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Old 2020-05-10, 23:49   #11
Dr Sardonicus
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Feb 2017

43·101 Posts

There is a fine little book (good luck finding a copy!) entitled The Gamma Function by Emil Artin. In it he shows that the Gamma function is distinguished by being "log convex."

As Retina has noted, x! = Γ(x+1) when x is a non-negative integer.

As to the derivative: There is a well-known asymptotic expansion [Stirling's asymptotic series] for ln(Γ(z)), z a complex variable. Taking the derivative term by term gives an asymptotic series for Γ'(z)/Γ(z).
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