Jensen's Formula/Proof 2

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

Let $f: S \to \C$ be holomorphic on $S$.

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

Proof
Write
 * $\displaystyle f \left({z}\right) = \frac{r^2-z \overline{\rho_1}}{r(z - \rho_1)} \cdots \frac{r^2-z \overline{\rho_n}}{r(z - \rho_n)} g \left({z}\right)$

, 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.

When $|z|=r$, we have $1/z=\overline{z}/r^2$ and $|z/r|=1$, so
 * $\displaystyle\left|\frac{r^2-z\overline{\rho_k}}{r(z-\rho_k)}\right|=\frac{|r^2-\overline{z}\rho_k|}{|r(z-\rho_k)|}=\frac{|1-\overline{z}\rho_k/r^2|}{|z/r-\rho_k/r|}=\frac{|1-\overline{z}\rho_k/r^2|}{|1-\rho_k/z|}=\frac{|1-\overline{z}\rho_k/r^2|}{|1-\overline{z}\rho_k/r^2|}=1$

so the left hand side is $0$.

Moreover, $\displaystyle \log \left|\frac{r^2-0 \overline{\rho_k}}{r(0 - \rho_k)}\right|=\log |\rho_k| - \log r$, so the right hand side is $0$.

Therefore, the formula holds for $\displaystyle\frac{r^2-z \overline{\rho_k}}{r(z - \rho_k)}$.

Since $r^2-z \overline{\rho_i}=0$ implies $|z| = r^2/|\overline{\rho_i}|>r$, $g(z)$ is holomorphic without zeros on $D_r$.

So the right hand side is $\log |g(0)|$.

On the other hand, by the mean value property, we see
 * $\displaystyle \frac{1}{2\pi} \int_0^{2\pi} \log |g(r e^{i \theta})| d\theta =\log g(0)$.

as required.