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兩個(gè)積分

2023-09-17 21:13 作者:艾琳娜的糖果屋  | 我要投稿

1.求第一類貝塞爾函數(shù)的梅林變換

利用積分表達(dá)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

根據(jù)漸近展開有

%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.求兩個(gè)貝塞爾函數(shù)相乘的梅林變換

%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

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%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


兩個(gè)積分的評(píng)論 (共 條)

分享到微博請(qǐng)遵守國(guó)家法律
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