Jensen's Formula

Theorem
Let $f: S \to \C$ with $S$ an open set containing the closed disk $D_r = \left\{{z \in \C : \left\vert{z}\right\vert \le r}\right\}$ of radius $r$ about $0$.

Suppose that $f$ has no zeros on the circle $\left\vert{z}\right\vert = r$, and $f \left({0}\right) \ne 0$.

Let $\rho_1, \ldots, \rho_n$ be the zeros of $f$ in $D_r$, counted with multiplicity.

Then:


 * $(1): \quad \displaystyle \frac 1 {2 \pi} \int_0^{2 \pi} \log \left\vert{f \left({r e^{i \theta} }\right)}\right\vert \ \mathrm d \theta = \log \left\vert{f \left({0}\right)}\right\vert + \sum_{k \mathop = 1}^n \left({\log r - \log \left\vert{\rho_k}\right\vert}\right)$