Beta is symmetric

Theorem

The following formula holds: $$B(x,y)=B(y,x),$$ where $B$ denotes the beta function.

Proof

Using beta in terms of gamma, we may calculate $$B(x,y)=\dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} = \dfrac{\Gamma(y)\Gamma(x)}{\Gamma(y+x)} = B(y,x),$$ as was to be shown.

References

• 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.5 (4)$
• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.2.2$