Beta

The $\beta$ function is defined by the formula $$B(x,y)=\displaystyle\int_0^1 t^{x-1}(1-t)^{y-1}dt.$$

• 
• 

Properties

Theorem

The following formula holds: $$B(x,y)=\dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)},$$ where $B$ denotes the beta function and $\Gamma$ denotes the gamma function.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.2.2$

Theorem: The following formula holds: $$B(x,y)=2 \displaystyle\int_0^{\frac{\pi}{2}} (\sin t)^{2x-1}(\cos t)^{2y-1}dt,$$ where $\sin$ and $\cos$ denote the sine and cosine functions.

Proof: █

Theorem: $B(x,y)=B(y,x)$

Proof: █

Theorem: (i) $B(x+1,y)=\dfrac{x}{x+y} B(x,y)$
(ii) $B(x,y+1)=\dfrac{y}{x+y}B(x,y)$

Proof: █

References

Bell. Special Functions
Special functions by Leon Hall