - 1,-V — dt, Rev<—§. 5.Show that theHermite functions satisfy theintegral equation x""*"’”Hv(X) =2Ian(Xy)1’”’J-V/2(2xy)y"”*"”H.(y) dy,O 0<x<oo, Rev<l. 6.Show thattheHermite functions ofhalf-integral degree canbeexpressed in terms ofthecylinder functions ofimaginary argument. Inparticular, prove therelation 1/Z 2 H_1/2(2) = €z2I2K1/4(€Z)1 |arg Z‘ <g‘ Hint. Use theintegral representation (10.5.1), and make thechange of variable t=2zsinh2 (0/4). 7.Prove theformula 2 2 w _1 n1“ Ko(L;_y) :2Ziii”e-“C”W”/2H2.(x)H_2._1(y),Tl 0 —oo<x<oo, 0<y<oo, where K0(z) isMacdonald’s function. Hint. Apply Theorem 2,p.71andtheresult ofProblem 1. PROBLEMS PARABOLIC CYLINDER FUNCTIONS 8.Consider thesystem ofparaboloidal coordinates at,[5,cprelated totherect- angular coordinates x,y,zbytheformulas x=cocﬁcosrp,y=caﬁsinq»,z=%(oz2 —B2), where0< at<oo,0< B<oo,—1r<<p<Tr,andc >0isascalefactor.In thiscoordinate system, thesurfaces on=const, [5=const areparaboloids of revolution instead ofparabolic cylinders, asin(10.1.1). Find thesquare of theelement ofarclength, themetric coefficients andLaplace’s equation inthe system oz,[3,cp.Show that separation ofvariables ispossible inLaplace’s equation written inthecoordinates at,[3,cp,andﬁndtheappropriate particular solutions, both fortheinterior andtheexterior problem. BIBLIOGRAPHY Bailey, W.N.,Generalized Hypergeometric Series, Cambridge Tracts inMathe- matics andMathematical Physics, No.32,Cambridge University Press, London (1935). Bateman, H.,Partial Dlﬂerential Equations ofMathematical Physics, Cambridge University Press, London (1959). Bateman, H.,TheMathematical Analysis ofElectrical andOptical Wave-Motion ontheBasis ofMaxwell ‘sEquations, Dover Publications, Inc., New York (1955). Bateman Manuscript Project, seeworks byA.Erdélyi etal.cited below. Bowman, F.,Introduction toBessel Functions, Dover Publications, Inc., New York (1958). Buchholz, H., Die Konﬂuente Hypergeometrische Funktion mit Besonderer Berucksichtigung Ihrer Anwendungen, Springer-Verlag, Berlin (1953). Buronova, N.M.,AGuide toMathematical Tables, Supplement No.ItoAGuide toMathematical Tables byA.V.Lebedev andR.M.Fedorova (translated by D.G.Fry), Pergamon Press, Inc.,New York (1960). Copson, E.T.,AnIntroduction totheTheory ofFunctions ofaComplex Variable, Oxford University Press, London (1935). Courant, R.andD.Hilbert, Methods ofMathematical Physics, Interscience Pub- lishers, New York, Volume I(1953), Volume II(1962). DeBruijn, N.G.,Asymptotic Methods inAnalysis, Interscience Publishers, Inc., New York (1958). Erdélyi, A.,Asymptotic Expansions, Dover Publications, Inc., New York (1956). Erdélyi, A.,W.Magnus, F.Oberhettinger andF.G.Tricomi, Higher Transcen- dental Functions (inthree volumes), based, inpart, onnotes leftbyHarry Bateman, McGraw-Hill Book Co., New York (1953). Erdélyi, A.,W.Magnus, F.Oberhettinger andF.G.Tricomi, Tables ofIntegral Transforms (intwovolumes), based, inpart, onnotes leftbyHarry Bateman, McGraw-Hill Book Co., New York (1954). 300 BIBLIOGRAPHY 301 Fletcher, A.,J.C.P.Miller, L.Rosenhead andL.J.Comrie, AnIndex ofMathe- matical Tables (intwovolumes), second edition, Addison-Wesley Publishing Co., Inc., Reading, Mass. (1962). Frank, P.and R.von Mises, Die Diﬁ"erential- und Integralgleichungen der Mechanik undPhysik (intwovolumes), second enlarged edition, Dover Pub- lications, Inc., New York (1961). Gray, A.andG.B.Mathews, ATreatise onBessel Functions andTheir Applica- tions toPhysics, second edition, prepared byA.Gray andT.M.MacRobert, Macmillan andCo., Ltd., London (1952). Heatley, A.H.,Some Integrals, Diﬁ'erential Equations, andSeries Related tothe Modiﬁed Bessel Function oftheFirst Kind, University ofToronto Studies, Mathematical Series, No.7,University ofToronto Press, Toronto (1939). Hobson, E.W.,TheTheory ofSpherical andEllipsoidal Harmonics, Cambridge University Press, London (1931). Hochstadt, H.,Special Functions ofMathematical Physics, Holt, Rinehart and Winston, Inc., New York (1961). Jackson, D.,Fourier Series and Orthogonal Polynomials, Carus Mathematical Monograph No.6,Mathematical Association ofAmerica, State University of New York, Buffalo, N.Y. (1941). Jahnke, E.andF.Emde, Tables ofHigher Functions, sixth edition, revised byF. Losch, McGraw-Hill Book Co., New York (1960). Jeffreys, H.,Asymptotic Approximations, Oxford University Press, London (1962). Jeffreys, H.andB.S.Jeffreys, Methods ofMathematical Physics, third edition, Cambridge University Press, London (1956). Kampé deFériet, J.,LaFonction Hypergéométrique, Memorial desSciences Mathématiques, Fascicule 85,Gauthier-Villars, Paris (1937). Klein, F.,Vorlesungen iiber dieHypergeometrische Funktion, Springer-Verlag, Berlin (1933). Lebedev, A.V.andR.M.Fedorova, AGuide toMathematical Tables (translated byD.G.Fry), Pergamon Press, Inc., New York (1960). Seesupplement by N.M.Buronova cited above. Lense, J.,Reihenentwicklungen inderMathematischen Physik, third edition, Walter deGruyter &Co., Berlin (1953). Lense, J.Kugelfunktionen, second edition, Akademische Verlagsgesellschaft, Geest &Portig K.-G., Leipzig (1954). Losch, F.andF.Schoblik, DieFakultdt (Gammafunktion) undVerwandte Funk- tionen mitBesonderer Berzicksichtigung Ihrer Anwendungen, B.G.Teubner, Leipzig (1951). Luke, Y.L.,Integrals ofBessel Functions, McGraw-Hill Book Co., New York (1962). MacRobert, T.M.,Spherical Harmonics, AnElementary Treatise onHarmonic Functions with Applications, second edition, Methuen andCo., Ltd., London (1947). MacRobert, T.M.,Functions ofaComplex Variable, ﬁfthedition, Macmillan and Co., Ltd., London (1962). 302 BIBLIOGRAPHY McLachlan, N.S.,Bessel Functions forEngineers, second edition, Oxford Univer- sityPress, London (1955). Magnus, W.andF.Oberhettinger, Formulas andTheorems fortheFunctions of Mathematical Physics (translated byJ.Wermer), Chelsea Publishing Co., New York (1954). Morse, P.M.andH.Feshbach, Methods ofTheoretical Physics (intwovolumes), McGraw-Hill Book Co., New York (1953). Nielsen, N.,Handbuch derTheorie derZylinderfunktionen, B.G.Teubner, Leipzig (1904). Nielsen, N.,Handbuch derTheorie derGammafunktion, B.G.Teubner, Leipzig (1906). Nielsen, N.,Theorie desIntegrallogarithmus und Verwandter Transzendenten, B.G. Teubner, Leipzig (1906). Petiau, G.,LaThéorie desFonctions deBessel, Centre National delaRecherche Scientiﬁque, Paris (1955). Rainville, E.D.,Special Functions, TheMacmillan Co., New York (1960). Relton, F.E.,Applied Bessel Functions, Blackie andSon, Ltd., London (1946). ReyPastor, J.andA.DeCastro Brzezicki, Funciones deBessel, Teoria Matematica yAplicaciones alaCienciay alaTécnica, Editorial Dossat, S.A.,Madrid (1958). Robin, L.,Fonctions Sphériques deLegendre etFonctions Sphéroidales, Gauthier- Villars, Paris, Volume I(1957), Volume I1(1958), Volume III(1959). 2 ll Rosser, J.B.,Theory and Application offae"‘2 dxandf e"’21’2 dyIe"‘2 dx, O 0 O Mapleton House, Brooklyn, N.Y. (1948). Ryshik, I.M.andI.S.Gradstein, Tables ofSeries, Products, andIntegrals, VEB Deutscher Verlag derWissenschaften, Berlin (1957). Sansone, G.,Orthogonal Functions (translated byA.H.Diamond), Interscience Publishers, New York (1959). Shohat, J.A.,E.Hille andJ.L.Walsh, ABibliography onOrthogonal Polynomials, Bulletin National Research Council, No. 103, National Research Council of theNational Academy ofSciences, Washington, D.C. (1940). Slater, L.J.,Conﬂuent Hypergeometric Functions, Cambridge University Press, London (1960). Smirnov, V.1.,Lehrgang derHiiheren Matematik, VEB Deutscher Verlag der Wissenschaften, Berlin, Volume III,Part2(1955), Volume IV(1958). Sneddon, I.N.,Fourier Transforms, McGraw-Hill Book Co., New York (1951). Sneddon, I.N.,Special Functions ofMathematical Physics andChemistry, second edition, Oliver andBoyd, London (1961). Snow, C.,The Hypergeometric and Legendre Functions with Applications to Integral Equations ofPotential Theory, National Bureau ofStandards Applied Mathematics Series, No. 19,U.S. Government Printing Ofﬁce, Washington, D.C. (1952). Sommerfeld, A.,Partial Diﬂerential Equations inPhysics (translated byE.G. Straus), Academic Press Inc., New York (1949). BIBLIOGRAPHY 303 Sternberg, W.J.andT.L.Smith, TheTheory ofPotential andSpherical Harmonics, University ofToronto Press, Toronto (1952). Szegii, G.,Orthogonal Polynomials, revised edition, American Mathematical Society, New York (1959). Tikhonov, A.N.andA.A.Samarski, Differentialgleichungen derMathematischen Physik, VEB Deutscher Verlag derWissenschaften, Berlin (1959). Tolstov, G.P.,Fourier Series (translated byR.A.Silverman), Prentice-Hall, Inc., Englewood Cliffs, N.J.(1962). Tricomi, F.G.,Funzioni Ipergeomerriche Conﬂuenti, Edizioni Cremonese, Rome (1954). Tricomi, F.G., Vorlesungen ﬁber Orthogonalreihen, Springer-Verlag, Berlin (1955). Tricomi, F.G.,Fonctions Hypergéométriques Conﬂuentes, Mémorial desSciences Mathématiques, Fascicule 140, Gauthier-Villars, Paris (1960). Truesdell, C.,AnEssay toward aUniﬁed Theory ofSpecial Functions Based upon 8 . . . the Functional Equation 32F(z,at)=F(z,on+1), Princeton University Press, Princeton, N.J. (1948). Watson, G.N.,ATreatise ontheTheory ofBessel Functions, second edition, Cambridge University Press, London (1962). Weyrich, R.,DieZylinderfunktionen undIhre Anwendungen, B.G.Teubner, Leipzig (1937). Whittaker, E.T.andG.N.Watson, ACourse o/"Modern Analysis, fourth edition, Cambridge University Press, London (1963). INDEX A Adamov, A.A.,44,66 Airy functions, 136-139 asymptotic representation of,138 integral representations of,138-139 oftheﬁrstkind, 137 ofthesecond kind, 137 Akhiezer, N.1.,44 Associated Legendre polynomials, 192-- 199,260 fortheinterval (-1,1), 193 integral representations of,199 oftheﬁrstkind, 193 ofthesecond kind, 193 recurrence relations for, 194-195 representation interms ofhypergeo- metric series, 197-199 Wronskians of,196 Asymptotic equality, 9 Asymptotic expansion, 9 Asymptotic representations, 8ff. Asymptotic series, 9 B Bailey, W.N.,300 Barnes, E.W.,171,176,189,202 Bateman, H.,25,27,44,97,109, 155, 176, 188, 199,242, 255, 258, 266 276,284,300 Bernstein, S.N.,44 Bessel function(s) : generating function of,101 modiﬁed, 109 oftheﬁrstkind, 109 ofthethird kind (see Macdonald’s function) ofimaginary argument, 108-111, 136 differential equation for,110 recurrence relations for, 110 304Bessel function(s) (Cont.): oftheﬁrstkind: ofarbitrary order, 102-103, 134 recurrence relations for,103 series expansion of,102- ofnonnegative integral order, 99- 101 recurrence relations for,100 series expansion of,99 ofthesecond kind, 101,104-107, 135 recurrence relations for,105 series expansion of,106-107 ofthethird kind, 107-108, 135 Bessel’s equation, 98ff. related equations, 106 Betafunction, 13 relation togamma function, 14 Bipolar coordinates, 222,230 Bishop, R.E.D.,207 Bohm, D.,297 Boundary conditions, 146ff. homogeneous, 146 inhomogeneous, 146 Boundary value problems ofpotential theory, 146ff. ﬁrst(seeDirichlet problem) second (seeNeumann problem) third (ormixed), 146 Bowman, F.,300 Buchholz, H.,300 Buronova, N.M.,300, 301 C Carslaw, H.S.,25,26 Chebyshev, P.L.,44,66 Chebyshev polynomials, 44,97 Cherry, T.M.,120, 136 Coddington, E.A.,51,112, 138, 162, 163 Comrie, L.J.,301 Conﬂuent hypergeometric function(s), 260-274 Conﬂuent hypergeometric function(s) (Cont.): asymptotic representations of,268-271 contiguous, 262 differential equation for,262 integral representations of,266-268 ofthesecond kind, 263 representation ofvarious functions in terms of,271-274 Convolution theorem, 26 Cooling ofaheated cylinder, 155-156 Cooling ofaheated object, 24-26 Copson, E.T.,300 Cosine integral, 33,273 asymptotic representation of,37 relation toexponential integral, 36 Courant, R.,146,208,300 Cylinder functions, 98-160, 274 addition theorems for,124-126 applications of,143-160 asymptotic representations of,120- 124 deﬁnite integrals involving, 131-134 deﬁnition of,98,104 expansions inseries andintegrals of, 128-131 integral representations of,113-120 ofhalf-integral order, 111-112 ofnonnegative argument and order, 134-136 theory of,98-142 Wronskians of,113 zeros of,126-128 D DeBruijn, N.G.,300 DeCastro Brzezicki, A.,302 Diffraction byacylinder, 156-157 Dini series, 130, 148, 156 Dirichlet problem, 146, 204 foracone, 210-213 foradomain bounded bytwo inter- secting spheres, 227-230 foradomain bounded bytwoparallel planes, 149-150 forahyperboloid ofrevolution, 220- 221 foraparabolic cylinder, 293-296 forasphere, 206-208 foraspheroid, 215-218 foratorus, 224-226 forawedge, 150-153 Dixon, A.L.,140monx 305 Domain, 2 closed, 2 Duplication formula, 4 E Eigenfunction method (see Superposition method) Electrostatic ﬁeld: ofacharged spherical bowl, 229-230 ofacharged torus, 225-226 oftwocharged spheres, 233-234 Elliptic integrals, 188, 259 Emde, F.,24,301 Erdélyi, A.,25,44,300 Error function, 17,272 complement of,17,272 Euler’s constant, 6,8 Exponential integral, 30ff.,272 asymptotic representation of,32-33 ofimaginary argument, 33-37 modiﬁed, 32 F Fedorova, R.M.,300, 301 Feller, W.,23 Ferrar, W.L.,140 Feshbach, H.,302 Field ofapoint charge: inside ahollow conducting sphere, 208-210 near theedge ofaconducting sheet, 153-155 ontheaxis ofahollow conducting cone, 212-213 Fletcher, A.,188,301 Fock, V.A.,136, 137, 221 Fomin, S.V.,158 Fourier-Bessel integral, 130, 150 Fourier-Bessel series, 129, 147 Fourier-Mellin inversion theorem, 25,92 Fourier method (see Superposition method) Frank, P.,146, 301 Freeman, I.,39 Fresnel integrals, 21-23, 272 asymptotic representation of,23 relation toprobability integral, 22 Frullani’s integral, 6 Fry, D.G.,300, 301 Fubini’s theorem, 6 Functions ofbounded variation, 129 306 INDEX G Gamma function, 1-15 asymptotic representation of,8-13 basic relations satisﬁed by,3 deﬁnite integrals related to,13-14 incomplete, 15 complement of,15 inﬁnite product representation of,8 logarithmic derivative of,5-8 integral representations of,6-7 poles of,2 Gauss, C.F.,255 Gegenbauer polynomials, 125 Gelfand, I.M.,158 Generating functions, 43 Goursat, E.,255 Gradstein, I.S.,302 Gravitational attraction ofaspheroid, 218-220 Gray, A.,301 Grinberg, G.A.,146, 235 H Hankel functions, 107 recurrence relations for,108 Hankel’s integral theorem, 130 Harmonic function, 146 Harmonic oscillator, 297 Harnack’s theorem, 208,218 Heatley, A.H.,301 Helmholtz’s equation, 145,157,234 solution inspherical coordinates, 234- 235 Hermite, C.,60 Hermite functions, 283-299 asymptotic representations of,291-293 integral representations of,290-291 recurrence relations for,288-289 relations satisﬁed by287-288 Hermite polynomials, 60-76, 273 addition theorem for,96 asymptotic representation of,66-68 expansion offunctions inseries of,68- 73 examples of,73-76 generating function of,60 generating function ofproducts of,61 integral equations satisﬁed by,64-65 integral representations of,63-64 orthogonality of,65-66 recurrence relations and differential equation for,61Hermite polynomials (Cont.): relation toLaguerre polynomials, 81 Hilbert, D.,146,208,300 Hildebrand, F.B.,46,205 Hille, E.,78,302 Hobson, E.W.,53,58,161, 171, 189, 192,195, 201,202,213,301 Hochstadt, H.,301 Hypergeometric equation, 162, 243 Hypergeometric function(s), 238-280 analytic continuation inexceptional cases, 256-258 asafunction ofitsparameters, 245- 246 conﬂuent (see Conﬂuent hypergeo- metric functions) contiguous, 242 elementary properties of,241-243 generalized, 275-276 linear transformations of,246-250 ofthesecond kind, 277 quadratic transformations of,250-255 representation ofvarious functions in terms of,258-260 symmetry property of,241 Hypergeometric polynomials, 277 Hypergeometric series, 163, 238 analytic continuation of,239 I Image charge, 210 Image point, 210 lngham, A.E.,38 Integral Bessel function, 142 J Jackson, D.,44 Jacobi polynomials, 44,96-97 Jaeger, J.C.,25,26 Jahnke, E.,24,301 Jeffreys, B.S.,6,139,301 Jeffreys, H.,6,139, 301 Johnson, D.C.,27 Joos, G.,39,40,155 K Kampé deFériet, J.,301 Kan, V.L.,50 Kestelman, H.,6 Klein, F.,301 Knopp, K.,11 Korous, I.,73,88 Koshlyakov, N.S.,94 Kummer, E.E.,255 L Laguerre, E.N.,76 Laguerre polynomials, 76-93, 273 asymptotic representation of,85-87 expansion offunctions inseries of,88 examples of,88-91 generating function of,77 generating function ofproducts of,78 integral equation satisﬁed by,82-83 integral representation of,80-81 orthogonality of,83-84 recurrence relations and differential equation for,78-80 relation toHermite polynomials, 81 Langer, R.E.,120,136 Laplace transform, 25 Laplace’s equation, solution of: inbipolar coordinates, 230-234 incylindrical coordinates, 143-145 inparabolic coordinates, 281-283 inparaboloidal coordinates, 299 inspherical coordinates, 205-206 inspheroidal coordinates, 213-215 intoroidal coordinates, 221-224 Laplace’s integral, 48 Lebedev, A.V.,300,301 Lebedev, N.N.,131, 221 Legendre functions: associated (see Associated Legendre functions) asymptotic representations of,189-192 behavior near :1,201 integral representations of,171-174 ofhalf-integral degree, 186-188 ofnonnegative integral degree, 184- 185 oftheﬁrstkind, 165 ofthesecond kind, 165 recurrence relations for,183-184 relations satisﬁed by,174-175 series representations of,176-181 Legendre polynomials, 44-60, 184-185, 260 asymptotic representation of,51-53 expansion offunctions inseries of,53- 58 examples of,58-60 generating function of,45 integral representations of,48-49INDEX 307 Legendre polynomials (Cont.): orthogonality of,50-51 recurrence relations and differential equation for,46-48 Legendre’s equation, 164 Wronskians ofpairs ofsolutions of, 181-183 Lense,J.,161,192, 301 Logarithmic integral, 37-38, 273 asymptotic representation of,38 modiﬁed, 38 relation toexponential integral, 38 Ltisch, F.,24,301 Luke, Y.L.,301 M Macdonald’s function, 14,109 recurrence relations for,110 series expansion of,110 MacRobert, T.M.,301 Magnus, W.,25,44,300, 302 Markushevich, A.I.,2,3,16,35,36,45 Mathews, G.B.,301 McLachlan, N.S.,302 Mehler-Dirichlet formula, 49 Mehler-Fock theorem, 221 Miller, J.C.P.,301 Mixed boundary value problem, 146, 208, 218 Morse, P.M.,302 N Neumann problem, 146,208,218 Nielsen, N.,302 Norlund, N.E.,276 Norrpal (Gaussian) random variable, 23 mean of,23 standard deviation of,23 O Oberhettinger, F.,25,44,132, 302 Orthogonal functions, 43 Orthogonal polynomials, 43-97 P Parabolic coordinates, 281 Parabolic cylinder functions, 281-299 Paraboloidal coordinates, 299 Petiau, G.,302 Poisson’s equation, 219 308 mnnx Probability integral, 16ff. asymptotic representation of,18-19 ofargument \/ix, 21-23 ofimaginary argument, 19-21 Probable error, 24 R Radiation conditions, 157 Radiation ofalinear half-wave oscillator, 39-40 Rainville, E.D.,302 Ramo, S.,91,92 Reﬂection from theendofatransmis- sionline,91-93 Region, 2 closed, 2 open, 2 Relton, F.E.,302 ReyPastor, I.,302 Riemann, B.,255 Robin, L.,161,302 Rodrigues' formula, 44 Rosenhead, L.,301 Rosser, J.B.,302 Ryshik, I.M.,302 S Samarski, A.A.,146,157,208,210,303 Sansone, G.,44,68,302 Schoblik, F.,301 Schrodinger equation, 297 Schwarz’s inequality, 54 Separation constants, 145,206 Shilov, G.E.,45 Shohat, J.A.,302 Silverman, R.A.,2,25,45,158,303 Sineintegral, 33,273 asymptotic representation of,37 relation toexponential integral, 36 Slater, L.J.,302 Smimov, V.I.,302 Smith, T.L.,208,218,303 Sneddon, I.N.,302 Snow, C.,302 Sommerfeld. A..302 Sonine, N.Y.,44,76 Spherical harmonics: applications of,204-237 theory of,161-203 Spheroidal coordinates, 213-215 oblate, 214 prolate, 213Steklov, V.A.,44,51 Sternberg, W.J.,208,218,303 Stirling’s formula, 12 Straus, E.G.,302 Surface temperature, 26 Superposition method, 148ff. Szego, G.,44,68,88,94,97,303 T Tikhonov, A.N.,146,157,208,210,303 Titchmarsh, E.C.,1,2,4,6,ll,63,82, 102,129,166,211,240 Tolstov, G.P.,25,54,55,57,95,129, 130,151,158, 303 Toroidal coordinates, 222 Transverse vibrations ofarod,26-28 Tricomi, F.G.,25,44,300,303 Truesdell, C.,303 U Uspensky, J.V.,44,88 V VonMises, R.,146,301 W Walsh, J.L.,302 Watson, G.N.,12,98,114,120,122,123, 125,126,127,129,130,132,139, 140,142,189,201,208,240,255, 274,277,279,280,303 Wave propagation along atransmission line,91-93 Weber’s integral, 132 Weierstrass’ theorem, 2,47 Weight, 43 Wermer, J.,302 Weyrich, R.,303 Whinnery, J.R.,91,92 Whittaker, E.T.,208,240,255,274,277, 279,280,303 Whittaker functions, 274,279 Whittaker’s equation, 279 Widder, D.V.,1,6,21,192 Wronskian, 112 Y Young, R.C.I-I.,ll