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數(shù)學(xué)物理方法公式(6):第一類Bessel函數(shù)

2023-03-21 13:35 作者:打電動的阿偉嘻嘻嘻  | 我要投稿

首先通過級數(shù)展開定義第一類Bessel函數(shù)

J_%7B%5Cnu%7D(x)%3D%5Csum%5E%7B%5Cinfty%7D_%7Bm%3D0%7D%5Cfrac%7B(-1)%5Em%7D%7Bm!%5CGamma(m%2B%5Cnu%2B1)%7D(%5Cfrac%7Bx%7D%7B2%7D)%5E%7B2m%2B%5Cnu%7D%2C

稱為?%5Cnu?階第一類Bessel函數(shù).

特別地對于整數(shù)階的第一類Bessel函數(shù)?%5Cnu%3Dn%5Cin%5C%7B0%2C1%2C2%2C%5Ccdots%5C%7D

(1):J_n(x)%3D%5Csum%5E%7B%5Cinfty%7D_%7Bm%3D0%7D%5Cfrac%7B(-1)%5Em%7D%7Bm!(m%2Bn)!%7D(%5Cfrac%7Bx%7D%7B2%7D)%5E%7B2m%2Bn%7D.

(2):J_%7B-n%7D(x)%3D(-1)%5EnJ_n(x).

若一個函數(shù)的級數(shù)展開式的系數(shù)為Bessel函數(shù),則稱該函數(shù)為Bessel函數(shù)的生成函數(shù)與母函數(shù),f(x%2Cr)%3D%5Csum_nJ_n(x)r%5En%EF%BC%9A

e%5E%7B%5Cfrac%7Bx%7D%7B2%7D(r-%5Cfrac%7B1%7D%7Br%7D)%7D%3D%5Csum%5E%7B%5Cinfty%7D_%7B-%5Cinfty%7DJ_n(x)r%5En.

遞推公式:

(1):%5Cfrac%7Bd%7D%7Bdx%7D%5Bx%5E%7B%5Cnu%7DJ_%7B%5Cnu%7D(x)%5D%3Dx%5E%7B%5Cnu%7DJ_%7B%5Cnu-1%7D(x).

(2):%5Cfrac%7Bd%7D%7Bdx%7D%5Bx%5E%7B-%5Cnu%7DJ_%7B%5Cnu%7D(x)%5D%3D-x%5E%7B-%5Cnu%7DJ_%7B%5Cnu%2B1%7D(x).

(3):J_%7B%5Cnu%7D%5E%7B%5Cprime%7D(x)%3D%5Cfrac%7B1%7D%7B2%7D%5BJ_%7B%5Cnu-1%7D(x)-J_%7B%5Cnu%2B1%7D(x)%5D.

(4):J_%7B%5Cnu-1%7D(x)%2BJ_%7B%5Cnu%2B1%7D(x)%3D%5Cfrac%7B2%5Cnu%7D%7Bx%7DJ(x).

半奇數(shù)階Bessel函數(shù):n%5Cin%5C%7B0%2C1%2C2%2C%5Ccdots%5C%7D

(1):J_%7Bn%2B%5Cfrac%7B1%7D%7B2%7D%7D(x)%3D(-1)%5En%5Csqrt%7B%5Cfrac%7B2%7D%7B%5Cpi%7D%7Dx%5E%7Bn%2B%5Cfrac%7B1%7D%7B2%7D%7D(%5Cfrac%7B1%7D%7Bx%7D%5Cfrac%7Bd%7D%7Bdx%7D)%5En(%5Cfrac%7B%7B%5Crm%20sin%7Dx%7D%7Bx%7D).

(2):J_%7B-(n%2B%5Cfrac%7B1%7D%7B2%7D)%7D(x)%3D%5Csqrt%7B%5Cfrac%7B2%7D%7B%5Cpi%7D%7Dx%5E%7Bn%2B%5Cfrac%7B1%7D%7B2%7D%7D(%5Cfrac%7B1%7D%7Bx%7D%5Cfrac%7Bd%7D%7Bdx%7D)%5En(%5Cfrac%7B%7B%5Crm%20cos%7Dx%7D%7Bx%7D).

積分表達(dá):

(1):e%5E%7Bix%7B%5Crm%20sin%7D%5Ctheta%7D%3D%5Csum%5E%7B%5Cinfty%7D_%7B-%5Cinfty%7DJ_n(x)e%5E%7Bin%5Ctheta%7D.

(2):J_n(x)%3D%5Cfrac%7B1%7D%7B%5Cpi%7D%5Cint%5E%7B%5Cpi%7D_0%20%7B%5Crm%20cos%7D(x%7B%5Crm%20sin%7D%5Ctheta-n%5Ctheta)d%5Ctheta.

數(shù)學(xué)物理方法公式(6):第一類Bessel函數(shù)的評論 (共 條)

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