Difference between revisions of "Gamma(z+1)=zGamma(z)"

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==References==
 
==References==
 
* {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=Gamma(1)=1|next=Gamma(n+1)=n!}}: Theorem 2.2
 
* {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=Gamma(1)=1|next=Gamma(n+1)=n!}}: Theorem 2.2
 +
* {{BookReference|Hypergeometric Orthogonal Polynomials and Their q-Analogues|2010|Roelof Koekoek|author2=Peter A. Lesky|author3=René F. Swarttouw|prev=Gamma|next=findme}}: $(1.2.2)$
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
 
[[Category:Proven]]
 
[[Category:Proven]]
 
[[Category:Justify]]
 
[[Category:Justify]]

Revision as of 11:53, 5 April 2018

Theorem

The following formula holds: $$\Gamma(z+1)=z\Gamma(z),$$ where $\Gamma$ denotes gamma.

Proof

Use integration by parts to compute $$\begin{array}{ll} \Gamma(z+1) &= \displaystyle\int_0^{\infty} \xi^z e^{-\xi} \mathrm{d}\xi \\ &= -\xi^z e^{-\xi}\Bigg|_0^{\infty}- \displaystyle\int_0^{\infty} z \xi^{z-1} e^{-\xi} \mathrm{d}\xi \\ &= z\Gamma(z), \end{array}$$ as was to be shown. █

References