Difference between revisions of "Gamma function written as a limit of a factorial, exponential, and a rising factorial"
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Revision as of 09:36, 4 June 2016
Theorem
The following formula holds: $$\Gamma(z) = \displaystyle\lim_{n \rightarrow \infty} \dfrac{n! n^z}{z(z+1) \ldots (z+n)} = \displaystyle\lim_{n \rightarrow \infty} \dfrac{n^z}{z(1+z)(1+\frac{z}{2}) \ldots (1+\frac{z}{n})},$$ where $\Gamma$ denotes the gamma function.
Proof
References
- 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous): §1.1 (2)