Gamma function written as infinite product

Theorem

The following formula holds: $$\Gamma(z) = \dfrac{1}{z} \displaystyle\prod_{k=1}^{\infty} \dfrac{(1+\frac{1}{k})^z}{1+\frac{z}{k}},$$ where $\Gamma$ denotes the gamma function.

References

• 1920: Edmund Taylor Whittaker and George Neville Watson: A course of modern analysis ... (previous) ... (next): $\S 12 \cdot 11$
• 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): §1.1 (2)