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WuChong-shi ( ) 1 !#$%&(’)§8.1()*+,-./01,234256789:;=?@ABCDd2wdz2+p(z)dwdz+q(z)w=0,(8.1)p(z)Eq(z)FGHIJKLM•HIJNOPQRHIJKLSTJM•UVOWHINJNXYOPQRHIKLJNXYSTJMZ[\]^]_‘abcdWefg]hijklmnz0gopqrsgtu[\Mbcv\p(z),q(z)wz0ng]xyz{m|[\]wz0ng]xyW}~ Wz{m|[\]g W WiTaylor[\ iLaurent[\M• p(z),q(z) z0 NXW z0 FGHIJ M• p(z),q(z) z0 NXW z0 FGHIJ M 8.1 HI(Hypergeometricequation)z(1−z)d2wdz2+γ−(1+α+β)zdwdz−αβw=0JKLOp(z)=γ−(1+α+β)zz(1−z)Eq(z)=−αβz(1−z). Wp(z)Eq(z) z=0Ez=1M W z=0Ez=1O HIJ W J¡¢ £OHIJ M 8.2LegendreHI 1−x2d2ydx2−2xdydx+l(l+1)y=0, J Gx=±1MWuChong-shi§8.1 ⁄¥ƒ⁄ 2 §¤'“« z=∞O OHI(8.1)J W ‹›fifl –J †z=1/tMdwdz=−t2dwdt,d2wdz2=t4d2wdt2+2t3dwdt.‡·WHI(8.1) Gd2wdt2+2t−1t2p1tdwdt+1t4q1tw=0.(8.2) t=0OHI(8.2)J ( )W F“« z=∞OHI(8.1)J ( )Mt=0( z=∞)GHI J¶•Op1t=2t+a2t2+a3t3+···,q1t=b4t4+b5t5+···, p(z)=2z+a2z2+a3z3+···,q(z)=b4z4+b5z5+···.“« O HIELegendreHIJ MWuChong-shi ( ) 3 §8.2?;‚„”»@…‰ W ¿ `´ˆ˜¯˘JT˙M¨ 8.1 p(z)Eq(z) ˚|z−z0|R¸ ˝NXW ·˚¸ ˛ˇHI—˝ d2wdz2+p(z)dwdz+q(z)w=0,w(z0)=c0,w0(z0)=c1(c0,c1G L) J Nw(z)W w(z) ˚¸ ˝NXM T˙W w(z) z0 J |z−z0|R¸ GTaylorÆLw(z)=∞Xk=0ck(z−z0)k. ªW (z−z0)0 (z−z0)1JKLc0 c1 E—˝¶• ŁMØ ŒºJÆLN ˛ˇHIW KLW æ KLckMT˙ `WKLck(k=2,3,···) ıc0,c1 M 8.3æLegendreHI 1−x2d2ydx2−2xdydx+l(l+1)y=0 x=0 ¸JNW¡ lO łLM…x=0OHIJ W‡·W øNy=∞Xk=0ckxk. HIW 1−x2∞Xk=0ckk(k−1)xk−2−2x∞Xk=0ckkxk−1+l(l+1)∞Xk=0ckxk=0,œ˙ß W ∞Xk=0n(k+2)(k+1)ck+2−k(k+1)−l(l+1)ckoxk=0. Taylor J YW (k+2)(k+1)ck+2−[k(k+1)−l(l+1)]ck=0, ck+2=k(k+1)−l(l+1)(k+2)(k+1)ck=(k−l)(k+l+1)(k+2)(k+1)ck. KL J M ı KW æ KLWuChong-shi§8.2 ⁄ 4 c2n=(2n−l−2)(2n+l−1)2n(2n−1)c2n−2=(2n−l−2)(2n−l−4)(2n+l−1)(2n+l−3)2n(2n−1)(2n−2)(2n−3)c2n−4=···=c0(2n)!(2n−l−2)(2n−l−4)···(−l)·(2n+l−1)(2n+l−3)···(l+1),c2n+1=(2n−l−1)(2n+l)(2n+1)(2n)c2n−1=(2n−l−1)(2n−l−3)(2n+l)(2n+l−2)(2n+1)(2n)(2n−1)(2n−2)c2n−3=···=c1(2n+1)!(2n−l−1)(2n−l−3)···(−l+1)·(2n+l)(2n+l−2)···(l+2). ıΓ LJY Γ(z+1)=zΓ(z),Γ(z+n+1)=(z+n)(z+n−1)···(z+1)zΓ(z), Øc2nEc2n+1 c2n=22n(2n)!Γn−l2Γ−l2Γn+l+12Γl+12c0,c2n+1=22n(2n+1)!Γn−l−12Γ−l−12Γn+1+l2Γ1+l2c1. WLegendreHIJN Oy(x)=c0y1(x)+c1y2(x),¡ y1(x)=∞Xn=022n(2n)!Γn−l2Γ−l2Γn+l+12Γl+12x2n,y2(x)=∞Xn=022n(2n+1)!Γn−l−12Γ−l−12Γn+1+l2Γ1+l2x2n+1.WuChong-shi ( ) 5 T˙ W T c0Ec1W T æ HIJ UNMUVOW• c0=1,c1=0W UNy1(x) • c0=0,c1=1W UNy2(x)M W ]y1(x) y2(x)i yt gM yt ]!Wz#$%bcg&]M• Nº J Lc0Ec1’ O (¿ LW)˘ J OHIJ*NM +…@,-8@./M)˘æ JUN Wy1(x)01 xJ234Wy2(x)01 xJ 34W y1(x)OxJ2 LWy2(x)OxJ LM5æNJ6I7’W OR8 K 0 9KLck+2EckW: ck+1“ W‡·c2nPQRc0STWc2n+1PQRc1STM5 ;)7 WHIJNJFY( =JO 2Y)Wª?@OHIJFYJ AM*6 BCW ’ ¸æÆLNJ DEFM O •Ø(HI ¸J)N GTaylorÆLW ˛ˇHI • KLW KL J K • ı KWæ KLckJGH Iº(ıc0Ec1 )W5:JK ÆLN R8 K TOLYJ(‡GHIOLYJ)W JKJÆLN T w(z)=c0w1(z)+c1w2(z)JŒºM KL J K W DMNO 9ck,ck+1,ck+2P Q JKLW‡·ckMNORS8c0Ec1WJKæ Jw1(z)Tw2(z) M01 zJ234T 34MWuChong-shi§8.2 ⁄ 6 ?ı ˛ˇHIJ4ÆLNUW HI TV ¸JNºMWXY Z§Wæ HI NV ¸JNºM `WHI NV ¸JNºW[GNX\]M‡·WY 5HI ^ V ¸JNº _W*6NX\]W HI ¡¢V ¸JNºM 8.4‘w1OHId2wdz2+p(z)dwdz+q(z)w=0(8.3)JNW V G1¸NXMaew1Ow1 V G2¸JNX\]W w1≡ew1,z∈G1TG2,(8.4)b ` ew1cOHI(8.3)JNMd‘d2ew1dz2+p(z)dew1dz+q(z)ew1=g(z),g(z) G2¸NXM‡Gw1OHI(8.3) V G1¸JNWe fV G1TG2¸WcghHId2w1dz2+p(z)dw1dz+q(z)w1=0.: ·fV ¸Ww1(z)≡ew1(z)Wed2ew1dz2+p(z)dew1dz+q(z)ew1=0,z∈G1TG2, g(z)≡0,z∈G1TG2M NX LJ YWi g(z)≡0,z∈G2,j ew1 G2¸ghHId2ew1dz2+p(z)dew1dz+q(z)ew1=0. 8.5‘w1Ew2£OHI(8.3)J LY“ NW V G1¸NXMaew1Eew2ˇVOw1Ew2 V G2¸JNX\]W z∈G1TG2 w1≡ew1,w2≡ew2.b ew1Eew2cLY“ MdRC8.4kWew1Eew2cOHI( G2¸)JNM‡Gw1Ew2LY“ WΔ[w1,w2]≡ w1w2w01w02 6=0,z∈G1.‘Δ[ew1,ew2]≡ ew1ew2ew01ew02 =g(z),g(z) G2¸NXMR8 z∈G1TG2 Ww1≡ew1,w2≡ew2,eg(z)6=0,z∈G1TG2Mcª NX LJ YW g(z)6=0,z∈G2. Wew1Eew2( G2¸)cLY“ M