北大数学物理方法(A)-数学物理方程教案07球函数2

Outline1Ôù¥¼ê()ÔnÆ2007cSC.S.Wu1Ôù¥¼ê()OutlineùLJ:1Legendreõ‘ª5Ÿ(Y)Legendreõ‘ª)¤¼êLegendreõ‘ª4í'X2Legendreõ‘ªA^þ!|¥N¥þ!‘‚·³C.S.Wu1Ôù¥¼ê()OutlineùLJ:1Legendreõ‘ª5Ÿ(Y)Legendreõ‘ª)¤¼êLegendreõ‘ª4í'X2Legendreõ‘ªA^þ!|¥N¥þ!‘‚·³C.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsReferencesÇÂÁ§5êÆÔn{6§§16.5,16.6,16.7ù&œ§5êÆÔn{6§§10.1nÎ!X1Á§5êÆÔn{6§§12.3C.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGeneratingFunctionRecurrenceFormulasforLegendrePolynomialsùLJ:1Legendreõ‘ª5Ÿ(Y)Legendreõ‘ª)¤¼êLegendreõ‘ª4í'X2Legendreõ‘ªA^þ!|¥N¥þ!‘‚·³C.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGeneratingFunctionRecurrenceFormulasforLegendrePolynomialsµ3å:r?˜k˜‡ü :Ö§:Ö¤3:•z¶•§ùž:Ö3(r0,θ,φ):³(w,†φÃ')=1√r2+r02−2rr0cosθ=1r1√1−2xt+t2t=r0r1r01√1−2xt+t2t=rr0Ù¥x=cosθ§¿5½1√1−2xt+t2t=0=1C.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGeneratingFunctionRecurrenceFormulasforLegendrePolynomialsLegendreõ‘ª)¤¼ê3d5½e§¼ê1/√1−2xt+t23t=0:9ٍS´)ۧό±ŠTaylorÐm1√1−2xt+t2=∞Xl=0cltl|t|x±px2−1Œ±y²ÐmXêclÒ´Legendreõ‘ªPl(x)1√1−2xt+t2=∞Xl=0Pl(x)tl|t|x±px2−1¼ê1/√1−2xt+t2=¡Legendreõ‘ª)¤¼êC.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGeneratingFunctionRecurrenceFormulasforLegendrePolynomialsLegendreõ‘ª)¤¼ê3d5½e§¼ê1/√1−2xt+t23t=0:9ٍS´)ۧό±ŠTaylorÐm1√1−2xt+t2=∞Xl=0cltl|t|x±px2−1Œ±y²ÐmXêclÒ´Legendreõ‘ªPl(x)1√1−2xt+t2=∞Xl=0Pl(x)tl|t|x±px2−1¼ê1/√1−2xt+t2=¡Legendreõ‘ª)¤¼êC.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGeneratingFunctionRecurrenceFormulasforLegendrePolynomialsLegendreõ‘ª)¤¼ê3d5½e§¼ê1/√1−2xt+t23t=0:9ٍS´)ۧό±ŠTaylorÐm1√1−2xt+t2=∞Xl=0cltl|t|x±px2−1Œ±y²ÐmXêclÒ´Legendreõ‘ªPl(x)1√1−2xt+t2=∞Xl=0Pl(x)tl|t|x±px2−1¼ê1/√1−2xt+t2=¡Legendreõ‘ª)¤¼êC.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGeneratingFunctionRecurrenceFormulasforLegendrePolynomialsLegendreõ‘ª)¤¼ê1√1−2xt+t2=∞Xl=0Pl(x)tl|t|x±√x2−1Proof†ò¼ê1/√1−2xt+t23t=0:ŠTaylorÐm1√1−2xt+t2=1p1−2t+t2−2(x−1)tC.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGeneratingFunctionRecurrenceFormulasforLegendrePolynomialsLegendreõ‘ª)¤¼ê1√1−2xt+t2=∞Xl=0Pl(x)tl|t|x±√x2−1Proof1√1−2xt+t2=1p1−2t+t2−2(x−1)t=11−t1−2(x−1)t(1−t)2−1/2C.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGeneratingFunctionRecurrenceFormulasforLegendrePolynomialsLegendreõ‘ª)¤¼ê1√1−2xt+t2=∞Xl=0Pl(x)tl|t|x±√x2−1Proof1√1−2xt+t2=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)2kC.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGeneratingFunctionRecurrenceFormulasforLegendrePolynomialsLegendreõ‘ª)¤¼ê1√1−2xt+t2=∞Xl=0Pl(x)tl|t|x±√x2−1Proof1√1−2xt+t2=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)C.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGeneratingFunctionRecurrenceFormulasforLegendrePolynomialsLegendreõ‘ª)¤¼ê1√1−2xt+t2=∞Xl=0Pl(x)tl|t|x±√x2−1Proof1√1−2xt+t2=11−t1−2(x−1)t(1−t)2−1/2=∞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)!tnC.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGeneratingFunctionRecurrenceFormulasforLegendrePolynomialsLegendreõ‘ª)¤¼ê1√1−2xt+t2=∞Xl=0Pl(x)tl|t|x±√x2−1Proof1√1−2xt+t2=11−t1−2(x−1)t(1−t)2−1/2=∞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#tlC.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGeneratingFunctionRecurrenceFormulasforLegendrePolynomialsLegendreõ‘ª)¤¼ê1√1−2xt+t2=∞Xl=0Pl(x)tl|t|x±px2−1Œ±íÑNõk^(J©~X-x=1§íØ11√1−2t+t2=11−t∞Xl=0Pl(1)tl=∞Xl=0tl=⇒Pl(1)=1C.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGeneratingFunctionRecurrenceFormulasforLegendrePolynomialsLegendreõ‘ª)¤¼ê1√1−2xt+t2=∞Xl=0Pl(x)tl|t|x±px2−1Œ±íÑNõk^(J©~X-x=1§íØ11√1−2t+t2=11−t∞Xl=0Pl(1)tl=∞Xl=0tl=⇒Pl(1)=1C.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGeneratingFunctionRecurrenceFormulasforLegendrePolynomialsLegendreõ‘ª)¤¼ê1√1−2xt+t2=∞Xl=0Pl(x)tl|t|x±px2−1Œ±íÑNõk^(J©Ï1√1−2xt+t2=1p1−2(−x)(−t)+(−t)2íØ2∞Xl=0Pl(x)tl=∞Xl=0Pl(−x)(−t)l⇒Pl(−x)=(−)lPl(x)C.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGeneratingFunctionRecurrenceFormulasforLegendrePolynomialsLegendreõ‘ª)¤¼ê1√1−2xt+t2=∞Xl=0Pl(x)tl|t|x±px2−1Œ±íÑNõk^(J©Ï1√1−2xt+t2=1p1−2(−x)(−t)+(−t)2íØ2∞Xl=0Pl(x)tl=∞Xl=0Pl(−x)(−t)l⇒Pl(−x)=(−)lPl(x)C.S.Wu1Ôù¥¼ê()PropertiesofLegendrePolynomials(cont.)ApplicationsofLegendrePolynomialsGener