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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.)

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    $\begingroup$ Have you done any research on this question, such as reading about the motivation for this definition? $\endgroup$ Commented Jan 23, 2019 at 23:13
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    $\begingroup$ en.wikipedia.org/wiki/Bohr-Mollerup_theorem $\endgroup$ Commented Jan 23, 2019 at 23:15
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    $\begingroup$ A good reference is Emil Artin's book "The Gamma Function" I think. $\endgroup$ Commented Jan 23, 2019 at 23:40
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    $\begingroup$ Possible duplicate of a highly voted (and older) post also tells you why Gamma is the "best" extension. $\endgroup$ Commented Jan 23, 2019 at 23:51
  • $\begingroup$ 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 $e^{F(s)} = \int_0^\infty t^{s-1} e^{-t}dt$. $\endgroup$ Commented Jan 24, 2019 at 0:48

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