 The behavior for non-positive formula_31 is more intricate.Euler's integral does not converge for formula_161, 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,
choosing formula_33 such that formula_164 is positive.