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WuChong-shi ( )§23.1Bessel Bessel !# $%&’()*+,-(./012 3456789:;=:?@ABCDE56FGHIJKLMNOPQ RSTU7VWXYZ[\]^_‘abcdefgehij$GHSTVkKlm nTMLD?@ABCfopPQqRKrs]^tuvlm$UwVxyGH56zIJKLMNOPQ R{U7|}M_‘~^7i =$ *+,- $ ∂2u∂t2−c21r∂∂rr∂u∂r+1r2∂2u∂φ2=0,(23.1a)u r=0. ,u r=a=0,(23.1b)u φ=0=u φ=2π,∂u∂φ φ=0=∂u∂φ φ=2π.(23.1c) % (23.1b) (23.1c) ¡¢ω# £⁄ (23.1a).¥ƒ§u(r,φ,t)=v(r,φ)eiωt.(23.2)¤'§“« (23.1)‹ (23.1b) (23.1c) ›fik=ω/c fl⁄ 1r∂∂rr∂v∂r+1r2∂2v∂φ2+k2v=0,v r=0. v r=a=0,v φ=0=v φ=2π,∂v∂φ φ=0=∂v∂φ φ=2π. fiv(r,φ)=R(r)Φ(φ) –†‡ ⁄ · !# Φ00(φ)+m2Φ(φ)=0,(23.3a)Φ(0)=Φ(2π),Φ0(0)=Φ0(2π)(23.3b) 1rddrrdR(r)dr+k2−m2r2R(r)=0,(23.4a)R(0). ,R(a)=0.(23.4b)WuChong-shi§23.1Bessel ¶•‚„”»…‰2 !# (23.3)¿ `´ˆ ˜ ¯˘˙¨!#m2,m=0,1,2,3,···,! ˚¸Φm(φ)=cosmφ,sinmφ. fl !# (23.4) ˝˚m2 ¿˛ ˇk2 !# —%$ fl k2Za0R(r)R∗(r)rdr=m2Za0R(r)R∗(r)drr+Za0dR(r)drdR∗(r)drrdr, fl ).!#k20$ ˜ † x=kr y(x)=R(r) fl¤ (23.4a) ¸Bessel ˇ%⁄¨ §R(r)=CJm(kr)+DNm(kr).(23.5) (23.4b) % R(0). D=0 ˙ %R(a)=0 ⁄ Jm(ka)=0.(23.6)¤m Bessel ˚Jm(x) i ƒ ( Æ ª) μ(m)i i=1,2,3,··· !# (23.4)§ !#k2mi=μ(m)ia!2,i=1,2,3,···,(23.7a)! ˚Rmi(r)=Jm(kmir).(23.7b)˙ %⁄ *+,-(. ŁØ/0ωmi=μ(m)iac,(23.8)Œ μ(m)i m Bessel ˚Jm(x) i ƒ $ º %§˜ º . Jν(x)ƒ æ F ν−1 ¸ ˚ı Jν(x). ` ƒ ¨łøœß ˚ ¯ º$WuChong-shi ( )‰3 ZerosofthefunctionsJν(z)&Nν(z)1.RealzerosWhenνisreal,thefunctionsJν(z)&Nν(z)eachhaveaninfinitenumberofzeros,allofwhicharesimplewiththepossibleexceptionofz=0.Fornon-negativeνthesthpositivezerosofthesefunctionsaredenotedbyjν,sandnν,srespectively.sj0,sj1,sn0,sn1,s12.404833.831710.893582.1971425.520087.015593.957685.4296838.6537310.173477.068058.59601411.7915313.3236910.2223511.74915514.9309216.4706313.3611014.89744618.0710619.6158616.5009218.04340721.2116422.7600819.6413121.18807824.3524725.9036722.7820324.33194927.4934829.0468325.9229627.475291030.6346132.1896829.0640330.618292.McMahon’sexpansionsforlargezerosjν,s,nν,s∼β−μ−18β−4(μ−1)(7μ−31)3(8β)3−32(μ−1)(83μ2−982μ+3779)15(8β)5−64(μ−1)(6949μ3−153855μ2+1585743μ−6277237)105(8β)7−······,s ν,μ=4ν2,β=s+ν2−14π,forjν,ss+ν2−34π,fornν,s3.ComplexzerosofJν(z)Whenν≥−1thezerosofJν(z)areallreal.Ifν−1andνisnotanintegerthenumberofcomplexzerosofJν(z)istwicetheintegerpartof(−ν);iftheintegerpartof(−ν)isoddtwoofthesezeroslieontheimaginaryaxis.4.ComplexzerosofNν(z)WhenνisrealthepatternofthecomplexzerosofNν(z)dependsonthenon-integerpartofν.Attentionisconfinedheretothecaseν=n,apositiveintegerorzero.WuChong-shi§23.1Bessel ¶•‚„”»…‰4 ZerosofNn(z)Thefigure23.1showstheapproximatedistributionofthecomplexzerosofNn(z)intheregion|argz|≤π.Thefigureissymmetricalabouttherealaxis.Thetwocurvesontheleftextendtoinfinity,havingtheasymptotesImz=±12ln3=±0.54931......Thereareaninfinitenumberofzerosneareachofthesecurves.Thetwocurvesextendingfromz=−ntoz=nandboundinganeye-shapeddomainintersecttheimaginaryaxisatthepoints±i(na+b),whereFigure23.1ZerosofNn(z)a=qt20−1=0.66274......b=12q1−t−20ln2=0.19146......andt0=1.19968......isthepositiverootofcotht=t.Therearenzerosneareachofthesecurves.ComplexzerosofN0(z)ComplexzerosofN1(z)RealpartImaginarypartRealpartImaginarypart−2.403020.53988−0.502740.78624−5.519880.54718−3.833530.56236−8.653670.54841−7.015900.55339WuChong-shi ( )‰5 ¸ –†‡ ˘ º ⁄ ! ˚ $ ¸ fl ı⁄ ! ˚ $ ! ˚Rmi(r)=Jm(kmir) 1rddrrdJm(kmir)dr+k2mi−m2r2Jm(kmir)=0,(23.9a)Jm(0). ,Jm(kmia)=0.(23.9b) ı ˚R(r)=Jm(kr) 1rddrrdJm(kr)dr+k2−m2r2Jm(kr)=0,(23.10a)Jm(0). .(23.10b) ˙Œ k¸ ˚ fl .Jm(ka)=0$ rJm(kr) rJm(kmir) ! (23.9a) (23.10a) Jm(kr)ddrrdJm(kmir)dr+k2mi−m2r2rJm(kmir)Jm(kr)=0,Jm(kmir)ddrrdJm(kr)dr+k2−m2r2rJm(kmir)Jm(kr)=0,# › $%[0,a]º& ⁄ k2mi−k2Za0Jm(kmir)Jm(kr)rdr=rJm(kmir)dJm(kr)dr−Jm(kr)dJm(kmir)dr r=ar=0.« (23.9b) (23.10b) fl¤º æ’ ¸ k2mi−k2Za0Jm(kmir)Jm(kr)rdr=−kmiaJm(ka)J0m(kmia).(23.11)(ł¯· )*++,-.$ ++ k=kmj6=kmi$ ı .Jm(kmja)=0 /'(23.11)“01¸0$2 ˙kmj6=kmi flZa0Jm(kmir)Jm(kmjr)rdr=0,kmi6=kmj,(23.12)3¯˘˙ !#! ˚ $%[0,a]ºfl45r $6 ++ k=kmi ı(23.11)“·17¸0$(ł fl ¤(23.11)“·1 8flk2mi−k2 9 k→kmi ⁄ Za0J2m(kmir)rdr=−limk→kmikmiak2mi−k2Jm(ka)J0m(kmia)=a22[J0m(kmia)]2.(23.13) ! ˚Jm(kmir): $;’¤!# (23.9) r=a1´ (23.9b)=¸ ? @? A fl?B $C º flD @ ++E F1rddrrdR(r)dr+k2−m2r2R(r)=0,(23.14a)WuChong-shi§23.1Bessel ¶•‚„”»…‰6 R(0). ,αR0(a)+βR(a)=0.(23.14b);’α6=0,β=0 ? ;’α=0,β6=0 ? ;’α β7 ¸0 ¸ @? $ ˙Bessel ˚GHIJ KLM æ F;’ ˚f(r) $%[0,a]ºNO PL.. Æ Q! ˚Jm(kir)RS f(r)=∞Xi=1biJm(kir),(23.15)Œ Jm(kir) !# (23.14)§ ˇRS ˚¸bi=Za0f(r)Jm(kir)rdrZa0J2m(kir)rdr.(23.16) ⁄ T˚ $%[δ,a−δ](δ0)º UVW$ ˆ˝ XY[15], 17.33Z$[X A\.]^_RS)‘$WuChong-shi ( )‰7 a23.1*b cd$e. f*b gh¸a$i (ł˘[j b z 3¸*b $;’b k lmno¸0 pl¸u0f(r) q%b rlm † $st lmuuφ,z 3u=u(r,t)$¨ )§ ∂u∂t−κr∂∂rr∂u∂r=0,(23.17a)u r=0. ,u r=a=0,(23.17b)u t=0=u0f(r)(23.17c)v)$wxy z{ ')§ §u(r,t)=∞Xi=1ciJ0μiraexp−κμia2t,(23.18)Œ μi J0(x) i ƒ $« p .u(r,t) t=0