What is the generalization for a convolution in $\mathbb C$?

Question

Since the integration range of "the" convolution is $\mathbb R$, what is a sensible generalization in complex numbers? Would one still integrate over $\mathbb R$, or some other path, or over the entirety of $\mathbb C$?

Answer 1

Convolution makes sense, and works the same way, on abelian groups with a measure; be they $\mathbb R$, $\mathbb Z$, $\mathbb C$, or something else. Locally compact abelian groups with a Radon measure provide the conventional setting. In particular, in the complex plane we have $$f*g(z) = \int_{\mathbb C} f(z-\zeta)g(\zeta)\,dA(\zeta)$$ where $dA(\zeta)$ is the area element.

Common example: the Cauchy potential of a measure or a function, which is the convolution with $1/z$.