Difference between revisions of "Reciprocal gamma written as an infinite product"

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* {{BookReference|A course of modern analysis|1920|Edmund Taylor Whittaker|edpage=Third edition|author2=George Neville Watson|prev=Euler-Mascheroni constant|next=Gamma function written as infinite product}}: $\S 12 \cdot 11$
 
* {{BookReference|A course of modern analysis|1920|Edmund Taylor Whittaker|edpage=Third edition|author2=George Neville Watson|prev=Euler-Mascheroni constant|next=Gamma function written as infinite product}}: $\S 12 \cdot 11$
 
* {{BookReference|Higher Transcendental Functions Volume I|1953|Harry Bateman|prev=Gamma function written as infinite product|next=Euler-Mascheroni constant}}: §1.1 $(3)$
 
* {{BookReference|Higher Transcendental Functions Volume I|1953|Harry Bateman|prev=Gamma function written as infinite product|next=Euler-Mascheroni constant}}: §1.1 $(3)$
 +
* {{BookReference|Special Functions|1960|Earl David Rainville|prev=findme|next=findme}}: $8.(1)$
 
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Gauss' formula for gamma function|next=Euler-Mascheroni constant}}: $6.1.3$
 
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Gauss' formula for gamma function|next=Euler-Mascheroni constant}}: $6.1.3$
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
 
[[Category:Unproven]]
 
[[Category:Unproven]]

Revision as of 05:18, 21 December 2017

Theorem

The following formula holds for all $z \in \mathbb{C}$: $$\dfrac{1}{\Gamma(z)} = ze^{\gamma z} \displaystyle\prod_{k=1}^{\infty} \left( 1 + \dfrac{z}{k}\right)e^{-\frac{z}{k}},$$ where $\dfrac{1}{\Gamma}$ is the reciprocal gamma function, and $\gamma$ is the Euler-Mascheroni constant.

Proof

References

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