# Equilibrium measure and capacity for a region in complex plane.

Abstract: Let $E\subset \mathbb{C}$ and $\mu$ be a probability measure such that $Supp(\mu)\subset E$. Consider the total energy $-\int \log|z-t|d\mu(z)d\mu(t)$ due to the measure $\mu$. We are interested to find the measure for which total energy is minimized and supported in $E$. We will address the existence and uniqueness of this measure (Equilibrium measure). We will discuss about capacity of a region in the complex plane. Finally we will introduce the notion of weighted equilibrium measure and weighted capacity for a region in the complex plane.