\[ \large \int_{-\infty}^{\infty} \dfrac{ \mathrm{d} x}{\Gamma(\alpha +x) \Gamma (\beta - x)} = \dfrac{2^{\alpha + \beta -2}}{\Gamma(\alpha + \beta -1)} \quad ; \quad \Re (\alpha + \beta) > 1\]

Prove the equation above without using the method of residues.

**Notation:** \( \Gamma(\cdot) \) denotes the gamma function.

This is a part of the set Formidable Series and Integrals.

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## Comments

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TopNewestI wonder if it is possible to prove it without contour, well here is its solution, just put \(n=0\) at the end. @Ishan Singh

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I haven't tried the general question, but this special case does have an elementary solution.

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