Difference between revisions of "Beta"
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$$B(x,y)=\dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)},$$ | $$B(x,y)=\dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)},$$ | ||
where $\Gamma$ denotes the [[gamma function]]. | where $\Gamma$ denotes the [[gamma function]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> 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. | ||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
Revision as of 06:58, 11 February 2015
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 $\Gamma$ denotes the gamma function.
Proof: █
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
