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 $\dfrac{1}{\Gamma}$ is the [[reciprocal gamma]] function, and $\gamma$ is the [[Euler-Mascheroni constant]]. | where $\dfrac{1}{\Gamma}$ is the [[reciprocal gamma]] function, and $\gamma$ is the [[Euler-Mascheroni constant]]. | ||
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==References== | ==References== | ||
* {{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|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Gauss' formula for gamma function|next=Euler-Mascheroni constant}}: 6.1.3 | ||
Revision as of 07:05, 8 June 2016
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
- 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): §1.1 (3)
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 6.1.3
