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


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

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.

First let $h(z) = s - \rho_k$ for some $k \in \{1,\ldots,n\}$.

Making use of the substitution $u = re^{i\theta} - \rho_k$ we find that


 * $\displaystyle \frac 1{2\pi} \int_0^{2\pi} \log |h(re^{i\theta})|\ d\theta = \frac 1{2\pi i} \int_\gamma \frac{\log |u|}{u + \rho_k}\ du$

where $\gamma$ is a circle of radius $r$ centred at $-\rho_k$, traversed anticlockwise.

On this circle, $\log |u| = \log r$ is constant, and we have that


 * $\displaystyle \int_\gamma \frac{1}{u + \rho_k}\ du = \int_{|z| = r} \frac{du}{u} = 2\pi i$

Therefore the left hand side of $(1)$ is $\log r$ as required.

To show equality for $g(z)$, first observe that by the Residue Theorem:


 * $\displaystyle \int_{|z| = r} \frac{\log g(z)}{z}\ dz = 2\pi i \log g(0)$

Therefore substituting $z = re^{i\theta}$ we have


 * $\displaystyle 2\pi i \log g(0) = i\int_0^{2\pi} \log g(re^{i\theta})\ d\theta$

Comparing the imaginary parts of this equality we see that


 * $\displaystyle \frac 1{2\pi} \int_0^{2\pi} \log |g(re^{i\theta})\ d\theta = \log|g(0)|$

as required.