Solutions in Vic rdinary Point &z Singularity 定义 如果p(x),q(x)均在30点解析,则点称为方程 的常点 °如果p(x),q(x)中至少有一个在0点不解析, 则20点称为方程的奇点 例91 Legendre方程 d2a du (1-2)2-22+(+1o=0 系数是 +1 C. S. Wu
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point Analytic Continuation Formulation Ordinary Point & Singularity ½Â XJp(z), q(z)þ3z0:)Û§Kz0:¡§ ~: XJp(z), q(z)¥k3z0:Ø)Û§ Kz0:¡§Û: ~9.1 Legendre§ (1 − z 2 ) d 2w dz 2 − 2z dw dz + l(l + 1)w = 0 Xê´ p(z) = − 2z 1 − z 2 q(z) = l(l + 1) 1 − z 2 3k?Û:z = ±1 C. S. Wu 1Êù ~©§?ê){()
Solutions in Vic rdinary Point &z Singularity 定义 如果p(x),q(x)均在30点解析,则点称为方程 的常点 °如果p(x),q(x)中至少有一个在0点不解析, 则20点称为方程的奇点 例9.1 Legendre方程 daqu du 22x+l(+1) 0 dz2 d 系数是 p() l(+1) 1 q(2) 故在有限远处的奇点为2=士1
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point Analytic Continuation Formulation Ordinary Point & Singularity ½Â XJp(z), q(z)þ3z0:)Û§Kz0:¡§ ~: XJp(z), q(z)¥k3z0:Ø)Û§ Kz0:¡§Û: ~9.1 Legendre§ (1 − z 2 ) d 2w dz 2 − 2z dw dz + l(l + 1)w = 0 Xê´ p(z) = − 2z 1 − z 2 q(z) = l(l + 1) 1 − z 2 3k?Û:z = ±1 C. S. Wu 1Êù ~©§?ê){()
Solutions in Vic rdinary Point &z Singularity 定义 如果p(x),q(x)均在30点解析,则点称为方程 的常点 °如果p(x),q(x)中至少有一个在0点不解析, 则20点称为方程的奇点 例9.1 Legendre方程 daqu du dz2 22x+l(+1) 0 d 系数是 p(z) q(2) l(+1) 故在有限远处的奇点为z=±1
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point Analytic Continuation Formulation Ordinary Point & Singularity ½Â XJp(z), q(z)þ3z0:)Û§Kz0:¡§ ~: XJp(z), q(z)¥k3z0:Ø)Û§ Kz0:¡§Û: ~9.1 Legendre§ (1 − z 2 ) d 2w dz 2 − 2z dw dz + l(l + 1)w = 0 Xê´ p(z) = − 2z 1 − z 2 q(z) = l(l + 1) 1 − z 2 3k?Û:z = ±1 C. S. Wu 1Êù ~©§?ê){()
Solutions in Vic rdinary Point &z Singularity 定义 如果p(z),q(z)均在0点解析,则点称为方程 的常点 如果p(),q()中至少有一个在0点不解析, 则0点称为方程的奇点 例9.2超几何( hypergeometric)方程 d2 du (1 +[-(1+a+)2] abu d dz 系数是 C. S. Wu
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point Analytic Continuation Formulation Ordinary Point & Singularity ½Â XJp(z), q(z)þ3z0:)Û§Kz0:¡§ ~: XJp(z), q(z)¥k3z0:Ø)Û§ Kz0:¡§Û: ~9.2 AÛ(hypergeometric)§ z(1 − z) d 2w dz 2 + [γ − (1 + α + β)z] dw dz − αβw = 0 Xê´ p(z) = γ − (1 + α + β)z z(1 − z) q(z) = − αβ z(1 − z) 3k?Û:z = 0z = 1 C. S. Wu 1Êù ~©§?ê){()
Solutions in Vic rdinary Point &z Singularity 定义 如果p(z),q(z)均在0点解析,则点称为方程 的常点 如果p(),q()中至少有一个在0点不解析, 则0点称为方程的奇点 例9.2超几何( hypergeometric)方程 d2 du abu d +[-(1+a+)2] dz 系数是 p() 7-(1+a+/) q( 2(1-2)k 故在有限远处的奇点为2=0与2=1 C. S. Wu
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point Analytic Continuation Formulation Ordinary Point & Singularity ½Â XJp(z), q(z)þ3z0:)Û§Kz0:¡§ ~: XJp(z), q(z)¥k3z0:Ø)Û§ Kz0:¡§Û: ~9.2 AÛ(hypergeometric)§ z(1 − z) d 2w dz 2 + [γ − (1 + α + β)z] dw dz − αβw = 0 Xê´ p(z) = γ − (1 + α + β)z z(1 − z) q(z) = − αβ z(1 − z) 3k?Û:z = 0z = 1 C. S. Wu 1Êù ~©§?ê){()