Outline1Êùμê()ÔnÆ2007cSC.S.Wu1Êùμê()OutlineùÇ:1Bessel¼êNeumann¼êBessel§Ä)4í'XìCÐm2êBessel¼ê5Bessel¼ê)¤¼êBessel¼êÈ©L«C.S.Wu1Êùμê()OutlineùÇ:1Bessel¼êNeumann¼êBessel§Ä)4í'XìCÐm2êBessel¼ê5Bessel¼ê)¤¼êBessel¼êÈ©L«C.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderReferencesÇÂÁ§5êÆÔn{6§§17.1—17.4ù&§5êÆÔn{6§§11.1,11.2nÎ!X1Á§5êÆÔn{6§§13.2C.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansionBessel¼êNeumann¼êC.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansionÚóHelmholtz§3ÎIXe©lCþ§1rddrrdR(r)dr+hk2−λ−μr2iR(r)=0ek2−λ6=0§Cx=√k2−λr,y(x)=R(r)§K§C(Ù¥μ=ν2)(ν)Bessel§1xddxxdy(x)dx+1−ν2x2y(x)=0ù9eù8¥?ØBessel§)9Ù5§±93©lCþ{¥A^C.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansionÚóHelmholtz§3ÎIXe©lCþ§1rddrrdR(r)dr+hk2−λ−μr2iR(r)=0ek2−λ6=0§Cx=√k2−λr,y(x)=R(r)§K§C(Ù¥μ=ν2)(ν)Bessel§1xddxxdy(x)dx+1−ν2x2y(x)=0ù9eù8¥?ØBessel§)9Ù5§±93©lCþ{¥A^C.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansionÚóHelmholtz§3ÎIXe©lCþ§1rddrrdR(r)dr+hk2−λ−μr2iR(r)=0ek2−λ6=0§Cx=√k2−λr,y(x)=R(r)§K§C(Ù¥μ=ν2)(ν)Bessel§1xddxxdy(x)dx+1−ν2x2y(x)=0ù9eù8¥?ØBessel§)9Ù5§±93©lCþ{¥A^C.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansionùÇ:1Bessel¼êNeumann¼êBessel§Ä)4í'XìCÐm2êBessel¼ê5Bessel¼ê)¤¼êBessel¼êÈ©L«C.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansion®k(JBessel§(½Reν≥0)1zddzzdw(z)dz+1−ν2z2w(z)=0Bessel§küÛ:µz=0(KÛ:)Úz=∞(KÛ:)3KÛ:z=0?§Iρ=±νν6=ê§Bessel§ü(5Ã')K)´J±ν(z)=∞Xk=0(−)kk!Γ(k±ν+1)z22k±νC.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansion®k(JBessel§(½Reν≥0)1zddzzdw(z)dz+1−ν2z2w(z)=0Bessel§küÛ:µz=0(KÛ:)Úz=∞(KÛ:)3KÛ:z=0?§Iρ=±νν6=ê§Bessel§ü(5Ã')K)´J±ν(z)=∞Xk=0(−)kk!Γ(k±ν+1)z22k±νC.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansion®k(JBessel§(½Reν≥0)1zddzzdw(z)dz+1−ν2z2w(z)=0Bessel§küÛ:µz=0(KÛ:)Úz=∞(KÛ:)3KÛ:z=0?§Iρ=±νν6=ê§Bessel§ü(5Ã')K)´J±ν(z)=∞Xk=0(−)kk!Γ(k±ν+1)z22k±νC.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansion®k(JBessel§(½Reν≥0)1zddzzdw(z)dz+1−ν2z2w(z)=0Bessel§küÛ:µz=0(KÛ:)Úz=∞(KÛ:)3KÛ:z=0?§Iρ=±νν6=ê§Bessel§ü(5Ã')K)´J±ν(z)=∞Xk=0(−)kk!Γ(k±ν+1)z22k±νC.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansion®k(JBessel§(½Reν≥0)1zddzzdw(z)dz+1−ν2z2w(z)=0Bessel§1)Jν(z)=∞Xk=0(−)kk!Γ(k+ν+1)z22k+ν1)J±ν(z)5|ÜNν(z)=cosνπJν(z)−J−ν(z)sinνπC.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansion®k(JBessel§(½Reν≥0)1zddzzdw(z)dz+1−ν2z2w(z)=0ν=ên§Jn(z)ÚJ−n(z)5'J−n(z)=(−)nJn(z)d§Bessel§1)E´Jn(z)§1)KNn(z)=limν→nNν(z)=limν→ncosνπJν(z)−J−ν(z)sinνπC.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansion®k(JBessel§(½Reν≥0)1zddzzdw(z)dz+1−ν2z2w(z)=0ν=ên§Jn(z)ÚJ−n(z)5'J−n(z)=(−)nJn(z)d§Bessel§1)E´Jn(z)§1)KNn(z)=limν→nNν(z)=limν→ncosνπJν(z)−J−ν(z)sinνπC.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansion®k(JBessel§(½Reν≥0)1zddzzdw(z)dz+1−ν2z2w(z)=0A^L’Hospital{K§Nn(z)=2πJn(z)lnz2−1πn−1Xk=0(n−k−1)!k!z22k−n−1π∞Xk=0(−)kk!(k+n)!ψ(n+k+1)+ψ(k+1)z22k+n¿ ½§n=0§IKLª¥1kÚC.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansionBessel¼êã/5¿Bessel¼êã/A:oNCzª³x=0NC1:©ÙC.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansionBessel¼êã/5¿Bessel¼êã/A:oNCzª³x=0NC1:©ÙC.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelationsAsymptoticExpansionBessel¼êã/5¿Bessel¼êã/A:oNCzª³x=0NC1:©ÙC.S.Wu1Êùμê()Besselftns&NeumannftnsPropertiesofBesselftnswithIntegerOrderFundamentalSolutionstoBesselEquationRecurrenceRelation