Chapter 12 Bessel Functions 12.1 BesselFunctionsoftheFirstKind, J (x) νν Besselfunctionsappearinawidevarietyofphysicalproblems.Whenonean- alyzes the sound vibrations of a drum, the partial differential wave equation (PDE)issolvedincylindricalcoordinates.Byseparatingtheradialandangu- larvariables, R(r)einϕ, oneisledtotheBesselordinarydifferentialequation (ODE) for R(r)involving the integer nas a parameter (see Example 12.1.4). TheWentzel-Kramers-Brioullin(WKB)approximationinquantummechanics involves Bessel functions. A spherically symmetric square well potential in quantummechanicsissolvedbysphericalBesselfunctions.Also,theextrac- tionofphaseshiftsfromatomicandnuclearscatteringdatarequiresspherical Besselfunctions.InSections8.5and8.6seriessolutionstoBessel’sequation weredeveloped.InSection8.9wehaveseenthattheLaplaceequationincylin- dricalcoordinatesalsoleadstoaformofBessel’sequation.Besselfunctions also appear in integral form—integral representations. This may result from integraltransforms(Chapter15). Besselfunctionsandcloselyrelatedfunctionsformarichareaofmathe- maticalanalysiswithmanyrepresentations,manyinterestingandusefulprop- erties,andmanyinterrelations.Someofthemajorinterrelationsaredeveloped inSection12.1andinsucceedingsections.NotethatBesselfunctionsarenot restrictedtoChapter12.TheasymptoticformsaredevelopedinSection7.3as wellasinSection12.3,andtheseriessolutionsarediscussedinSections8.5 and8.6. 589 590 Chapter12 BesselFunctions BiographicalData Bessel, Friedrich Wilhelm. Bessel, a German astronomer, was born in Minden, Prussia, in 1784 and died in Ko¨nigsberg, Prussia (now Russia) in 1846.Attheageof20,herecalculatedtheorbitofHalley’scomet,impressing thewell-knownastronomerOlberssufficientlytosupporthimin1806fora postatanobservatory.Therehedevelopedthefunctionsnamedafterhim inrefinementsofastronomicalcalculations.Thefirstparallaxmeasurement ofastar,61Cygniabout6light-yearsawayfromEarth,duetohimin1838, proveddefinitivelythatEarthwasmovinginaccordwithCopernicantheory. His calculations of irregularities in the orbit of Uranus paved the way for the later discovery of Neptune by Leverrier and J. C. Adams, a triumph of Newton’stheoryofgravity. GeneratingFunctionforIntegralOrder Although Bessel functions Jν(x) are of interest primarily as solutions of Bessel’sdifferentialequation,Eq.(8.62), x2d2Jν +xdJν +(x2−ν2)Jν =0, dx2 dx itisinstructiveandconvenienttodevelopthemfromageneratingfunction,just asforLegendrepolynomialsinChapter11.1Thisapproachhastheadvantages offindingrecurrencerelations,specialvalues,andnormalizationintegralsand focusingonthefunctionsthemselvesratherthanonthedifferentialequation theysatisfy.Sincethereisnophysicalapplicationthatprovidesthegenerating functioninclosedform,suchastheelectrostaticpotentialforLegendrepoly- nomialsinChapter11,wehavetofinditfromasuitabledifferentialequation. WethereforestartbyderivingfromBessel’sseries[Eq.(8.70)]forinteger indexν =n, (cid:3) (cid:4) (cid:2)∞ (−1)s x 2s+n J (x)= , n s!(s+n)! 2 s=0 thatconvergesabsolutelyforallx,therecursionrelation d (x−nJn(x))=−x−nJn+1(x). (12.1) dx Thiscanalsobewrittenas n xJn(x)− Jn+1(x)= Jn(cid:6)(x). (12.2) Toshowthis,wereplacethesummationindexs→s−1intheBesselfunction series[Eq.(8.70)]for Jn+1(x), (cid:3) (cid:4) (cid:2)∞ (−1)s x 2s+n+1 Jn+1(x)= s!(s+n+1)! 2 , (12.3) s=0 1GeneratingfunctionswerealsousedinChapter5.InSection5.6,thegeneratingfunction(1+x)n definesthebinomialcoefficients;x/(ex−1)generatestheBernoullinumbersinthesamesense. 12.1 BesselFunctionsoftheFirstKind, J (x) 591 νν inordertochangethedenominator(s+n+1)!to(s+n)!.Thus,weobtainthe series (cid:3) (cid:4) 1 (cid:2)∞ (−1)s2s x n+2s Jn+1(x)=−x s!(s+n)! 2 , (12.4) s=0 whichisalmosttheseriesfor J (x), exceptforthefactors.Ifwedividebyxn n anddifferentiate,thisfactorsisproducedsothatwegetfromEq.(12.4) (cid:3) (cid:4) d (cid:2) (−1)s x 2s d x−nJn+1(x)=−dx s!(s+n)! 2 2−n=−dx[x−nJn(x)], (12.5) n thatis,Eq.(12.1). AsimilarargumentforJn−1,withsummationindexsreplacedfirstbys−n andthenbys→s+1, yields d (xnJn(x))= xnJn−1(x), (12.6) dx whichcanbewrittenas n Jn−1(x)− xJn(x)= Jn(cid:6)(x). (12.7) Eliminating J(cid:6) fromEqs.(12.2)and(12.7),weobtaintherecurrence n 2n Jn−1(x)+ Jn+1(x)= Jn(x), (12.8) x whichwesubstituteintothegeneratingseries (cid:2)∞ g(x,t)= J (x)tn. (12.9) n n=−∞ ThisgivestheODEint(withxaparameter) (cid:3) (cid:4) (cid:2)∞ 1 2t∂g tn(Jn−1(x)+ Jn+1(x))= t+ t g(x,t)= x ∂t. (12.10) n=−∞ Writingitas (cid:3) (cid:4) 1∂g x 1 = 1+ , (12.11) g ∂t 2 t2 andintegratingweget (cid:3) (cid:4) x 1 lng = t− +lnc, 2 t which,whenexponentiated,leadsto g(x,t)=e(x/2)(t−1/t)c(x), (12.12) wherecistheintegrationconstantthatmaydependontheparameterx.Now taking x = 0 and using J (0) = δ (from Example 12.1.1) in Eq. (12.9) n n0 gives g(0,t) = 1 and c(0) = 1. To determine c(x) for all x, we expand the 592 Chapter12 BesselFunctions Figure12.1 BesselFunctions, J (x), 0 J1(x),and J2(x) 1.0 J0(x) J(x) 1 J (x) 2 x 1 2 3 4 5 6 7 8 9 exponential in Eq. (12.12). The 1/t term leads to a Laurent series (see Sec- tion6.5).Incidentally,weunderstandwhythesummationinEq.(12.9)hasto runovernegativeintegersaswell.SowehaveaproductofMaclaurinseries inxt/2and−x/2t, (cid:3) (cid:4) (cid:3) (cid:4) (cid:2)∞ x rtr (cid:2)∞ x st−s ext/2·e−x/2t = (−1)s . (12.13) 2 r! 2 s! r=0 s=0 Foragivenswegettn(n≥0)fromr =n+s (cid:3) (cid:4) (cid:3) (cid:4) x n+s tn+s x st−s (−1)s . (12.14) 2 (n+s)! 2 s! Thecoefficientoftnisthen2 (cid:3) (cid:4) (cid:2)∞ (−1)s x n+2s xn xn+2 J (x)= = − +··· (12.15) n s!(n+s)! 2 2nn! 2n+2(n+1)! s=0 sothatc(x)≡1forallxinEq.(12.12)bycomparingthecoefficientoft0, J (x), 0 withEq.(12.3)forn=−1.Thus,thegeneratingfunctionis (cid:2)∞ g(x,t)=e(x/2)(t−1/t) = J (x)tn. (12.16) n n=−∞ Thisseriesform,Eq.(12.15),exhibitsthebehavioroftheBesselfunctionJ (x) n forall x, convergingeverywhere,andpermitsnumericalevaluationof J (x). n TheresultsforJ , J ,andJ areshowninFig.12.1.FromSection5.3,theerror 0 1 2 inusingonlyafinitenumberoftermsinnumericalevaluationislessthanthe first term omitted. For instance, if we want J (x) to ±1% accuracy, the first n term alone of Eq. (12.15) will suffice, provided the ratio of the second term to the first is less than 1% (in magnitude) or x < 0.2(n+1)1/2. The Bessel 2Fromthestepsleadingtothisseriesandfromitsabsoluteconvergencepropertiesitshouldbe clearthatthisseriesconvergesabsolutely,withxreplacedbyzandwithzanypointinthefinite complexplane. 12.1 BesselFunctionsoftheFirstKind, J (x) 593 νν functionsoscillatebutarenot periodic;however,inthelimitasx→ ∞the zerosbecomeequidistant(Section12.3).TheamplitudeofJ (x)isnotconstant n butdecreasesasymptoticallyasx−1/2.[SeeEq.(12.106)forthisenvelope.] Equation(12.15)actuallyholdsforn<0,alsogiving (cid:3) (cid:4) (cid:2)∞ (−1)s x 2s−n J−n(x)= s!(s−n)! 2 , (12.17) s=0 whichamountstoreplacingnby−ninEq.(12.15).Sincenisaninteger(here), (s−n)! → ∞fors = 0,...,(n−1).Hence,theseriesmaybeconsideredto startwiths=n.Replacingsbys+n,weobtain (cid:3) (cid:4) (cid:2)∞ (−1)s+n x n+2s J−n(x)= s!(s+n)! 2 , s=0 showingthat Jn(x)and J−n(x)arenotindependentbutarerelatedby J−n(x)=(−1)nJn(x) (integraln). (12.18) Theseseriesexpressions[Eqs.(12.15)and(12.17)]maybeusedwithnreplaced byν todefine Jν(x)and J−ν(x)fornonintegralν (compareExercise12.1.11). EXAMPLE12.1.1 SpecialValues Settingx=0inEq.(12.12),usingtheseries[Eq.(12.9)]yields (cid:2)∞ 1= J (0)tn, n n=−∞ fromwhichweinfer(uniquenessofLaurentexpansion) J0(0)=1, Jn(0)=0= J−n(0), n≥1. Fromt =1wefindtheidentity (cid:2)∞ (cid:2)∞ 1= J (x)= J (x)+2 J (x) n 0 n n=−∞ n=1 usingthesymmetryrelation[Eq.(12.18)]. Finally,theidentityg(−x,t)=g(x,−t)implies (cid:2)∞ (cid:2)∞ J (−x)tn= J (x)(−t)n, n n n=−∞ n=−∞ andagaintheparityrelations J (−x)=(−1)nJ (x).Theseresultscanalsobe n n extractedfromtheidentityg(−x,1/t)=g(x,t). (cid:2) ApplicationsofRecurrenceRelations We have already derived the basic recurrence relations Eqs. (12.1), (12.2), (12.6), and (12.7) that led us to the generating function. Many more can be derivedasfollows. 594 Chapter12 BesselFunctions EXAMPLE12.1.2 AdditionTheorem Thelinearityofthegeneratingfunctionintheexponent xsuggeststheidentity g(u+v,t)=e(u+v)/2(t−1/t) =e(u/2)(t−1/t)e(v/2)(t−1/t) =g(u,t)g(v,t), whichimpliestheBesselexpansions (cid:2)∞ (cid:2)∞ (cid:2)∞ (cid:2)∞ J (u+v)tn = J(u)tl · J (v)tk = J(u)J (v)tl+k n l k l k n=−∞ l=−∞ k=−∞ k,l=−∞ (cid:2)∞ (cid:2)∞ = tm Jl(u)Jm−l(v) m=−∞ l=−∞ denotingm=k+l.Comparingcoefficientsyieldstheadditiontheorem (cid:2)∞ Jm(u+v)= Jl(u)Jm−l(v). (12.19) l=−∞ (cid:2) DifferentiatingEq.(12.16)partiallywithrespecttox,wehave (cid:3) (cid:4) ∂ 1 1 (cid:2)∞ g(x,t)= t− e(x/2)(t−1/t) = J(cid:6)(x)tn. (12.20) ∂x 2 t n n=−∞ Again,substitutinginEq.(12.16)andequatingthecoefficientsoflikepowers oft,weobtain Jn−1(x)− Jn+1(x)=2Jn(cid:6)(x), (12.21) whichcanalsobeobtainedbyaddingEqs.(12.2)and(12.7).Asaspecialcase ofthisrecurrencerelation, J(cid:6)(x)=−J (x). (12.22) 0 1 Bessel’s DifferentialEquation SupposeweconsiderasetoffunctionsZν(x)thatsatisfiesthebasicrecurrence relations [Eqs. (12.8) and (12.21)], but with ν not necessarily an integer and Zν not necessarily given by the series [Eq. (12.15)]. Equation (12.7) may be rewritten(n→ν)as xZν(cid:6)(x)= xZν−1(x)−νZν(x). (12.23) Ondifferentiatingwithrespecttox,wehave xZν(cid:6)(cid:6)(x)+(ν+1)Zν(cid:6) −xZν(cid:6)−1−Zν−1 =0. (12.24) MultiplyingbyxandthensubtractingEq.(12.23)multipliedbyν givesus x2Zν(cid:6)(cid:6)+xZν(cid:6) −ν2Zν +(ν−1)xZν−1−x2Zν(cid:6)−1 =0. (12.25) 12.1 BesselFunctionsoftheFirstKind, J (x) 595 νν NowwerewriteEq.(12.2)andreplacenbyν−1: xZν(cid:6)−1 =(ν−1)Zν−1−xZν. (12.26) UsingEq.(12.26)toeliminate Zν−1and Zν(cid:6)−1fromEq.(12.25),wefinallyget x2Zν(cid:6)(cid:6)+xZν(cid:6) +(x2−ν2)Zν =0. (12.27) ThisisBessel’sODE.Hence,anyfunctions,Zν(x),thatsatisfytherecurrence relations[Eqs.(12.2)and(12.7),(12.8)and(12.21),or(12.1)and(12.6)]satisfy Bessel’sequation;thatis,theunknown Zν areBesselfunctions.Inparticular, we have shown that the functions J (x), defined by our generating function, n satisfyBessel’sODE.Undertheparitytransformation,x→−x, Bessel’sODE staysinvariant,therebyrelating Zν(−x)to Zν(x), uptoaphasefactor.Ifthe argumentiskρratherthanx,whichisthecaseinmanyphysicsproblems,then Eq.(12.27)becomes d2 d ρ2dρ2Zν(kρ)+ρdρZν(kρ)+(k2ρ2−ν2)Zν(kρ)=0. (12.28) IntegralRepresentations AparticularlyusefulandpowerfulwayoftreatingBesselfunctionsemploys integralrepresentations.Ifwereturntothegeneratingfunction[Eq.(12.16)] andsubstitutet =eiθ,weget eixsinθ = J (x)+2[J (x)cos2θ + J (x)cos4θ +···] 0 2 4 +2i[J (x)sinθ + J (x)sin3θ +···], (12.29) 1 3 inwhichwehaveusedtherelations J1(x)eiθ + J−1(x)e−iθ = J1(x)(eiθ −e−iθ) =2iJ (x)sinθ, (12.30) 1 J2(x)e2iθ + J−2(x)e−2iθ =2J2(x)cos2θ, and so on. In summation notation, equating real and imaginary parts of Eq.(12.29),wehave (cid:2)∞ cos(xsinθ)= J (x)+2 J (x)cos(2nθ), 0 2n n=1 (cid:2)∞ sin(xsinθ)=2 J2n−1(x)sin[(2n−1)θ]. (12.31) n=1 Itmightbenotedthatangleθ (inradians)hasnodimensions,justasx.Like- wise,sinθ hasnodimensionsandthefunctioncos(xsinθ)isperfectlyproper fromadimensionalstandpoint. 596 Chapter12 BesselFunctions Ifnandmarepositiveintegers(zeroisexcluded),3 werecalltheorthog- onalitypropertiesofcosineandsine:4 (cid:5) π π cosnθcosmθdθ = δ , (12.32) nm 2 0 (cid:5) π π sinnθsinmθ dθ = δ . (12.33) nm 2 0 MultiplyingEq.(12.31)bycosnθ andsinnθ, respectively,andintegratingwe obtain (cid:5) (cid:6) 1 π J (x) n even, cos(xsinθ)cosnθdθ = n (12.34) π 0, n odd, 0 (cid:5) (cid:6) 1 π 0, n even, sin(xsinθ)sinnθdθ = (12.35) π J (x), n odd, 0 n upon employing the orthogonality relations Eqs. (12.32) and (12.33). If Eqs. (12.34)and(12.35)areaddedtogether,weobtain (cid:5) 1 π J (x)= [cos(xsinθ)cosnθ +sin(xsinθ)sinnθ]dθ n π (cid:5)0 1 π = cos(nθ −xsinθ)dθ, n=0,1,2,3,.... (12.36) π 0 Asaspecialcase, (cid:5) 1 π J (x)= cos(xsinθ)dθ. (12.37) 0 π 0 Noting that cos(xsinθ) repeats itself in all four quadrants (θ = θ,θ = 1 2 π −θ,θ = π +θ,θ = −θ),cos(xsinθ ) = cos(xsinθ), etc., we may write 3 4 2 Eq.(12.37)as (cid:5) 1 2π J (x)= cos(xsinθ)dθ. (12.38) 0 2π 0 On the other hand, sin(xsinθ) reverses its sign in the third and fourth quadrantssothat (cid:5) 1 2π sin(xsinθ)dθ =0. (12.39) 2π 0 AddingEq.(12.38)anditimesEq.(12.39),weobtainthecomplexexponential representation (cid:5) (cid:5) 1 2π 1 2π J (x)= eixsinθdθ = eixcosθdθ. (12.40) 0 2π 2π 0 0 3Equations(12.32)and(12.33)holdforeithermorn=0.Ifbothmandn=0,theconstantinEq. (12.32)becomesπ;theconstantinEq.(12.33)becomes0. 4Theyareeigenfunctionsofaself-adjointequation(oscillatorODEofclassicalmechanics)and satisfyappropriateboundaryconditions(compareSection9.2). 12.1 BesselFunctionsoftheFirstKind, J (x) 597 νν Figure12.2 Fraunhofer Incident waves y Diffraction-Circular Aperture r q x a Thisintegralrepresentation[Eq.(12.40)]maybeobtainedmoredirectlyby employingcontourintegration.5 Manyotherintegralrepresentationsexist. EXAMPLE12.1.3 Fraunhofer Diffraction, Circular Aperture In the theory of diffraction throughacircularapertureweencountertheintegral (cid:5) (cid:5) a 2π (cid:13)∼ eibrcosθdθrdr (12.41) 0 0 for(cid:13),theamplitudeofthediffractedwave.Here,theparameterb isdefined as 2π b= sinα, (12.42) λ where λ is the wavelength of the incident wave, α is the angle defined by a pointonascreenbelowthecircularaperturerelativetothenormalthrough thecenterpoint,6 andθ isanazimuthangleintheplaneofthecircularaper- tureofradiusa.TheothersymbolsaredefinedinFig.12.2.FromEq.(12.40) weget (cid:5) a (cid:13)∼2π J (br)rdr. (12.43) 0 0 5Forn=0asimpleintegrationoverθfrom0to2πwillconvertEq.(12.29)intoEq.(12.40). 6Theexponentibrcosθgivesthephaseofthewaveonthedistantscreenatangleαrelativetothe phaseofthewaveincidentontheapertureatthepoint(r,θ).Theimaginaryexponentialformof thisintegrandmeansthattheintegralistechnicallyaFouriertransform(Chapter15).Ingeneral, theFraunhoferdiffractionpatternisgivenbytheFouriertransformoftheaperture. 598 Chapter12 BesselFunctions Table12.1 Number ZerosoftheBessel ofZeros J0(x) J1(x) J2(x) J3(x) J4(x) J5(x) FunctionsandTheir 1 2.4048 3.8317 5.1356 6.3802 7.5883 8.7715 FirstDerivatives 2 5.5201 7.0156 8.4172 9.7610 11.0647 12.3386 3 8.6537 10.1735 11.6198 13.0152 14.3725 15.7002 4 11.7915 13.3237 14.7960 16.2235 17.6160 18.9801 5 14.9309 16.4706 17.9598 19.4094 20.8269 22.2178 J(cid:2)(x)a J(cid:2)(x) J(cid:2)(x) J(cid:2)(x) 0 1 2 3 1 3.8317 1.8412 3.0542 4.2012 2 7.0156 5.3314 6.7061 8.0152 3 10.1735 8.5363 9.9695 11.3459 aJ0(cid:6)(x)=−J1(x). Equation(12.6)enablesustointegrateEq.(12.43)immediatelytoobtain (cid:3) (cid:4) 2πab λa 2πa (cid:13)∼ J (ab)∼ J sinα . (12.44) b2 1 sinα 1 λ Theintensityofthelightinthediffractionpatternisproportionalto(cid:13)2and (cid:6) (cid:7) J [(2πa/λ)sinα] 2 (cid:13)2 ∼ 1 . (12.45) sinα FromTable12.1,whichlistssomezerosoftheBesselfunctionsandtheir firstderivatives,7 Eq.(12.45)willhaveitssmallestzeroat 2πa sinα =3.8317... (12.46) λ or 3.8317λ sinα = . (12.47) 2πa Forgreenlightλ=5.5×10−7m.Hence,ifa=0.5cm, α ≈sinα =6.7×10−5(radian) ≈14secofarc, (12.48) whichshowsthatthebendingorspreadingofthelightrayisextremelysmall, becausemostoftheintensityoflightisintheprincipalmaximum.Ifthisanaly- sishadbeenknowninthe17thcentury,theargumentsagainstthewavetheory oflightwouldhavecollapsed.Inthemid-20thcenturythissamediffractionpat- ternappearsinthescatteringofnuclearparticlesbyatomicnuclei—astriking demonstrationofthewavepropertiesofthenuclearparticles. (cid:2) 7AdditionalrootsoftheBesselfunctionsandtheirfirstderivativesmaybefoundinC.L.Beattie, Tableoffirst700zerosofBesselfunctions.BellSyst.Tech.J.37,689(1958),andBellMonogr. 3055.