Reciprocal gamma written as an infinite product

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.

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): $6.1.3$