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WuChong-shi ( )§20.1Legendre Legendre !r#$%&’()*+,-*+. )/0z123,45*+ (r0,θ,φ) * (678φ9:);01√r2+r02−2rr0cosθ=1r1√1−2xt+t2,t=r0r,1r01√1−2xt+t2,t=rr0, x=cosθ,=? @AB1/√1−2xt+t2 (@CD01√1−2xt+t2 t=0=1 4E ?F,AB1/√1−2xt+t2 t=0GHIJ KL ,MNOPQTaylorRS1√1−2xt+t2=∞Xl=0cltl,|t||x±px2−1|.FTUVRSWBclX Legendre Pl(x),;1√1−2xt+t2=∞Xl=0Pl(x)tl,|t||x±px2−1|.AB1/√1−2xt+t2;Y0Legendre Z[AB \]^_AB1/√1−2xt+t2 t=0QTaylorRS1√1−2xt+t2=1p1−2t+t2−2(x−1)t=11−t1−2(x−1)t(1−t)2−1/2=11−t∞Xk=01k!−12−32···12−k−2(x−1)t(1−t)2k=∞Xk=0(2k−1)!!k!(x−1)ktk(1−t)−(2k+1)=∞Xk=0(2k−1)!!k!(x−1)ktk∞Xn=0(2k+n)!n!(2k)!tn=∞Xl=0lXk=0(l+k)!k!k!(l−k)!x−12k#tl.‘aLegendre Z[AB,bOPcde %a fg hi,jx=1,Xcd1√1−2t+t2=11−t=∞Xl=0tl=∞Xl=0Pl(1)tl=⇒Pl(1)=1.ki,1√1−2xt+t2=1p1−2(−x)(−t)+(−t)2,∞Xl=0Pl(x)tl=∞Xl=0Pl(−x)(−t)l,=⇒Pl(−x)=(−)lPl(x).WuChong-shi§20.2Legendrelmnopqrst2u§20.2Legendre vwxyzLegendre Z[AB{|,}~ {H Legendre :W,;Legendre :W Legendre Z[AB1√1−2xt+t2=∞Xl=0Pl(x)tl, t ,%−12−2x+2t(1−2xt+t2)3/2=∞Xl=0lPl(x)tl−1,;x−t(1−2xt+t2)1/2= 1−2xt+t2∞Xl=0lPl(x)tl−1=(x−t)∞Xl=0Pl(x)tl. tl WB,%xPl(x)−Pl−1(x)=(l+1)Pl+1(x)−2xlPl(x)+(l−1)Pl−1(x), ;c(2l+1)xPl(x)=(l+1)Pl+1(x)+lPl−1(x).(z)4EXcdLegendre &’ :W, Legendre ¡¢‘a4’ :W,XOP£⁄¥ Legendre aƒ Legendre P0(x)=1§& Legendre P1(x)=x¤'{“ _Legendre Z[AB1√1−2xt+t2=∞Xl=0Pl(x)tl, x« ,k‹cd−12−2t(1−2xt+t2)3/2=∞Xl=0P0l(x)tl,› t∞Xl=0Pl(x)tl= 1−2xt+t2∞Xl=0P0l(x)tl. tl+1 WB,cPl(x)=P0l+1(x)−2xP0l(x)+P0l−1(x).(#)4’ :W ,{fi fl’H Legendre G B WuChong-shi –†‡· ¶(–)t3u£(z) x« ,•OPcd(2l+1)Pl(x)+(2l+1)xP0l(x)=(l+1)P0l+1(x)+lP0l−1(x),§(#) ‚„,”»P0l−1(x)…P0l+1(x),kOPcd :WP0l+1(x)=xP0l(x)+(l+1)Pl(x),P0l−1(x)=xP0l(x)−lPl(x).4 ’ :W,‰ £P0l±1(x)aPl(x)G B¤'{“ £4 :W¿ `´,•OP &ˆcd˜¯ :W :W &’a˘ ˙¨ ˚¸ C,hiZ1−1xPk(x)Pl(x)dx. :W(z),X‹˝˙¨{Z1−1xPk(x)Pl(x)dx=l+12l+1Z1−1Pk(x)Pl+1(x)dx+l2l+1Z1−1PkPl−1(x)dx=l+12l+122l+3δl+1,k+l2l+122l−1δl−1,k.WuChong-shi§20.3Legendrelmn˛ˇ— t4u§20.3Legendre 20.1 * * 0E0 * $ &’^ , 0a « Æ⁄¥& * $ ª, › * , TŁXد[&? ZT*+CŒ,Nº [0 F Æ⁄¥& * X !% * * § Z*+ * F ^ ,¥ æ * 00 FM0 Æ## %*+,.P Æ * Laplace2ı F a łW, ł!8 ø¿´,œ1ß!“* 23 F d * PG Y , TŁ Z*+&? œ1 ,MN, › Æ⁄¥&,9 Z*+ Z * ,… * ,b œ1 u(r,θ) Æ&(r,θ,φ) * ,u1(r,θ)§u2(r,θ)C * § Z*+ * ,u1(r,θ)=−E0z+u0=−E0rcosθ+u0, Bu0;0 ł!# * u2(r,θ)‰ ?K 1r2∂∂rr2∂u2∂r+1r2sinθ∂∂θsinθ∂u2∂θ=0,u2 θ=0% ,u2 θ=π% ,u2 r=a=E0acosθ−u0,u2 r→∞→0. ? u2(r,θ) .P Laplace2ı,z Ł , › Z*+ CŒ TŁ,N Æ## 9 Z*+ zB Ł ,M0u(r,θ)=u1(r,θ)+u2(r,θ)§( u1(r,θ) Laplace2ı E › Z*+ CŒ TŁ,.P r→∞5u2(r,θ) ›0 «K?K _2ı§% C ª,OPcd1sinθddθhsinθdΘ(θ)dθi+λΘ(θ)=0,Θ(0)% ,Θ(π)% ,ddrhr2dR(r)dri−λR(r)=0,WuChong-shi –†‡· ¶(–)t5u λ C 5 ? B 18 2 ! 4’#$@ ,K #$@λl=l(l+1),l=0,1,2,3,···,#$ABΘl(θ)=Pl(cosθ).0%«K:›R(r) 2ı,&7OPQ ’t=lnr,_2ı 0d2Rldt2+dRldt−l(l+1)Rl=0.› Rl(r)=Alelt+Ble−(l+1)t=Alrl+Blr−l−1.M(, Laplace2ı§% &)KX u2(r,θ)=∞Xl=0 Alrl+Blr−l−1Pl(cosθ). d9*+ u2 r→∞→0, ,%Al=0.-./ Tr=aŁ 0 ,u2(r,θ) r=a=∞Xl=0Bla−l−1Pl(cosθ)=E0acosθ−u0=E0aP1(cosθ)−u0P0(cosθ),.P%B0=−u0a,B1=E0a3,§Bl=0,l≥2.4EX«cu2(r,θ)=−u0ar+E0a3r2cosθ.12345u2(r,θ)6789:;=?@AB5CDEF GHIAJ5KLM,NO=5?@ABP6QRQSTUV5VABWAXYZ5[\ VAB5A]^−4πε0u0a_AXYZ5XY‘^4πε0E0a3,abcHIAJ5abPd _u1(r,θ)§u2(r,θ) ,Xcd Æ⁄¥& * eu(r,θ)=u01−ar−E01−a3r3rcosθ.f20.1g{%œ1 ⁄¥&’hTŁ* i CŒf WuChong-shi§20.3Legendrelmn˛ˇ— t6uf20.1 * 20.2 j*klm * %& klm, 0a, *+0M,« n ⁄¥& * &- łW, ł!$ mø,Nlm‰# opTŁ 45,n ⁄¥&(r,θ,φ) * ,8φ9:,u=u(r,θ) OPq{u. 2ı§rC?K 1r2∂∂rr2∂u∂r+1r2sinθ∂∂θsinθ∂u∂θ=0,(r,θ)6=a,π2,u θ=0% ,u θ=π% ,u r=0% ,u r→∞→0.1stuvwxy5z{|}(~ ,1wz{|}tu z5), ^ 9: @ A 5 (ABCD)5EF q{ klmŁ *+ , adδAB 2ıX 01r2∂∂rr2∂u∂r+1r2sinθ∂∂θsinθ∂u∂θ=−1ε0f(r)δ(r−a)δθ−π2,(#)ABf(r)OP ZZZf(r)δ(r−a)δθ−π2r2sinθdrdθdφ=1?{,N ›f(r)δ(r−a)=f(a)δ(r−a),.P%f(r)=f(a)=12πa2. δAB OP p, r6=a5,2ı(#)X 0Laplace2ı 4E,-f´% §9*+ ,Xcdu(r,θ)=∞Xl=0AlrlPl(cosθ),ra,∞Xl=0Blr−l−1Pl(cosθ),ra.7ªX ,‘alm CŒ(;2ı(#) )?{WBAl§Bl WuChong-shi –†‡· ¶(–)t7uF dδAB , AB B,.Pu(r,θ) Tr=aŁ&? ,u(r,θ) r=a+0r=a−0=0,F∂u(r,θ)/∂r Tr=aŁ&? , Tr=a OP 2ı(#) r Ccder2∂u∂r r=a+0r=a−0=−12πε0δθ−π2;∂u∂r r=a+0r=a−0=−12πε0a2δθ−π2._δ(θ−π/2)b Legendre RSδθ−π2=∞Pl=0clPl(cosθ),cl=2l+12Zπ0δθ−π2Pl(cosθ)sinθdθ=2l+12Pl(0).M(Alal=Bla−l−1,Allal+1+Bl(l+1)a−l=2l+14πε0Pl(0).K ;cAl=14πε0a−l−1Pl(0),Bl=14πε0alPl(0)..Pu(r,θ)=14πε0a∞Xl=0ralPl(0)Pl(cosθ),ra,14πε0a∞Xl=0arl+1Pl(0)Pl(cosθ),ra../Pl(0)@,;cu(r,θ)=14πε0a∞Xl=0(−)l(2l)!22ll!l!ra2lP2l(cosθ),ra,14πε0a∞Xl=0(−)l(2l)!22ll!l!ar2l+1P2l(cosθ),ra. ¡¢5u£|}5T⁄{¥eƒxy5z{|} §,¤ v'{,7“«‹›fifl 5–†‡z [\· G1w|} ,a¶5•‚„”» …‰‡e G¿ r=a,θ=π/2t^0,`´,•‚„”G¿ 5 ˆ^∞(1˜¯˘˙¨ AB]^ ˚ˆ) P¸O,G3{ ˝8˛L;…‰5ˇ¥,— •‚„a¶(#) ^‚„a¶W=r=a5 N 12 u¡¢;v ¤ N 5a¥ WuChong-shi§20.3Legendrelmn˛ˇ— t8u# •%& , Æ ª (ŁØ Œº ), , æ ı ÆLegendre łøœ ß, u(r,θ) r=0 r=∞ Taylor