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数学物理方法公式（7）：第二类Bessel函数（Neumann函数）

2023-03-21 19:14 作者:打电动的阿伟嘻嘻嘻  | 我要投稿

首先定义第二类Bessel函数：

Y_%7B%5Cnu%7D(x)%3D%5Cfrac%7BJ_%7B%5Cnu%7D(x)%7B%5Crm%20cos%7D%5Cnu%5Cpi-J_%7B-%5Cnu%7D(x)%7D%7B%7B%5Crm%20sin%7D%5Cnu%5Cpi%7D%2C%5C%20%5Cnu%5Cnotin%5Cmathbb%7BZ%7D.

Y_%7B%5Cnu%7D(x)%3D%5Clim_%7B%5Calpha%5Cto%5Cnu%7D%5Cfrac%7BJ_%7B%5Calpha%7D(x)%7B%5Crm%20cos%7D%5Calpha%5Cpi-J_%7B-%5Calpha%7D(x)%7D%7B%7B%5Crm%20sin%7D%5Calpha%20%5Cpi%7D%2C%5C%20%5Cnu%5Cin%5Cmathbb%7BZ%7D.

级数展开：

Y_%7Bn%7D(x)%3D%5Cfrac%7B2%7D%7B%5Cpi%7D(%5Cln%5Cfrac%7Bx%7D%7B2%7D%2B%5Cgamma)J_%7Bn%7D(x)-%5Cfrac%7B1%7D%7B%5Cpi%7D%5Csum%5E%7Bn-1%7D_%7Bk%3D0%7D%5Cfrac%7B(n-k-1)!%7D%7Bk!%7D(%5Cfrac%7Bx%7D%7B2%7D)%5E%7B2k-n%7D-%5Cfrac%7B1%7D%7B%5Cpi%7D%5Csum%5E%7Bn-1%7D_%7Bk%3D0%7D%5Cfrac%7B(-1)%5Ek%7D%7Bk!(n%2Bk)!%7D%5B%5CPhi(k)%2B%5CPhi(n%2Bk)%5D(%5Cfrac%7Bx%7D%7B2%7D)%5E%7B2k%2Bn%7D%2C

其中n%5Cin%5C%7B0%2C1%2C2%2C%5Ccdots%5C%7D%2C%5C%20%5CPhi(p)%3D1%2B%5Cfrac%7B1%7D%7B2%7D%2B%5Cfrac%7B1%7D%7B3%7D%2B%5Ccdots%2B%5Cfrac%7B1%7D%7Bp%7D%2C%5C%20%5CPhi(0)%3D0%2C%5C%20%5Cgamma 是欧拉常数.

积分表示：

Y_n(x)%3D%5Cfrac%7B1%7D%7B%5Cpi%7D%5Cint%5E%7B%5Cpi%7D_%7B0%7D%7B%5Crm%20sin%7D(x%7B%5Crm%20sin%7D%5Ctheta-n%5Ctheta)d%5Ctheta-%5Cfrac%7B1%7D%7B%5Cpi%7D%5Cint%5E%7B%5Cinfty%7D_%7B0%7D%5Be%5E%7Bnt%7D%2B(-1)%5Ene%5E%7B-nt%7D%5De%5E%7B-x%5Csinh%20t%7Ddt.

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