Difference between revisions of "Reciprocal gamma written as an infinite product"
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==Theorem== | ==Theorem== | ||
| − | The following formula holds: | + | The following formula holds for all $z \in \mathbb{C}$: |
$$\dfrac{1}{\Gamma(z)} = ze^{\gamma z} \displaystyle\prod_{k=1}^{\infty} \left( 1 + \dfrac{z}{k}\right)e^{-\frac{z}{k}},$$ | $$\dfrac{1}{\Gamma(z)} = ze^{\gamma z} \displaystyle\prod_{k=1}^{\infty} \left( 1 + \dfrac{z}{k}\right)e^{-\frac{z}{k}},$$ | ||
| − | where | + | where $\dfrac{1}{\Gamma}$ is the [[reciprocal gamma]] function, and $\gamma$ is the [[Euler-Mascheroni constant]]. |
==Proof== | ==Proof== | ||
==References== | ==References== | ||
| − | * {{BookReference|Higher Transcendental Functions Volume I|1953| | + | * {{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|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|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}}: $6.1.3$ | ||
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 20:56, 3 March 2018
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
- 1920: Edmund Taylor Whittaker and George Neville Watson: A course of modern analysis ... (previous) ... (next): $\S 12 \cdot 11$
- 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): §1.1 $(3)$
- 1960: Earl David Rainville: Special Functions ... (previous) ... (next): $8.(1)$
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.1.3$
