Where does the gamma function come from

Asked Jan 23 '19 at 23:11

Active Jan 23 '19 at 23:16

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It has long been known that the gamma function is an extension (shift) of the factorial function defined on integers.

There were an infinite numbers of ways to continue the factorial, so what property does the gamma function have that makes/made us use it? (Also, this question is not about history, but the unique features of a gamma function.)

edited Jan 23 '19 at 23:16

David G. Stork

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asked Jan 23 '19 at 23:11

Pineapple Fish

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Have you done any research on this question, such as reading about the motivation for this definition? – Jan 23 '19 at 23:13
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https://en.wikipedia.org/wiki/Bohr-Mollerup_theorem – J.G. Jan 23 '19 at 23:15
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A good reference is Emil Artin's book "The Gamma Function" I think. – Nosrati Jan 23 '19 at 23:40
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Possible duplicate of a highly voted (and older) post also tells you why Gamma is the "best" extension. – Lee David Chung Lin Jan 23 '19 at 23:51
Once $f(s) =C+ \sum_{n=0}^\infty \frac{1}{s+n}- \frac{1}{n+1}$ converges and is analytic away from negative integers then it is clear that $f(s+1) = f(s)+\frac1s$. Letting $ F(s) = 1+\int_1^s f(z)dz $ and choosing $C$ such that $F(1) = F(2)$ then it is clear that $ F(s+1) = F(s)+ \log s, e^{F(s+1)}= s e^{F(s)}, e^{F(n+1)}= n!$. From there, the impressive and non-trivial fact is that $e^{F(s)} = \int_0^\infty t^{s-1} e^{-t}dt$. – reuns Jan 24 '19 at 00:48
Read this – J. W. Tanner Jan 24 '19 at 01:54

Where does the gamma function come from

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