= Poisson-Jensen formula
{c}
{title2=$\log|f(z)|=\operatorname{Poisson}(\log|f|)(z)-\sum m_aG_R(z,a)+\sum m_pG_R(z,p)$}
For a <meromorphic function> with no zero or <pole> on $|w|=R$, set $G_R(z,a)=\log|(R^2-\overline az)/(R(z-a))|$. At source points that are neither zeros nor <poles>, the displayed identity expresses its logarithmic modulus as the Poisson integral of its boundary values, minus its zero Green potentials plus its <pole> Green potentials. The Poisson kernel is $(R^2-|z|^2)/|Re^{i\theta}-z|^2$ with normalized angular measure. Subtracting the corresponding logarithmic singularities leaves a harmonic function, so the harmonic Poisson formula proves the identity. At zero it reduces to <Jensen's formula>.
Back to article page
