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Gamma Integral

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

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.

Note by Ishan Singh
1 year, 2 months ago

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I wonder if it is possible to prove it without contour, well here is its solution, just put \(n=0\) at the end. @Ishan Singh Tanishq Varshney · 1 year, 1 month ago

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@Tanishq Varshney I haven't tried the general question, but this special case does have an elementary solution. Ishan Singh · 1 year, 1 month ago

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