Reciprocal gamma written as an infinite product
From specialfunctionswiki
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): $\S 12 \cdot 1$
- 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
