Does the notion of Mellin transform make sense for complex functions as well?

Question

In most of the sources I looked regarding the Mellin transform, they say let $f$ be a function defined on the positive real axis $0 < t < \infty$. And then say the Mellin transform is $$ M(f;s) = \int_0^{\infty} f(t) t^{s-1} dt. $$ And all the examples I have seen is where $f$ is a real function. I was just wondering does the notion make sense for $f$ is a complex function as well? and people use it? thank you.