Beta as improper integral

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.

References

• 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.5 (2)$