Difference between revisions of "Beta"
From specialfunctionswiki
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$$B(x,y)=\displaystyle\int_0^1 t^{x-1}(1-t)^{y-1}dt$$ | $$B(x,y)=\displaystyle\int_0^1 t^{x-1}(1-t)^{y-1}dt$$ | ||
| − | [[File:Beta1.png]] | + | [[File:Beta1.png|500px]] |
| − | [[File:Beta2.png]] | + | [[File:Beta2.png|500px]] |
=Properties= | =Properties= | ||
Revision as of 17:05, 14 October 2014
$$B(x,y)=\displaystyle\int_0^1 t^{x-1}(1-t)^{y-1}dt$$
Properties
Theorem: $B(x,y)=\dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$
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
