Difference between revisions of "Beta as improper integral"

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(Created page with "==Theorem== The following formula holds for $\mathrm{Re}(x)>0$ and $\mathrm{Re}(y)>0$: $$B(x,y)=\displaystyle\int_0^{\infty} \xi^{x-1}(1+\xi)^{-x-y} \mathrm{d}\xi,$$ where $B$...")
 
 
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==References==
 
==References==
* {{BookReference|Higher Transcendental Functions Volume I|1953|Harry Bateman|prev=Beta|next=findme}}: $\S 1.5 (2)$
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* {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=Beta|next=B(x,y)=integral (t^(x-1)+t^(y-1))(1+t)^(-x-y) dt}}: $\S 1.5 (2)$
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
 
[[Category:Unproven]]
 
[[Category:Unproven]]

Latest revision as of 20:57, 3 March 2018

Theorem

The following formula holds for $\mathrm{Re}(x)>0$ and $\mathrm{Re}(y)>0$: $$B(x,y)=\displaystyle\int_0^{\infty} \xi^{x-1}(1+\xi)^{-x-y} \mathrm{d}\xi,$$ where $B$ denotes the beta function.

Proof

References