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  "body": "\n## Statement\n\nLet $0 \u003c R \u003c \\infty$ and let $f$ be a [[meromorphic function|0ZZ7ZYDR8W]] on a neighbourhood of the closed disc $\\abs{z} \\le R$, not identically zero.\nAssume that $f$ has no zeros and no poles on the circle $\\abs{z} = R$.\nLet $a_1, \\dots, a_m$ be the zeros and $b_1, \\dots, b_n$ the poles of $f$ in $\\abs{z} \u003c R$, each listed as often as its multiplicity.\nThen, for every $z$ with $\\abs{z} \u003c R$ that is neither a zero nor a pole,\n$$\\log\\abs{f(z)} = \\frac{1}{2\\pi} \\int_0^{2\\pi} \\frac{R^2 - \\abs{z}^2}{\\abs{R e^{i\\theta} - z}^2} \\log\\abs{f(R e^{i\\theta})} \\, d\\theta - \\sum_{\\mu = 1}^{m} \\log\\abs{\\frac{R^2 - \\bar a_\\mu z}{R (z - a_\\mu)}} + \\sum_{\\nu = 1}^{n} \\log\\abs{\\frac{R^2 - \\bar b_\\nu z}{R (z - b_\\nu)}}.$$\n\n## Remarks\n\nProof idea: each factor $B_a(z) = R(z - a)/(R^2 - \\bar a z)$ is a [[Blaschke factor|9N6R5ADJXT]] for the disc of radius $R$: it vanishes only at $a$ and has modulus $1$ on $\\abs{z} = R$.\nSo $g = f \\prod_\\nu B_{b_\\nu} / \\prod_\\mu B_{a_\\mu}$ is holomorphic and zero-free near the closed disc, and $\\log\\abs{g}$ is [[harmonic|0DEB0JMN0W]].\nThe Poisson formula, with the [[Poisson kernel|TCYSWCTYYM]] as in the integral, represents $\\log\\abs{g}$, and $\\log\\abs{g} = \\log\\abs{f}$ on the circle.\nAt $z = 0$ (if $f(0) \\ne 0, \\infty$) the formula becomes [[Jensen's formula|497K99BQ4Y]]: $\\log\\abs{f(0)}$ equals the [[circle average|Y7A1TP40MB]] of $\\log\\abs{f}$ minus $\\sum_\\mu \\log(R/\\abs{a_\\mu})$ plus $\\sum_\\nu \\log(R/\\abs{b_\\nu})$.\nExample: $f(z) = z$ and $R = 1$; the integral vanishes and the only zero is $a_1 = 0$, so the right side is $-\\log\\abs{1/z} = \\log\\abs{z}$.\nThe 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.\n\n## Explanations\n\n### L1 human\n\nIf one knows the size of a meromorphic function on a circle and where its zeros and poles are inside, one knows its size everywhere inside.\nThe integral is the harmonic part, and each zero pulls $\\log\\abs{f}$ down while each pole pushes it up.\n\n### L2 human\n\nThe formula is the basic identity of Nevanlinna theory, where it is used to bound the number of zeros of a function by its growth.\nIt 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.\nThis 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.\n",
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