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WuChong-shi ( ) Helmholtz 1rddrrdR(r)dr+hk2−λ−μr2iR(r)=0. k2−λ6=0 x=√k2−λr,y(x)=R(r) ! (ν#)Bessel 1xddxxdy(x)dx+1−ν2x2y(x)=0,$%μ=ν2&’()’2*%+ ,-.Bessel x=0/01!2& 3456789 + 0: &§22.1Bessel;=Neumann;Bessel?@ABCDEx=0Fx=∞Gx=0HIJCD x=∞HKIJCD&LIJCDx=0M NOρ=±ν&Pν6=QRS Bessel?TAB(UVWX)IJYHJ±ν(x)=∞Xk=0(−)kk!Γ(k±ν+1)x22k±ν.Z[ν=QRn JJn(x)FJ−n(x)UV\X J−n(x)=(−)nJn(x),]S Bessel?T^_Y‘HJn(x) ^aYJbcdNn(x)=limν→ncosνπJν(x)−J−ν(x)sinνπ=2πJn(x)lnx2−1πn−1Xk=0(n−k−1)!k!x22k−n−1π∞Xk=0(−)kk!(k+n)!ψ(n+k+1)+ψ(k+1)x22k+n,efgh Pn=0S ijklmno^apT@qF&WuChong-shi§22.1BesselrstNeumannrsu2vw22.1oxyz{|}d~RS BJn(x)Tw & 22.1Bessel BNn(x)Tw w22.2& 22.2Neumann RlmnHBessel RT lmn b yBessel RT_ V Z X ( _ )& Bessel RT Rlmn b T Z¡¢£⁄¥ƒ⁄£⁄§Bessel£⁄¤'¢¤¢“&WuChong-shi«‹›‹fiflrs( )u3v–22.1 Z∞0e−axJ0(bx)dx,Rea0.†‡·Bessel RT Rl e¶p &Z∞0e−axJ0(bx)dx=Z∞0e−ax∞Xk=0(−)k(k!)2bx22kdx=∞Xk=0(−)k(k!)2b22kZ∞0e−axx2kdx=∞Xk=0(−)k(k!)2b22k(2k)!a2k+1=1a∞Xk=01k!−12−32−52···−2k−12ba2k=1a1+ba2#−1/2=1√a2+b2.]•‚„T”DH R»F »FS ……‰@_hTq ¿ & ZL`´»FSˆ‰»|b/a|1&˜ˆ ¯˘˙ ¨ ˚¸ ˝T LRea0T˛ˇ— o_ ˘LRea0T˛ˇ Y G˘ yT [ L _ Y & Y T ˆb jk]Bq ¿ &WuChong-shi§22.2BesselrsÆ ª u4v§22.2Bessel; ŁØŒBessel RJ±ν(x)T X Hddx[xνJν(x)]=xνJν−1(x),ddxx−νJν(x)=−x−νJν+1(x).º ˚¸ddx[xνJν(x)]=xνJν−1(x).d Bessel RT RlmnJν(x)=∞Xk=0(−)kk!Γ(k+ν+1)x22k+νy & æ RL ´ b ¶pı &ddx[xνJν(x)]=ddx∞Xk=0(−)kk!Γ(k+ν+1)x2k+2ν22k+ν=∞Xk=0(−)kk!Γ(k+ν)x2k+2ν−122k+ν−1=xνJν−1(x). ddxx−νJν(x)=ddx∞Xk=0(−)kk!Γ(k+ν+1)x2k22k+ν=∞Xk=0(−)k+1k!Γ(k+ν+2)x2k+122k+ν+1=−x−νJν+1(x).()L]AB X ołjJν(x)øJ0ν(x) œb ß AB T X EJν−1(x)−Jν+1(x)=2J0ν(x),Jν−1(x)+Jν+1(x)=2νxJν(x). ] X b y ˛ˇQR TBessel R b J0(x)FJ1(x)l y˝& H Lddxx−νJν(x)=−x−νJν+1(x)o ν=0 ß J00(x)=−J1(x). Nν(x)Th Nν(x)=cosνπJν(x)−J−ν(x)sinνπ Jν(x)T X b yNν(x)T X nFJν(x) \ &ddx[xνNν(x)]=xνNν−1(x),WuChong-shi«‹›‹fiflrs( )u5vddxx−νNν(x)=−x−νNν+1(x). X ddx[xνCν(x)]=xνCν−1(x),ddxx−νCν(x)=−x−νCν+1(x)T R{Cν(x)} d R&b ˚¸E R_hHBessel?TY&Bessel RH^_ R Neumann RH^a R&Bessel R X T _ H Bessel RT & ‰ æ¡¢£⁄¥ £⁄§Bessel£⁄¤'¢T &–22.2 Z10 1−x2J0(μx)xdx oJ0(μ)=0&† X ddx[xνJν(x)]=xνJν−1(x). @Z10 1−x2J0(μx)xdx=Z10 1−x21μddx[xJ1(μx)]dx= 1−x21μ[xJ1(μx)] 10+2μZ10x2J1(μx)dx=2μ2x2J2(μx) 10=2μ2J2(μ). X Jν−1(x)+Jν+1(x)=2νxJν(x)oν=1 J0(x)+J2(x)=2xJ1(x),e J0(μ)=0 ˆ@J2(μ)=2μJ1(μ).‡· ßZ10 1−x2J0(μx)xdx=4μ3J1(μ).WuChong-shi§22.3BesselrsÆ u6v§22.3Bessel; Bessel RT!#$@A• &_•% æx→0 Jν(x)=1Γ(ν+1)x2ν+O xν+2.]b Bessel RT Rlmnß &&_•!#$% æx→∞ Jν(x)∼r2πxcosx−νπ2−π4,|argx|π.]B’nT ()‰ ˛ˇ Bessel RT l ‰ _• *T+,(-D„ ø ./ 0„)&12T b 3 45[3]T^76&L3 45[1]o xyzQR Bessel R!#$T˚¸&Px→0,Reν0S Nν(x)T!7d J−ν(x)8h Nν(x)∼−Γ(ν)πx2−ν.˘9æN0(x) b Nn(x)=2πJn(x)lnx2−1πn−1Xk=0(n−k−1)!k!x22k−n−1π∞Xk=0(−)kk!(k+n)!ψ(n+k+1)+ψ(k+1)x22k+n, ß N0(x)∼2πlnx2. :;νHdQR Nν(x)Lx=0D=H T& b ˚¸ Px→∞S Neumann RT!lmnHNν(x)∼r2πxsinx−νπ2−π4,|argx|π.WuChong-shi«‹›‹fiflrs( )u7v§22.4?@Bessel; AB;=CDEFBessel?1xddxxdy(x)dx+1−ν2x2y(x)=0oTν2≡μ ()H GHI¯Φ00+μΦ=0,Φ(0)=Φ(2π),Φ0(0)=Φ0(2π)8hT μ=m2,m=0,1,2,···& JKQR Bessel R @TV &1.Jn(x)¤LM£⁄NOP( ^7Q 7.4)expx2t−1t=∞Xn=−∞Jn(x)tn,0|t|∞.2.Jn(x)¤¢“RSJn(x)=1πZπ0cos(xsinθ−nθ)dθ.ºLBessel RTTU R#$no t=eiθ ˆß eixsinθ=∞Xn=−∞Jn(x)einθ.]ˆH ReixsinθTVWX#$n(YR n)&æH VWX#$T R’n ˆ ˚ßJn(x),=12πZπ−πeixsinθ einθ∗dθ=12πZπ−π[cos(xsinθ−nθ)+isin(xsinθ−nθ)]dθ.LZ[ T\ Ro ] HC R d0 ˆ˚ßJn(x)T l &Jn(x)T l b ˝ Bessel RT & Z9æ 22.1oT @Z∞0e−axJ0(bx)dx=Z∞0e−ax12πZπ−πeibxsinθdθdx=12πZπ−πdθZ∞0e−(a−ibsinθ)xdx=12πZπ−πdθa−ibsinθ. ^Rh ]B Z∞0e−axJ0(bx)dx=12πiI|z|=12dz−bz2+2az+b=1−bz+a z=(a−√a2+b2)/b=1√a2+b2.ˆ ¯˘˙ ]•‚„‰_‡·Bessel RT Rlmn‘¨ daLT bc¸d :e R»F‘f@+,V&]•‚„T&_BgMH:ihY &WuChong-shi§22.4isjBesselrsÆklrstmnopu8vZ[LBessel RTTU R#$nexpx2t−1t=∞Xn=−∞Jn(x)tn,0|t|∞.o t=ieiθ b ß eixcosθ=∞Xn=−∞Jn(x)ineinθ=J0(x)+2∞Xn=1inJn(x)cosnθ. H Z[ x=kr æHˆ@eikrcosθ=J0(kr)+2∞Xn=1inJn(kr)cosnθ.q`noTrFθ Yd rO oTrO|} efqk YdsR Sc\tTSu vde−iωt J`nA[= 9 æswx?\t vTyu Ez[H{Ix|}~ T ´s d T \t´Hkrcosθ−ωt=)R;˘Z[ poTJ0(kr)FJn(kr) TH ´s& ]B#$nTˇ ˆH ´s ´s#$&aL˝Y d Jν(kr) TH ´s& Bessel RT!#$Jν(x)∼r2πxcosx−νπ2−π4,|argx|π.(z)b y Pr S Jν(kr) Tswx?T\tˆHcoskr−νπ2−π4e−iωt=12exphikr−νπ2−π4−ωti+exph−ikr−νπ2−π4+ωti, \t´H ´kr−νπ2−π4∓ωt=)R, TH \t´ Su: ø T ø T ´s&˘f æ(z)no @ √rU _T v swx?T ˆ rU _ bH æ T ´ rUI_ tSu (x B ´ xT }:|&]ˆH (z)n T H_B: T ´s& Nν(x) b ˝ ¡ ´s H T ´sF T ´sT¢£& ⁄¥ƒ§¤'“«‹›fifl« –†‡· ¶•‹›fifl‚eiωt„”»…‰ ¿ §`x´ƒ§⁄¥«·