= Annular spherical Jensen formula
{c}
{title2=$T(r)=I(a)\log(1/r)+N(r;a)+\overline m(r;a)$}
For a <meromorphic function> near $0<|z|\le1$, and a target $a$ off its image of the unit circle, let $u_a=-\log k(f,a)$ for the <chordal metric>, $M_a(s)$ its normalized angular mean, and $I(a)=M_a'(1)$. The weighted spherical area $T(r)=(4\pi)^{-1}\int_{r<|z|<1}\log(|z|/r)(f^{\#}_{\mathrm{round}})^2\,dA$ satisfies the displayed formula, where $N$ counts $a$-points with <multiplicity> and weight $\log(|z|/r)$ and $\overline m=M_a(r)-M_a(1)$. Indeed $\Delta u_a=(f^{\#}_{\mathrm{round}})^2/2-2\pi\sum\deg_z(f)\delta_z$, and <Green's second identity> with the radial logarithmic weight proves the formula. The angular mean must include $1/(2\pi)$.
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