Calculating Incomplete Gamma Function for Complex Arguments

[bruno67]

How do I calculate the integral

$$\int_{ix}^{i\infty} e^{-t} t^{-s-1}dt,$$
where $x>0$, $s>0$? Mathematica gives $\Gamma(-s,ix)$, where $\Gamma(\cdot,\cdot)$ is the incomplete gamma function, but I am not sure how to justify this formally.

 

[mathman]

Substitute u = -it, so the integral is from x to inf.

 

[Mute]

The incomplete Gamma function is defined by the integral

$$\Gamma(s,x) = \int_{x}^\infty dt~t^{s-1}e^{-t}.$$

Replacing x with ix formally gives $\Gamma(s,ix)$. However, the meaning of the integral with lower bound ix is really just formal, I think. You identify the integral with the incomplete Gamma function, and then you determine the "integral's" value by using the analytic continuation of the incomplete Gamma function for complex arguments.