Difference between revisions of "Gamma function written as infinite product"
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==Theorem== | ==Theorem== | ||
The following formula holds: | The following formula holds: | ||
| − | $$\Gamma(z) = \dfrac{1}{z} \displaystyle\prod_{k=1}^{\infty} \dfrac{(1+\frac{1}{k})^z}{1+\frac{z}{ | + | $$\Gamma(z) = \dfrac{1}{z} \displaystyle\prod_{k=1}^{\infty} \dfrac{(1+\frac{1}{k})^z}{1+\frac{z}{k}},$$ |
where $\Gamma$ denotes the [[gamma]] function. | where $\Gamma$ denotes the [[gamma]] function. | ||
Revision as of 01:45, 1 July 2017
Theorem
The following formula holds: $$\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.
Proof
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)
