兩個(gè)積分

2023-09-17 21:13 作者:艾琳娜的糖果屋 0人讀過 | 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

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

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

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

根據(jù)漸近展開有

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2.求兩個(gè)貝塞爾函數(shù)相乘的梅林變換

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

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標(biāo)簽：

兩個(gè)積分的評(píng)論 (共條)

艾琳娜的糖果屋
發(fā)短消息
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