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两个积分

2023-09-17 21:13 作者:艾琳娜的糖果屋  | 我要投稿

1.求第一类贝塞尔函数的梅林变换

利用积分表达式

%0A%5Cmathrm%7BJ%7D_%7B%5Cnu%7D%5Cleft(%20z%20%5Cright)%20%3D%5Cfrac%7B2%7D%7B%5Csqrt%7B%5Cpi%7D%5CGamma%20%5Cleft(%20%5Cnu%20%2B%5Cfrac%7B1%7D%7B2%7D%20%5Cright)%7D%5Cleft(%20%5Cfrac%7Bz%7D%7B2%7D%20%5Cright)%20%5E%7B%5Cnu%7D%5Cint_0%5E1%7B%5Ccos%20%5Cleft(%20zt%20%5Cright)%20%5Cleft(%201-t%5E2%20%5Cright)%20%5E%7B%5Cnu%20-%5Cfrac%7B1%7D%7B2%7D%7D%5Cmathrm%7Bd%7Dt%7D%5C%2C%5C%2C%5Cmathrm%7BRe%7D%5Cleft(%20%5Cnu%20%5Cright)%20%3E-%5Cfrac%7B1%7D%7B2%7D%0A%0A

%0A%5Cmathscr%7BM%7D%20%5Cleft%5C%7B%20%5Cmathrm%7BJ%7D_%7B%5Cnu%7D%5Cleft(%20z%20%5Cright)%20%5Cright%5C%7D%20%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B%5Cpi%7D2%5E%7B%5Cnu%20-1%7D%5CGamma%20%5Cleft(%20%5Cnu%20%2B%5Cfrac%7B1%7D%7B2%7D%20%5Cright)%7D%5Cint_0%5E1%7B%5Cleft(%201-t%5E2%20%5Cright)%20%5E%7B%5Cnu%20-%5Cfrac%7B1%7D%7B2%7D%7D%5Cmathrm%7Bd%7Dt%5Cint_0%5E%7B%5Cinfty%7D%7Bz%5E%7B%5Cnu%20%2Bs-1%7D%5Ccos%20%5Cleft(%20zt%20%5Cright)%20%5Cmathrm%7Bd%7Dz%7D%7D

%0A%3D%5Cfrac%7B%5CGamma%20%5Cleft(%20%5Cnu%20%2Bs%20%5Cright)%20%5Ccos%20%5Cleft(%20%5Cfrac%7Bs%2B%5Cnu%7D%7B2%7D%5Cpi%20%5Cright)%7D%7B%5Csqrt%7B%5Cpi%7D2%5E%7B%5Cnu%7D%5CGamma%20%5Cleft(%20%5Cnu%20%2B%5Cfrac%7B1%7D%7B2%7D%20%5Cright)%7D%5Cint_0%5E1%7Bt%5E%7B-%5Cfrac%7B%5Cnu%20%2Bs%7D%7B2%7D-%5Cfrac%7B1%7D%7B2%7D%7D%5Cleft(%201-u%20%5Cright)%20%5E%7B%5Cnu%20-%5Cfrac%7B1%7D%7B2%7D%7D%5Cmathrm%7Bd%7Du%7D%3D%5Cfrac%7B%5CGamma%20%5Cleft(%20%5Cnu%20%2Bs%20%5Cright)%20%5Ccos%20%5Cleft(%20%5Cfrac%7Bs%2B%5Cnu%7D%7B2%7D%5Cpi%20%5Cright)%7D%7B%5Csqrt%7B%5Cpi%7D2%5E%7B%5Cnu%7D%5CGamma%20%5Cleft(%20%5Cnu%20%2B%5Cfrac%7B1%7D%7B2%7D%20%5Cright)%7D%5Cfrac%7B%5CGamma%20%5Cleft(%20%5Cfrac%7B1-%5Cnu%20-s%7D%7B2%7D%20%5Cright)%20%5CGamma%20%5Cleft(%20%5Cnu%20%2B%5Cfrac%7B1%7D%7B2%7D%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%201%2B%5Cfrac%7B%5Cnu%20-s%7D%7B2%7D%20%5Cright)%7D%0A%0A

%0A%3D%5Cfrac%7B%5CGamma%20%5Cleft(%20%5Cfrac%7B%5Cnu%20%2Bs%7D%7B2%7D%20%5Cright)%20%5CGamma%20%5Cleft(%20%5Cfrac%7B1%2B%5Cnu%20%2Bs%7D%7B2%7D%20%5Cright)%20%5CGamma%20%5Cleft(%20%5Cfrac%7B1-%5Cnu%20-s%7D%7B2%7D%20%5Cright)%202%5E%7B%5Cnu%20%2Bs-1%7D%7D%7B%5Cpi%202%5E%7B%5Cnu%7D%5CGamma%20%5Cleft(%201%2B%5Cfrac%7B%5Cnu%20-s%7D%7B2%7D%20%5Cright)%7D%5Ccos%20%5Cleft(%20%5Cfrac%7Bs%2B%5Cnu%7D%7B2%7D%5Cpi%20%5Cright)%20%3D2%5E%7Bs-1%7D%5Cfrac%7B%5CGamma%20%5Cleft(%20%5Cfrac%7B%5Cnu%20%2Bs%7D%7B2%7D%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%201%2B%5Cfrac%7B%5Cnu%20-s%7D%7B2%7D%20%5Cright)%7D%0A%0A

根据渐近展开有

%0A%5Cmathrm%7BJ%7D_%7B%5Cnu%7D%5Cleft(%20z%20%5Cright)%20%5Csim%20z%5E%7B%5Cnu%7D%5Cleft(%20z%5Crightarrow%200%20%5Cright)%20%5C%2C%5C%2C%20%5Cmathrm%7BJ%7D_%7B%5Cnu%7D%5Cleft(%20z%20%5Cright)%20%5Csim%20%5Csqrt%7B%5Cfrac%7B2%7D%7B%5Cpi%20z%7D%7D%5Ccos%20%5Cleft(%20z-%5Cfrac%7B%5Cpi%20%5Cnu%7D%7B2%7D-%5Cfrac%7B%5Cpi%7D%7B4%7D%20%5Cright)%20%5Cleft(%20z%5Crightarrow%20%5Cinfty%20%5Cright)%20%0A%5C%5C%0A%5Cmathscr%7BM%7D%20%5Cleft%5C%7B%20%5Cmathrm%7BJ%7D_%7B%5Cnu%7D%5Cleft(%20z%20%5Cright)%20%5Cright%5C%7D%20%3D%5Cint_0%5E%7B%5Cinfty%7D%7B%5Cmathrm%7BJ%7D_%7B%5Cnu%7D%5Cleft(%20z%20%5Cright)%20z%5E%7Bs-1%7D%5Cmathrm%7Bd%7Dz%7D%3D2%5E%7Bs-1%7D%5Cfrac%7B%5CGamma%20%5Cleft(%20%5Cfrac%7B%5Cnu%20%2Bs%7D%7B2%7D%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%201%2B%5Cfrac%7B%5Cnu%20-s%7D%7B2%7D%20%5Cright)%7D%5C%2C%5C%2C%20%20%20%20-%5CRe%20%5Cleft(%20%5Cnu%20%5Cright)%20%3C%5CRe%20%5Cleft(%20s%20%5Cright)%20%3C%5Cfrac%7B1%7D%7B2%7D%0A%0A

2.求两个贝塞尔函数相乘的梅林变换

%0A%5Cmathscr%7BM%7D%20%5Cleft%5C%7B%20%5Cmathrm%7BJ%7D_%7B%5Cnu%7D%5Cleft(%20z%20%5Cright)%20%5Cmathrm%7BJ%7D_%7B%5Cmu%7D%5Cleft(%20z%20%5Cright)%20%5Cright%5C%7D%20%3D%5Cint_0%5E%7B%5Cinfty%7D%7B%5Cmathrm%7BJ%7D_%7B%5Cnu%7D%5Cleft(%20z%20%5Cright)%20%5Cmathrm%7BJ%7D_%7B%5Cmu%7D%5Cleft(%20z%20%5Cright)%20z%5E%7Bs-1%7D%5Cmathrm%7Bd%7Dz%7D%5C%2C%5C%2C%20%20%20%20%20-%5CRe%20%5Cleft(%20%5Cnu%20%5Cright)%20-%5CRe%20%5Cleft(%20%5Cmu%20%5Cright)%20%3C%5CRe%20%5Cleft(%20s%20%5Cright)%20%3C1%2C%5CRe%20%5Cleft(%20%5Cnu%20%5Cright)%20%2C%5CRe%20%5Cleft(%20%5Cmu%20%5Cright)%20%3E-%5Cfrac%7B1%7D%7B2%7D%0A%0A

%0A%5Cint_0%5E%7B%5Cinfty%7D%7B%5Cmathrm%7BJ%7D_%7B%5Cnu%7D%5Cleft(%20z%20%5Cright)%20%5Cmathrm%7BJ%7D_%7B%5Cmu%7D%5Cleft(%20z%20%5Cright)%20z%5E%7Bs-1%7D%5Cmathrm%7Bd%7Dz%7D%3D%5Cfrac%7B1%7D%7B2%5Cpi%20i%7D%5Cint_%7Bc-i%5Cinfty%7D%5E%7Bc%2Bi%5Cinfty%7D%7B2%5E%7Bt-1%7D%5Cfrac%7B%5CGamma%20%5Cleft(%20%5Cfrac%7B%5Cnu%20%2Bt%7D%7B2%7D%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%201%2B%5Cfrac%7B%5Cnu%20-t%7D%7B2%7D%20%5Cright)%7D2%5E%7Bs-t-1%7D%5Cfrac%7B%5CGamma%20%5Cleft(%20%5Cfrac%7B%5Cmu%20%2Bs-t%7D%7B2%7D%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%201%2B%5Cfrac%7B%5Cmu%20-s%2Bt%7D%7B2%7D%20%5Cright)%7D%5Cmathrm%7Bd%7Dt%7D%0A%0A

%0A%0A%3D2%5E%7Bs-1%7D%5Cfrac%7B1%7D%7B2%5Cpi%20i%7D%5Cint_%7B%5Cfrac%7Bc-%5Cmu%20-s%7D%7B2%7D-i%5Cinfty%7D%5E%7B%5Cfrac%7Bc-%5Cmu%20-s%7D%7B2%7D%2Bi%5Cinfty%7D%7B%5Cfrac%7B%5CGamma%20%5Cleft(%20%5Cfrac%7B%5Cnu%20%2B%5Cmu%20%2Bs%2B2u%7D%7B2%7D%20%5Cright)%20%5CGamma%20%5Cleft(%20-u%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%201%2B%5Cfrac%7B%5Cnu%20-%5Cmu%20-s-2u%7D%7B2%7D%20%5Cright)%20%5CGamma%20%5Cleft(%201%2B%5Cfrac%7B%5Cmu%20-s%2B%5Cmu%20%2Bs%2B2u%7D%7B2%7D%20%5Cright)%7D%5Cmathrm%7Bd%7Du%7D%5C%2C%5C%2C%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%5Cleft(%20%5Cmu%20%2Bs-t%3D-2u%20%5Cright)%20%0A%0A%0A

%0A%3D-2%5E%7Bs-1%7D%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%7B%5Cfrac%7B%5Cleft(%20-1%20%5Cright)%20%5E%7Bn%2B1%7D%7D%7Bn!%7D%5Cfrac%7B%5CGamma%20%5Cleft(%20n%2B%5Cfrac%7B%5Cnu%20%2B%5Cmu%20%2Bs%7D%7B2%7D%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%201%2B%5Cfrac%7B%5Cnu%20-%5Cmu%20-s%7D%7B2%7D-n%20%5Cright)%20%5CGamma%20%5Cleft(%201%2B%5Cmu%20%2Bn%20%5Cright)%7D%7D%3D2%5E%7Bs-1%7D%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%7B%5Cfrac%7B%5Cleft(%20-1%20%5Cright)%20%5En%7D%7Bn!%7D%5Cfrac%7B%5CGamma%20%5Cleft(%20n%2B%5Cfrac%7B%5Cnu%20%2B%5Cmu%20%2Bs%7D%7B2%7D%20%5Cright)%20%5CGamma%20%5Cleft(%20n%2B%5Cfrac%7Bs%2B%5Cmu%20-%5Cnu%7D%7B2%7D%20%5Cright)%7D%7B%5Cpi%20%5CGamma%20%5Cleft(%201%2B%5Cmu%20%2Bn%20%5Cright)%7D%5Csin%20%5Cleft(%20%5Cfrac%7Bs%2B%5Cmu%20-%5Cnu%7D%7B2%7D%2Bn%20%5Cright)%20%5Cpi%7D%0A

%0A%3D2%5E%7Bs-1%7D%5Cfrac%7B%5Csin%20%5Cleft(%20%5Cfrac%7Bs%2B%5Cmu%20-%5Cnu%7D%7B2%7D%5Cpi%20%5Cright)%7D%7B%5Cpi%7D%5Cfrac%7B%5CGamma%20%5Cleft(%20%5Cfrac%7B%5Cnu%20%2B%5Cmu%20%2Bs%7D%7B2%7D%20%5Cright)%20%5CGamma%20%5Cleft(%20%5Cfrac%7Bs%2B%5Cmu%20-%5Cnu%7D%7B2%7D%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%201%2B%5Cmu%20%5Cright)%7DF%5Cleft(%20%5Cfrac%7B%5Cnu%20%2B%5Cmu%20%2Bs%7D%7B2%7D%2C%5Cfrac%7Bs%2B%5Cmu%20-%5Cnu%7D%7B2%7D%3B1%2B%5Cmu%20%3B1%20%5Cright)%20%0A%0A

%0A%3D%5Cfrac%7B2%5E%7Bs-1%7D%7D%7B%5CGamma%20%5Cleft(%20%5Cfrac%7Bs%2B%5Cmu%20-%5Cnu%7D%7B2%7D%20%5Cright)%20%5CGamma%20%5Cleft(%201-%5Cfrac%7Bs%2B%5Cmu%20-%5Cnu%7D%7B2%7D%20%5Cright)%7D%5Cfrac%7B%5CGamma%20%5Cleft(%20%5Cfrac%7B%5Cnu%20%2B%5Cmu%20%2Bs%7D%7B2%7D%20%5Cright)%20%5CGamma%20%5Cleft(%20%5Cfrac%7Bs%2B%5Cmu%20-%5Cnu%7D%7B2%7D%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%201%2B%5Cmu%20%5Cright)%7D%5Cfrac%7B%5CGamma%20%5Cleft(%201%2B%5Cmu%20%5Cright)%20%5CGamma%20%5Cleft(%201-s%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%201%2B%5Cfrac%7B%5Cmu%20-%5Cnu%20-s%7D%7B2%7D%20%5Cright)%20%5CGamma%20%5Cleft(%201%2B%5Cfrac%7B%5Cmu%20%2B%5Cnu%20-s%7D%7B2%7D%20%5Cright)%7D%0A%0A

%0A%3D%5Cfrac%7B2%5E%7Bs-1%7D%5CGamma%20%5Cleft(%20%5Cfrac%7B%5Cnu%20%2B%5Cmu%20%2Bs%7D%7B2%7D%20%5Cright)%20%5CGamma%20%5Cleft(%201-s%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%201%2B%5Cfrac%7B%5Cnu%20-%5Cmu%20-s%7D%7B2%7D%20%5Cright)%20%5CGamma%20%5Cleft(%201%2B%5Cfrac%7B%5Cmu%20-%5Cnu%20-s%7D%7B2%7D%20%5Cright)%20%5CGamma%20%5Cleft(%201%2B%5Cfrac%7B%5Cmu%20%2B%5Cnu%20-s%7D%7B2%7D%20%5Cright)%7D%5C%2C%5C%2C%20%20-%5CRe%20%5Cleft(%20%5Cnu%20%5Cright)%20-%5CRe%20%5Cleft(%20%5Cmu%20%5Cright)%20%3C%5CRe%20%5Cleft(%20s%20%5Cright)%20%3C1%2C%5CRe%20%5Cleft(%20%5Cnu%20%5Cright)%20%2C%5CRe%20%5Cleft(%20%5Cmu%20%5Cright)%20%3E-%5Cfrac%7B1%7D%7B2%7D%0A%0A


标签：

两个积分的评论 (共 条)

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