Difference between revisions of "Gamma(1)=1"

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==References==
 
==References==
* {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=Beta|next=Gamma(x+1)=xGamma(x)}}: $(2.1)$
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* {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=Beta|next=Gamma(z+1)=zGamma(z)}}: Theorem 2.1
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
 
[[Category:Proven]]
 
[[Category:Proven]]

Latest revision as of 19:47, 15 March 2018

Theorem

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

Proof

Compute using the fundamental theorem of calculus, $$\begin{array}{ll} \Gamma(1) &= \displaystyle\int_0^{\infty} \xi^{0} e^{-\xi} \mathrm{d}\xi \\ &= \displaystyle\int_0^{\infty} e^{-\xi} \mathrm{d}\xi \\ &= \left[ -e^{-\xi} \right.\Bigg|_{0}^{\infty} \\ &= 1, \end{array}$$ as was to be shown. █

References