Jensen's Formula

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

Suppose that $f$ has no zeros on the circle $\left|{z}\right| = r$, 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:


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

Proof
Write $f \left({z}\right) = (z - \rho_1) \cdots (z - \rho_n) g \left({z}\right)$, so $g \left({z}\right) \ne 0$ for $z \in D_r$.

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

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

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


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

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:


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

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

To show equality for $g \left({z}\right)$, first observe that by the Residue Theorem:


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

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


 * $\displaystyle 2 \pi i \log g \left({0}\right) = i \int_0^{2 \pi} \log g(r e^{i \theta})\ \mathrm d \theta$

Comparing the imaginary parts of this equality we see that:


 * $\displaystyle \frac 1 {2 \pi} \int_0^{2 \pi} \log |g (r e^{i \theta})|\ \mathrm d \theta = \log|g \left({0}\right)|$

as required.