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WuChong-shi ( ) 1 !#$%( )§9.1&’()*+,-./01234567859:;785=?@AB7859CD=AB745EF685GH=AB7I59JKL:;MN85OPQRB7STGUVCWXYZ[\]^_‘abcd9.1efz0A:;d2wdz2+p(z)dwdz+q(z)w=0785GNLp(z)gq(z)hBi7jklP0|z−z0|RQG:;7m_n6opBAw1(z)=(z−z0)ρ1∞Xk=−∞ck(z−z0)k,w2(z)=gw1(z)ln(z−z0)+(z−z0)ρ2∞Xk=−∞dk(z−z0)k,qrρ1,ρ2gghAst9Fefρ1Eρ2AutGvg=0GNz05K:;7B745EF6859Fefρ1Eρ2CAutGEg6=0GN:;7BKwxytGz05KqI59z{|}~ G G ¡ 9¢£⁄¥ ƒ §G¤'“«G‹› fifl –†‡· †‡G¶•‚„ ”»… ‰ 9ef¿tBr1 `_´ˆ˜G¯?˘˙¨u ˚7ρxG¸ ¿tBr˝ ´ˆ˜Gw1(z)=(z−z0)ρ1∞Xk=0ck(z−z0)k,w2(z)=gw1(z)ln(z−z0)+(z−z0)ρ2∞Xk=0dk(z−z0)k.˛AGˇ— p ˙R t7 9 GH ‘Æρx9⁄ ª £()09 g6=0¥Gw2(z) ª ·w1(z) (Ł›Ø ‡)G¢Œº 9 g=0¥Gw2(z) ¡ § ŁØ ‡G ª 9:;85OPQm_n6opBhAMNB7 æ G ı 7‘a(CX)bWuChong-shi§9.1 łøœß 2 cd9.2:;d2wdz2+p(z)dwdz+q(z)w=0,L 785z07OP0|z−z0|R m_MNBw1(z)=(z−z0)ρ1∞Xk=0ck(z−z0)k,c06=0,(9.1)w2(z)=gw1(z)ln(z−z0)+(z−z0)ρ2∞Xk=0dk(z−z0)k,gEd06=0(9.2)7 Az05Ap(z)7C ^ 745G (z−z0)p(z)Lz05Bi q(z)7C 745G (z−z0)2q(z)Lz05Bi9 z0K:;7MN859ρ1gρ2 KMNB7 9 9.1 Gz=0gz=1hA :;z(1−z)d2wdz2+[γ−(1+α+β)z]dwdz−αβw=07MN85 x=±1@hALegendre:; 1−x2d2ydx2−2xdydx+l(l+1)y=07MN859K o 5A KMN85G J z=1/tGeft=0A 7:;7MN85G t=05A 7:;785Gvt2t−1t2p1t=2−1tp1tgt2·1t4q1t=1t2q1tLt=05BiG z=∞5A :;785Gvzp(z)gz2q(z)Lz=∞5BiGN z=∞5A 7:;7MN859 ˙Go 5z=∞@hA :;gLegendre:;7MN859WuChong-shi ( ) 3 &’()*+,-. 0 b• MNBw1(z)Ew2(z) :;• ! tGRÆ#$g p •%&RÆ t7 ’(7RB ;G˘A) w1(z)k 7B :;GFef=*?R m_n6opBG +,-./0G˝ U w2(z)k 7B :;9Fef¯?1=R ^_B(1eρ1=ρ2?)G23G H U w2(z)k 7B(¯?7g^‘CK0) :;RB94Ts5æ:;7 a3G6˛^_ n6s5æ:;d2wdz2+p(z)dwdz+q(z)w=0,ef78RÆ ^_Bw1(z)G23G˘˙ 9æw2(z)=Aw1(z)Zz1[w1(z)]2exp−Zzp(ζ)dζdz:RÆ; B9¯AK¯m_Bh=:;d2w1dz2+p(z)dw1dz+q(z)w1=0,d2w2dz2+p(z)dw2dz+q(z)w2=0. w2(z)gw1(z)æ?@¯m_:;GU AG- Bw1d2w2dz2−w2d2w1dz2+p(z)w1dw2dz−w2dw1dz=0, ddzw1dw2dz−w2dw1dz+p(z)w1dw2dz−w2dw1dz=0.9æG w1dw2dz−w2dw1dz=Aexp−Zzp(ζ)dζ.mCD˙w21GE˙ Bddzw2w1=Aw21exp−Zzp(ζ)dζ.(9.3)U9æ^VG BF 7Gf9WuChong-shi§9.1 łøœß 4 9.2RLegendre:; 1−x2d2ydx2−2xdydx+l(l+1)y=0Lx=1OPQ7 HB90x=1ALegendre:;7MN85GI˚Jy(x)=(x−1)ρ∞Xn=0cn(x−1)n. :;G ∞Xn=0cn(n+ρ)(n+ρ+1)−l(l+1)(x−1)n+1+2∞Xn=0cn(n+ρ)2(x−1)n=0.KL˙ B#$:;ρ(ρ−1)+ρ=0g p cn=−n(n−1)−l(l+1)2n2cn−1.#$:;7BAρ1=ρ2=0.⁄“MLegendre {x=1NOPQ R¤ ST U{VP|x−1|2Q W G {x=1N›X ŒRY Z¤[Ł›Ø ‡G\x=1(·x=−1)£]NG¢Œ{x=1(·x=−1)N^_9‘aº R¤ 9K p G˙RÆLegendre:;Lx=15OPQ;^B7 t7˜b cn=(l+n)(l+1−n)2n2cn−1=(l+n)(l+1−n)2n2(l+n−1)(l+2−n)2(n−1)2cn−2=······=(l+n)(l+1−n)2n2(l+n−1)(l+2−n)2(n−1)2···(l+1)l2·12c0=1(n!)2Γ(l+n+1)Γ(l−n+1)12nc0.cc0=1G RÆ Legendre:;7;^BPl(x)=∞Xn=01(n!)2Γ(l+n+1)Γ(l−n+1)x−12n, KlV;^dLegendreyt9ef efR; BGN˚Jy2(x)=gPl(x)ln(x−1)+∞Xn=0dn(x−1)nWuChong-shi ( ) 5 =g∞Xn=01(n!)2Γ(l+n+1)Γ(l−n+1)x−12nln(x−1)+∞Xn=0dn(x−1)n.ghs5æ:;¿tBi7$jklG‘Æ tg(^‘CK0)gdn 9m’ 7niA4T; Bo;^Bpq7p GrÆy2(x)=gPl(x)Zx(1[Pl(ξ)]2expZξ2ζ1−ζ2dζ#)dξ=gPl(x)Zx1[Pl(ξ)]2dξ1−ξ2=gPl(x)Zxdξ1−ξ2+gPl(x)Zx1[Pl(ξ)]2−1dξ1−ξ2,st uC; ˜L|x−1|2QBiGL˙ ; BJKy2(x)=g2Pl(x)lnx+1x−1+∞Xn=0dn(x−1)n.cg=1Gv‘ÆdnGw ˙RÆLegendre:;7; BQl(x)=12Pl(x)lnx+1x−1−2γ−2ψ(l+1)+∞Xn=01(n!)2Γ(l+n+1)Γ(l−n+1)1+12+···+1nx−12n, KlV; dLegendreytGqrγAEulertGψ(z)AΓyt76t5x9K˛ytPl(x)(yzB{| G A˙x=−1gx=∞KI57wxyt)gQl(x)7wx67 }‘67~‘G¸ ? ? 9WuChong-shi§9.1 łøœß 6 ı ˘G^ıRs5æ:;d2wdz2+p(z)dwdz+q(z)w=0LMN85OPQ7B7^ klG? bL 3 ıG:;7; BC 6t˜ L 3 ıG:;7; B= 6t˜ L 3 ıG:;7; B^‘ 6t˜9 ~‘:;LMN85 7m_#$Reρ1≥Reρ2GN ρ1−ρ26=ut?G; B^‘C 6t˜ ρ1=ρ2?G; B^‘ 6t˜ ρ1−ρ2=Mut?G; B= 6t˜9K GC Jz=05A 7MN859˛AGLz=057OPQG :;7 tJLaurent p(z)=∞Xl=0alzl−1,q(z)=∞Xl=0blzl−2.JBKw(z)=zρ∞Xk=0ckzk. :;G ∞Xk=0ck(k+ρ)(k+ρ−1)zk+ρ−2+∞Xl=0alzl−1∞Xk=0ck(k+ρ)zk+ρ−1+∞Xl=0blzl−2∞Xk=0ckzk+ρ=0, ∞Xk=0ck(k+ρ)(k+ρ−1)zk+ρ−2+∞Xk=0kXl=0al(k+ρ−l)+blck−lzk=0.! mCw VˆG z07 tG c0[ρ(ρ−1)+a0ρ+b0]=0.K˛c06=0G ˙ρ(ρ−1)+a0ρ+b0=0.¯ A#$:;G qr7a0gb0Ka0=limz→0zp(z),b0=limz→0z2q(z).4T#$:;˙RÆm_#$Gρ1gρ29 [Reρ1≥Reρ29WuChong-shi ( ) 7 U!zn7 tG (n+ρ)(n+ρ−1)cn+nXl=0al(n+ρ−l)+blcn−l=0, (n+ρ)(n+ρ−1)+a0(n+ρ)+b0cn+nXl=1al(n+ρ−l)+blcn−l=0.¯ - Æ tpq7 p 9¶•‚„ G \ cn ¡ 9 G{cn ¡ §¤[Ł›ρ9„ρ=ρ1 G w1(z)9 „ρ=ρ2 G w2(z)9 ρ1−ρ26= ¥G ( )¡ 9 ρ1=ρ2?G ¯ 1= B^_B9 ˙G¯?; B^‘ 6t˜9 ρ1−ρ2=Mutm?G6˛; B7 tc(2)mG (m+ρ2)(m+ρ2−1)+a0(m+ρ2)+b0c(2)m+mXl=1al(m+ρ2−l)+blc(2)m−l=0. m+ρ2=ρ1G ˙ 0·c(2)m+mXl=1hal(ρ1−l)+blic(2)m−l=0.L mXl=1hal(ρ1−l)+blic(2)m−l6=0?Gc(2)moB mXl=1hal(ρ1−l)+blic(2)m−l=0?Gc(2)m+ 9F6˛;^¢ kG:;7; B@^‘ 6t˜9F6˛; ¢ kG:;7; B^‘C 6t˜G H=efRB91A¯?˙ 7£˜ tc(2)n(nm)⁄?S¥˛c0(2)gc(2)m9; Bw2(z)- m˜G^˜M!˛c(2)0G^˜M!˛c(2)m9Uƒ§æi^ıG ⁄¤'Gc(2)m+ngc(2)mpq7p oc(1)ngc(1)0pq7p /{^ GLGoc(2)m0M!7˜M“ A;^B(ww=«^_st‹t)G&C cc(2)m=09WuChong-shi§9.2Bessel 8 §9.2Bessel›fifl Bessel:;d2wdz2+1zdwdz+1−ν2z2w=0As 7s5æ:;p^GqrνAstGReν≥09st Gz=0A:;7MN85Gz=∞A:;7–MN859F†23Bessel:;Lz=057OP|z|0Q7B9Jw(z)=zρ∞Xk=0ckzk,c06=0, Bessel:;G ∞Xk=0ck(k+ρ)(k+ρ−1)zk+ρ−2+∞Xk=0ck(k+ρ)zk+ρ−2+∞Xk=0ckzk+ρ−ν2∞Xk=0ckzk+ρ−2=0,}‡zρ−2G ∞Xk=0ck(k+ρ)2−ν2zk+∞Xk=0ckzk+2=0.4T¿t 7·^6G ! t9Kw Vˆz0˜7 tGvKc06=0G B &’Gρ