Difference between revisions of "Gamma function written as infinite product"
From specialfunctionswiki
| (2 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
==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}},$$ |
where $\Gamma$ denotes the [[gamma]] function. | where $\Gamma$ denotes the [[gamma]] function. | ||
| Line 7: | Line 7: | ||
==References== | ==References== | ||
| − | * {{BookReference|A course of modern analysis|1920|Edmund Taylor Whittaker|author2=George Neville Watson|prev=Reciprocal gamma written as an infinite product|next=findme}}: $\S 12 \cdot 11$ | + | * {{BookReference|A course of modern analysis|1920|Edmund Taylor Whittaker|author2=George Neville Watson|edpage=Third edition|prev=Reciprocal gamma written as an infinite product|next=findme}}: $\S 12 \cdot 11$ |
| − | * {{BookReference|Higher Transcendental Functions Volume I|1953| | + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=Gamma function written as a limit of a factorial, exponential, and a rising factorial|next=Reciprocal gamma written as an infinite product}}: §1.1 (2) |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 20:56, 3 March 2018
Theorem
The following formula holds: $$\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: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): §1.1 (2)
