# B(x,y)=2^(1-x-y)integral (1+t)^(x-1)(1-t)^(y-1)+(1+t)^(y-1)(1-t)^(x-1) dt

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## Theorem

The following formula holds for $\mathrm{Re}(x)>0$ and $\mathrm{Re}(y)>0$: $$B(x,y) = 2^{1-x-y} \displaystyle\int_0^1 (1+t)^{x-1}(1-t)^{y-1} + (1+t)^{y-1}(1-t)^{x-1} \mathrm{d}t,$$ where $B$ denotes the beta function.

## Proof

## References

- 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi:
*Higher Transcendental Functions Volume I*... (previous) ... (next): $\S 1.5 (10)$