# Integer and Complex Values for the Gamma Function:

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 $(3)$.

• complex numbers with integer $a$ and $b$.? – Nosrati Feb 27 '17 at 4:45
• @MyGlasses Uh... yes. $a,b\in\mathbb{R}$ – Crescendo Feb 27 '17 at 15:03
• 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: en.wikipedia.org/wiki/… no closed forms are given, which may have motivated the question – Harry Richman Oct 24 '17 at 13:56

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.