Poisson–Jensen formula

Remarks

Proof idea: each factor $B_a(z) = R(z - a)/(R^2 - \bar a z)$ is a Blaschke factor for the disc of radius $R$: it vanishes only at $a$ and has modulus $1$ on $\abs{z} = R$. So $g = f \prod_\nu B_{b_\nu} / \prod_\mu B_{a_\mu}$ is holomorphic and zero-free near the closed disc, and $\log\abs{g}$ is harmonic. The Poisson formula, with the Poisson kernel as in the integral, represents $\log\abs{g}$, and $\log\abs{g} = \log\abs{f}$ on the circle. At $z = 0$ (if $f(0) \ne 0, \infty$) the formula becomes Jensen's formula: $\log\abs{f(0)}$ equals the circle average of $\log\abs{f}$ minus $\sum_\mu \log(R/\abs{a_\mu})$ plus $\sum_\nu \log(R/\abs{b_\nu})$. Example: $f(z) = z$ and $R = 1$; the integral vanishes and the only zero is $a_1 = 0$, so the right side is $-\log\abs{1/z} = \log\abs{z}$. The formula still holds when $f$ has zeros or poles on the circle, since $\log\abs{f}$ remains integrable there, but the proof needs an extra limit argument.

Going deeper

For a student

The formula is the basic identity of Nevanlinna theory, where it is used to bound the number of zeros of a function by its growth. It shows that $\log\abs{f}$ for holomorphic $f$ is a harmonic function minus a sum of Green's functions of the disc, one for each zero. This is a special case of the Riesz decomposition of the subharmonic function $\log\abs{f}$, whose Riesz measure is the sum of point masses at the zeros.