简单搞个积

2023-03-23 21:00 作者:艾琳娜的糖果屋 0人读过 | 我要投稿

$%0A%5Cint_0%5E%7B%5Cinfty%7D%7B_2F_1%5Cleft(%20%5Cfrac%7B3%7D%7B4%7D%2C%5Cfrac%7B5%7D%7B6%7D%3B1%3B-x%20%5Cright)%20%5E2%5Cmathrm%7Bd%7Dx%7D%3D%5Cfrac%7B3%5Csqrt%5B4%5D%7B3%7D%5Cleft(%20%5Csqrt%7B3%7D-1%20%5Cright)%20%5CGamma%20%5E3%5Cleft(%20%5Cfrac%7B1%7D%7B3%7D%20%5Cright)%7D%7B2%5E%7B%5Cfrac%7B5%7D%7B6%7D%7D%5Cpi%20%5E2%7D%0A$

这里是个超几何函数的积分，实际上我们可以考虑一般点的情况

考虑2F1的梅林变换

$%0A%5Cint_0%5E%7B%5Cinfty%7D%7B%7Bx%5E%7Bs-1%7D%7D_2F_1%5Cleft(%20a%2Cb%3Bc%3B-x%20%5Cright)%20%5Cmathrm%7Bd%7Dx%7D%3D%5Cfrac%7B%5CGamma%20%5Cleft(%20c%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%20a%20%5Cright)%20%5CGamma%20%5Cleft(%20b%20%5Cright)%7D%5Cfrac%7B%5CGamma%20%5Cleft(%20s%20%5Cright)%20%5CGamma%20%5Cleft(%20a-s%20%5Cright)%20%5CGamma%20%5Cleft(%20b-s%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%20c-s%20%5Cright)%7D%0A%5Cleft(%200%3C%5Cmathrm%7BRe%7Ds%3C%5Cmin%20%5Cleft%5C%7B%20%5Cmathrm%7BRe%7Da%2C%5Cmathrm%7BRe%7Db%20%5Cright%5C%7D%20%5Cright)%20%0A%0A%0A$

这个结果利用2F1的积分表达或者拉马努金主定理就能得到

其次是梅林变换的卷积定理 $%0A%5Cint_0%5E%7B%5Cinfty%7D%7Bx%5E%7Bs-1%7Df%5Cleft(%20x%20%5Cright)%20g%5Cleft(%20x%20%5Cright)%20%5Cmathrm%7Bd%7Dx%7D%3D%5Cfrac%7B1%7D%7B2%5Cpi%20i%7D%5Cint_%7Bc-i%5Cinfty%7D%5E%7Bc%2Bi%5Cinfty%7D%7BF%5Cleft(%20%5Cmu%20%5Cright)%20G%5Cleft(%20s-%5Cmu%20%5Cright)%20%5Cmathrm%7Bd%7D%5Cmu%7D%0A$

那么有

$%0A%5Cint_0%5E%7B%5Cinfty%7D%7B%7Bx%5E%7Bs-1%7D%7D_2F_1%5Cleft(%20a%2Cb%3Bc%3B-x%20%5Cright)%20%5E2%5Cmathrm%7Bd%7Dx%7D%5C%5C%3D%5Cfrac%7B1%7D%7B2%5Cpi%20i%7D%5Cint_%7B%5Ceta%20-i%5Cinfty%7D%5E%7B%5Ceta%20%2Bi%5Cinfty%7D%7B%5Cleft(%20%5Cfrac%7B%5CGamma%20%5Cleft(%20c%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%20a%20%5Cright)%20%5CGamma%20%5Cleft(%20b%20%5Cright)%7D%20%5Cright)%20%5E2%5Cfrac%7B%5CGamma%20%5Cleft(%20%5Cmu%20%5Cright)%20%5CGamma%20%5Cleft(%20a-%5Cmu%20%5Cright)%20%5CGamma%20%5Cleft(%20b-%5Cmu%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%20c-%5Cmu%20%5Cright)%7D%5Cfrac%7B%5CGamma%20%5Cleft(%20s-%5Cmu%20%5Cright)%20%5CGamma%20%5Cleft(%20a-s%2B%5Cmu%20%5Cright)%20%5CGamma%20%5Cleft(%20b-s%2B%5Cmu%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%20c-s%2B%5Cmu%20%5Cright)%7D%5Cmathrm%7Bd%7D%5Cmu%7D%0A%0A$

再取s=c

$%0A%5Cint_0%5E%7B%5Cinfty%7D%7B%7Bx%5E%7Bc-1%7D%7D_2F_1%5Cleft(%20a%2Cb%3Bc%3B-x%20%5Cright)%20%5E2%5Cmathrm%7Bd%7Dx%7D%3D%5Cleft(%20%5Cfrac%7B%5CGamma%20%5Cleft(%20c%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%20a%20%5Cright)%20%5CGamma%20%5Cleft(%20b%20%5Cright)%7D%20%5Cright)%20%5E2%5Cfrac%7B1%7D%7B2%5Cpi%20i%7D%5Cint_%7B%5Ceta%20-i%5Cinfty%7D%5E%7B%5Ceta%20%2Bi%5Cinfty%7D%7B%5CGamma%20%5Cleft(%20a-c%2B%5Cmu%20%5Cright)%20%5CGamma%20%5Cleft(%20b-c%2B%5Cmu%20%5Cright)%20%5CGamma%20%5Cleft(%20a-%5Cmu%20%5Cright)%20%5CGamma%20%5Cleft(%20b-%5Cmu%20%5Cright)%20%5Cmathrm%7Bd%7D%5Cmu%7D%0A$

$%0A%3D%5Cleft(%20%5Cfrac%7B%5CGamma%20%5Cleft(%20c%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%20a%20%5Cright)%20%5CGamma%20%5Cleft(%20b%20%5Cright)%7D%20%5Cright)%20%5E2%5Cfrac%7B%5CGamma%20%5Cleft(%20a-c%2Ba%20%5Cright)%20%5CGamma%20%5Cleft(%20a-c%2Bb%20%5Cright)%20%5CGamma%20%5Cleft(%20b-c%2Ba%20%5Cright)%20%5CGamma%20%5Cleft(%20b-c%2Bb%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%20a-c%2Bb-c%2Ba%2Bb%20%5Cright)%7D%0A%5C%5C%0A%3D%5Cleft(%20%5Cfrac%7B%5CGamma%20%5Cleft(%20c%20%5Cright)%20%5CGamma%20%5Cleft(%20a%2Bb-c%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%20a%20%5Cright)%20%5CGamma%20%5Cleft(%20b%20%5Cright)%7D%20%5Cright)%20%5E2%5Cfrac%7B%5CGamma%20%5Cleft(%202a-c%20%5Cright)%20%5CGamma%20%5Cleft(%202b-c%20%5Cright)%7D%7B%5CGamma%20%5Cleft(%202a%2B2b-c%20%5Cright)%7D%0A%0A$

这里需要a-b不是整数，再带入数据我们就能得到

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