I have been studying a definite integral that I found out to be a particular (and possibly one of the simplest) case(s) of the arcane Mellin-Barnes integral. Solving this problem would lead to a breakthrough in my current research.

For any complex $z$, $e^{-z}$ can be represented by a line integral taken over a contour $C$,

$e^{-z}=\frac{1}{2\pi i}\int_{C}z^{-x}\Gamma(x)dx$

as long as the curve $C$ is chosen conveniently depending on the argument $z$. An almost complete discussion of this type of integral can be found in Paris and Kaminski's "Asymptotics and Mellin-Barnes Integrals" (pp. 89-90, Cambridge University Press, 2001).

For certain arguments, the curve $C$ is trivial. For example, if $z$ is a positive real number, then $C$ is a vertical line in the complex plane, and the integral can be expressed simply as,

$e^{-z}= \frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{\Gamma(1+i x)}{z^{1+i x}}dx$

The big challenge I am having with this type of integral though is that I am not able to figure out what the contour $C$ should be for an arbitrary non-real complex number (let's say $z=i$, or $z=1+i$), and the reference I posted doesn't have any examples. Ideally I'd like to be able to come up with an explicit parametrization of $C$ so that I can turn the Mellin-Barnes integral into a regular Riemann integral for any complex parameter (or at least some). I'm not sure if that is even possible, as evidenced by the lack of mention to such results in the aforementioned reference (or in the lengthy video tutorials that I watched on the Wolfram website).

Edit: I think I need to make the solution I'm looking for more explicit. As in the definition of line integral, and assuming $(-\infty,+\infty)$ as the domain of the curve, I'm looking for $h(x)$ such that,

$\int_{C}f(x)dx=\int_{-\infty}^{+\infty}f(h(x))*h'(x)dx$

PS Feel free to point out any inaccuracies in this description of Mellin-Barnes integrals, I am by no means an expert.