Jensen's Formula

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

Suppose that $f$ has no zeros on the circle $|z| = r$, and $f(0) \neq 0$.

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

Then


 * $\displaystyle \frac 1{2\pi} \int_0^{2\pi} \log |f(re^{i\theta})\ d\theta = \log|f(0)| + \sum_{k=1}^n (\log r - \log |\rho_k|)$

Proof
Write $f(z) = (z-\rho_1)\cdots (z-\rho_n) g(z)$, so $g(z) \neq 0$ for $z \in D_r$.

It is sufficient to check the equality for each factor of $f$ in this expansion,