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I want to compute integrals of the following form

[tex] I= \int_{-i \infty}^{i\infty} (1-x^2)^\frac{d-1}{2} \prod_{i=1}^4 _2F_1(a_i,b_i,c_i;\frac{1-x}{2}) dx [/tex]

where [tex]a_i,b_i,c_i[/tex] are constants and [tex]c_i\in \mathbb{N}[/tex].

d is a positive integer.

For odd d I know that the integral will be zero by calculus of residues, with the right complex half plane as integration contour, as the constants are such that the Hypergeometric functions decay fast enough.

For even d I cannot do this as I get a branch cut from 1 to infinity on the positiv real axis.

Does somebody see a way, how i can explicitly integrate this?

Thanks

betel

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# Difficult integral involving Hypergeometric functions

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