Difference between revisions of "Gamma function written as a limit of a factorial, exponential, and a rising factorial"
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==References== | ==References== | ||
| − | * {{BookReference|Higher Transcendental Functions Volume I|1953| | + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=Gamma(z) as integral of a power of log(1/t) for Re(z) greater than 0|next=Gamma function written as infinite product}}: §1.1 (2) |
Latest revision as of 20:56, 3 March 2018
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: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): §1.1 (2)
