Beta as improper integral

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Theorem

The following formula holds for $\mathrm{Re}(x)>0$ and $\mathrm{Re}(y)>0$: $$B(x,y)=\displaystyle\int_0^{\infty} \xi^{x-1}(1+\xi)^{-x-y} \mathrm{d}\xi,$$ where $B$ denotes the beta function.

Proof

References

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