Page "Gamma function" Paragraph 15

The behavior for nonpositive z is more intricate.

Euler's integral does not converge for z ≤ 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,