Difference between revisions of "Beta"
From specialfunctionswiki
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| − | $$B(x,y)=\displaystyle\int_0^1 t^{x-1}(1-t)^{y-1}dt$$ | + | The $\beta$ function is defined by the formula |
| − | + | $$B(x,y)=\displaystyle\int_0^1 t^{x-1}(1-t)^{y-1}dt.$$ | |
[[File:Beta2.png|500px]] | [[File:Beta2.png|500px]] | ||
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=Properties= | =Properties= | ||
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| − | <strong>Theorem:</strong> $B(x,y)=\dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ | + | <strong>Theorem:</strong> The following formula holds: |
| + | $$B(x,y)=\dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)},$$ | ||
| + | where $\Gamma$ denotes the [[gamma function]]. | ||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
Revision as of 00:03, 19 October 2014
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: $B(x,y)=B(y,x)$
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
