I'm trying to understand what the phrase means "If $\mu$ is a distribution on the complex plane with compact support, then their restriction to the entire functions is analytic functional".
From what I understand, a distribution $T$ is a continuous linear functional defined on the test function space $\mathcal{C}_c^\infty(\Omega)$ with $\Omega$ an open subset of $\mathbb{R}^n$.
I also know that the support of a distribution $T$ is defined as the complement of the largest open set of $\Omega$ in which $T$ vanishes
Question 1. What is a distribution on the complex plane? Is $T:\mathcal{C}^{\infty}_{c}(\mathbb{C})\to \mathbb{C}$ with $\mathbb{C}\approx \mathbb{R}^2$?