Outline1ù~©§?ê){()ÔnÆêÆÔn{§|2007cSC.S.Wu1ù~©§?ê){()OutlineùÇ:1§KÛ:?)§Û:?)/ªKÛ:2¦)g´(ئ)g´Ú½(Ø3Bessel§)Bessel§Bessel§1)Bessel§1)C.S.Wu1ù~©§?ê){()OutlineùÇ:1§KÛ:?)§Û:?)/ªKÛ:2¦)g´(ئ)g´Ú½(Ø3Bessel§)Bessel§Bessel§1)Bessel§1)C.S.Wu1ù~©§?ê){()OutlineùÇ:1§KÛ:?)§Û:?)/ªKÛ:2¦)g´(ئ)g´Ú½(Ø3Bessel§)Bessel§Bessel§1)Bessel§1)C.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquationReferencesÇÂÁ§5êÆÔn{6§§6.3—6.4ù&§5êÆÔn{6§§9.3nÎ!X1Á§5êÆÔn{6§§8.2C.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquation§Û:?)?Ø4:5Û:§Û:UÓ´)Û:U´){:C.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquation§Û:?)?Ø4:5Û:§Û:UÓ´)Û:U´){:C.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquation§Û:?)?Ø4:5Û:§Û:UÓ´)Û:U´){:C.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquationSolutionsinVicinityofSingularityRegularSingularityùÇ:1§KÛ:?)§Û:?)/ªKÛ:2¦)g´(ئ)g´Ú½(Ø3Bessel§)Bessel§Bessel§1)Bessel§1)C.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquationSolutionsinVicinityofSingularityRegularSingularity½n(Øy)XJz0´§d2wdz2+p(z)dwdz+q(z)w=0Û:§K3p(z)Úq(z)Ñ)Û/«0|z−z0|RS§§ü5Ã')´w1(z)=(z−z0)ρ1∞Pk=−∞ck(z−z0)kw2(z)=gw1(z)ln(z−z0)+(z−z0)ρ2∞Pk=−∞dk(z−z0)kÙ¥ρ1,ρ2ÚgÑ´~êC.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquationSolutionsinVicinityofSingularityRegularSingularity§Û:S)w1(z)=(z−z0)ρ1∞Pk=−∞ck(z−z0)kw2(z)=gw1(z)ln(z−z0)+(z−z0)ρ2∞Pk=−∞dk(z−z0)kXJρ1½ρ2´ê§ g=0§Kz0:§)4:½5Û:XJρ1½ρ2شꧽg6=0§K§)õ¼ê§z0:Ù{:C.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquationSolutionsinVicinityofSingularityRegularSingularity§Û:S)w1(z)=(z−z0)ρ1∞Pk=−∞ck(z−z0)kw2(z)=gw1(z)ln(z−z0)+(z−z0)ρ2∞Pk=−∞dk(z−z0)kXJρ1½ρ2´ê§ g=0§Kz0:§)4:½5Û:XJρ1½ρ2شꧽg6=0§K§)õ¼ê§z0:Ù{:C.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquationSolutionsinVicinityofSingularityRegularSingularity§Û:S)w1(z)=(z−z0)ρ1∞Pk=−∞ck(z−z0)kw2(z)=gw1(z)ln(z−z0)+(z−z0)ρ2∞Pk=−∞dk(z−z0)kXJρ1½ρ2´ê§ g=0§Kz0:§)4:½5Û:XJρ1½ρ2شꧽg6=0§K§)õ¼ê§z0:Ù{:C.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquationSolutionsinVicinityofSingularityRegularSingularity§Û:S)w1(z)=(z−z0)ρ1∞Pk=−∞ck(z−z0)kw2(z)=gw1(z)ln(z−z0)+(z−z0)ρ2∞Pk=−∞dk(z−z0)ky3XJ·rþ¡)ª\§§¦+E,UXêm4í'X§%Ã{¦ÑXêÊHLª©Ïù?ê)¥§`5§ÑkáõÚK§·Ã{½Xê/Ð0C.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquationSolutionsinVicinityofSingularityRegularSingularity§Û:S)w1(z)=(z−z0)ρ1∞Pk=−∞ck(z−z0)kw2(z)=gw1(z)ln(z−z0)+(z−z0)ρ2∞Pk=−∞dk(z−z0)kXJ?ê)¥kkK§ùo±NAρ§¦?ê)¥vkKw1(z)=(z−z0)ρ1∞Pk=0ck(z−z0)kw2(z)=gw1(z)ln(z−z0)+(z−z0)ρ2∞Pk=0dk(z−z0)kC.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquationSolutionsinVicinityofSingularityRegularSingularityw1(z)=(z−z0)ρ1∞Pk=−0ck(z−z0)kw2(z)=gw1(z)ln(z−z0)+(z−z0)ρ2∞Pk=0dk(z−z0)kE|^4í'X=¦XêÊHLª¢3½XêUküù«/ª)¡K)g=0§w2(z)Lª¥Ø¹éê§ü)/ªÓg6=0§w2(z)/ªÚw1(z)ØÓ(¹kéê)§ÏI©O¦)C.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquationSolutionsinVicinityofSingularityRegularSingularityw1(z)=(z−z0)ρ1∞Pk=−0ck(z−z0)kw2(z)=gw1(z)ln(z−z0)+(z−z0)ρ2∞Pk=0dk(z−z0)kE|^4í'X=¦XêÊHLª¢3½XêUküù«/ª)¡K)g=0§w2(z)Lª¥Ø¹éê§ü)/ªÓg6=0§w2(z)/ªÚw1(z)ØÓ(¹kéê)§ÏI©O¦)C.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquationSolutionsinVicinityofSingularityRegularSingularityw1(z)=(z−z0)ρ1∞Pk=−0ck(z−z0)kw2(z)=gw1(z)ln(z−z0)+(z−z0)ρ2∞Pk=0dk(z−z0)kE|^4í'X=¦XêÊHLª¢3½XêUküù«/ª)¡K)g=0§w2(z)Lª¥Ø¹éê§ü)/ªÓg6=0§w2(z)/ªÚw1(z)ØÓ(¹kéê)§ÏI©O¦)C.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquationSolutionsinVicinityofSingularityRegularSingularityw1(z)=(z−z0)ρ1∞Pk=−0ck(z−z0)kw2(z)=gw1(z)ln(z−z0)+(z−z0)ρ2∞Pk=0dk(z−z0)kE|^4í'X=¦XêÊHLª¢3½XêUküù«/ª)¡K)g=0§w2(z)Lª¥Ø¹éê§ü)/ªÓg6=0§w2(z)/ªÚw1(z)ØÓ(¹kéê)§ÏI©O¦)C.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquationSolutionsinVicinityofSingularityRegularSingularityw1(z)=(z−z0)ρ1∞Pk=−0ck(z−z0)kw2(z)=gw1(z)ln(z−z0)+(z−z0)ρ2∞Pk=0dk(z−z0)kE|^4í'X=¦XêÊHLª¢3½XêUküù«/ª)¡K)g=0§w2(z)Lª¥Ø¹éê§ü)/ªÓg6=0§w2(z)/ªÚw1(z)ØÓ(¹kéê)§ÏI©O¦)C.S.Wu1ù~©§?ê){()SolutionsinVicinityofRegularSingularityOutlines&Conclusions)Example:BesselEquationSolutionsinVicinityofSingularityRegularSingularityw1(