3.6 Analytic continuation for gamma function, 6

A First Course in String Theory

Residues

The behavior for non-positive $\displaystyle{z}$ is more intricate. Euler’s integral does not converge for $\displaystyle{\Re (z)\leq 0}$ , but the function it defines in the positive complex half-plane has a unique analytic continuation to the negative half-plane. One way to find that analytic continuation is to use Euler’s integral for positive arguments and extend the domain to negative numbers by repeated application of the recurrence formula,

$\displaystyle{\Gamma (z)={\frac {\Gamma (z+n+1)}{z(z+1)\cdots (z+n)}}}$ ,

choosing $\displaystyle{n}$ such that $\displaystyle{z+n}$ is positive. The product in the denominator is zero when $\displaystyle{z}$ equals any of the integers $\displaystyle{0,-1,-2,\ldots}$ . Thus, the gamma function must be undefined at those points to avoid division by zero; it is a meromorphic function with simple poles at the non-positive integers.