Another form of Jensen's Theorem In Jensen's Theorem, we have that if $f(z)$ is analytic in a closed disk with radius $R$ and centre $a$. We assume that the function is non zero on the boundary and at the centre. If $z_i$ are the zeroes of the function in the interior of the disk and n(R) is the number of zeroes then we have the formula $log |f(a)| = -\Sigma_{i=1}^{n(R)}log \frac {R}{|z_i|}+ \frac{1} {2\pi}\int_0^{2\pi}log|f(a+Re^{i\theta})|d\theta$. I have come across an application of this formula where $\Sigma_{i=1}^{n(R)}log \frac {R}{|z_i|}$ has been replaced by $\int_0^{R} \frac {n(r)} {r} dr$. How do we get this? Is this always true or are there extra conditions required? In the case where I have come across this, the given function is real on the real axis.
Edit: On further consultations with books, I find that this always holds and is given as an exercise in Basic Complex Analysis by Barry Simon.
 A: Yes, if  $z_1, \ldots, z_n$ are the zeros of $f$ in $B(R, a)$, counted with multiplicity, and $f(a) \ne 0$, then
$$
\int_0^R \frac{n(r)}{r} \, dr = \sum_{k=1}^{n} \log \frac{R}{|z_k|} \, .
$$
For a proof we can assume that the $z_k$ are sorted in increasing order of absolute value:
$$
  0 < |z_1| \le |z_2| \le \cdots \le |z_n| < R \, .
$$
Then $n(r) = k$ for $|z_k| \le r < |z_{k+1}|$, and therefore
$$
\begin{align}
 \int_0^R \frac{n(r)}{r} \, dr &= \sum_{k=1}^{n-1} \int_{|z_k|}^{|z_{k+1}|} \frac k r \, dr + \int_{|z_n|}^R \frac n r \, dr \\
&= \sum_{k=1}^{n-1} k \bigl( \log |z_{k+1}| - \log |z_{k}|\bigr)
+ n \bigl( \log R -  \log |z_{n}|\bigr) \\
&= n \log R - \sum_{k=1}^{n}\log |z_{k}|\\
& = \sum_{k=1}^{n} \log \frac{R}{|z_k|} \, .
\end{align}
$$
If you are familiar with the Riemann-Stieltes integral then this can also be obtained with integration by parts:
$$
\begin{align}
\int_0^R \frac{n(r)}{r} \, dr &= \int_0^R n(r) \, d\log r \\
 &= n(R) \log R - \int_0^R \log r\, dn(r) \\
 &= n \log R - \sum_{k=1}^{n}\log |z_{k}| \, .
\end{align}
$$
Remark: If $f(a) = 0$ then the identity becomes
$$
\int_0^R \frac{n(r)-n(0)}{r} \, dr = \sum_{k=1}^{n} \log \frac{R}{|z_k|} \, .
$$
where $z_1, \ldots, z_n$ are the zeros of $f$ in $B(R, a) \setminus \{ a \}$, counted with multiplicity.
