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While reading the Wikipedia page on Particular values of the Gamma Function, it listed a formula:$$\Gamma\left(\dfrac z2\right)=\sqrt{\pi}\dfrac {(z-2)!!}{2^{(z-1)/2}}\tag{1}$$ Where $z\in\mathbb{Z}$ for positive half integers. $(1)$ can be used to compute $\Gamma\left(\frac 12\right)$ by setting $z=1$ to get$$\Gamma\left(\dfrac 12\right)=\sqrt{\pi}\dfrac {(1-2)!!}{2^{(1-1)/2}}=\sqrt\pi\tag{2}$$ Extending this, I'm wondering

Questions:

  1. Can forumula $(1)$ be generalized to include complex numbers?$$\Gamma\left(a+bi\right)=\text{something}\tag{3}$$
  2. If so, how would you prove such formula?

Running it for WolframAlpha, it says that the Gamma function of a complex number is defined and is possible. But I'm just not sure how to derive a formula for $(3)$.

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    complex numbers with integer $a$ and $b$.? – 2017-02-27
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    @MyGlasses Uh... yes. $a,b\in\mathbb{R}$ – 2017-02-27
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    not sure if this was up when the question was posted, but the Wikipedia page does have some numeric data on complex values of Gamma: https://en.wikipedia.org/wiki/Particular_values_of_the_Gamma_function#Imaginary_and_complex_arguments no closed forms are given, which may have motivated the question – 2017-10-24

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There is no corresponding formula for complex values of $z$: unlike adding ordinary integers, there is no way to relate $\Gamma(a+i)$ to $\Gamma(a)$ (except that one has absolute value smaller than the other, by trivially bounding the integral definition). One can find the absolute value of $\Gamma(yi)$ for real $y$ by using the reflection formula (which comes out as $\sqrt{\frac{\pi}{y\sinh{(\pi y)}}}$), but there is no equivalent to $\Gamma(1+z)=z\Gamma(z)$. Hence one can relate the values of $\Gamma$ along a horizontal string of points separated by integers, but there is no way to move between these.

This is, in effect, a worse version of the problem with expressing $\Gamma(1/n)$ in finite form for $n>2$: there is no nice formula containing just one of these quantities: you only have things like $$ \Gamma(1/3)\Gamma(2/3) = \frac{2\pi}{\sqrt{3}} $$ from the reflection formula, or $$ \frac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(2/3)} = \sqrt[3]{2}\sqrt{\pi} $$ from the duplication formula.