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  "body": "\n## Statement\n\nLet $r \u003e 0$, and let $f$ be a [[meromorphic function|0NSD1TPHR0]] on an open set that contains the closed disc $\\abs{z} \\le r$.\nAssume that $f(0) \\ne 0$, that $0$ is not a pole of $f$, and that $f$ has no zeros and no poles on the circle $\\abs{z} = r$.\nLet $a_1, \\dots, a_m$ be the [[zeros|0CEGEASY49]] and $b_1, \\dots, b_n$ the [[poles|0WPQYW8MTE]] of $f$ in the disc $\\abs{z} \u003c r$, each repeated according to its multiplicity.\nThen\n$$\\log \\abs{f(0)} = \\frac{1}{2\\pi} \\int_0^{2\\pi} \\log \\abs{f(r e^{i\\theta})} \\, d\\theta - \\sum_{j=1}^{m} \\log \\frac{r}{\\abs{a_j}} + \\sum_{k=1}^{n} \\log \\frac{r}{\\abs{b_k}}.$$\n\n## Remarks\n\nDivide $f$ by the [[Blaschke factors|9N6R5ADJXT]] $r(z - a_j)/(r^2 - \\overline{a_j} z)$ and multiply by the analogous factors for the poles.\nThe quotient $g$ has no zeros or poles on the closed disc, and $\\abs{g} = \\abs{f}$ on the circle $\\abs{z} = r$.\nThen $\\log \\abs{g}$ is [[harmonic|0W5NJQ8MKY]], and its mean value property gives the formula, since each factor has modulus $\\abs{a_j}/r$ or $\\abs{b_k}/r$ at $0$.\nFor example, take $f(z) = z - a$ with $0 \u003c \\abs{a} \u003c 1$ and $r = 1$.\nThe integral is $0$, because $\\abs{e^{i\\theta} - a} = \\abs{1 - \\overline{a} e^{i\\theta}}$ and $\\log \\abs{1 - \\overline{a} z}$ is harmonic on the closed unit disc with value $0$ at $0$; both sides then equal $\\log \\abs{a}$.\nThe formula remains true when $f$ has zeros or poles on the circle, since the integral still converges.\n\n## Explanations\n\n### L1 human\n\nJensen's formula relates the size of a function on a circle to its zeros inside the circle.\nFor a function without zeros or poles, $\\log \\abs{f}$ is harmonic and its value at the centre is its average on the circle.\nEach zero inside the disc pulls the value at the centre down, and each pole pushes it up, by an exact amount.\nSo a function that is not large on the circle cannot have many zeros inside.\n\n### L2 human\n\nFor a holomorphic $f$, the formula gives $\\sum_j \\log(r/\\abs{a_j}) \\le \\log M(r) - \\log \\abs{f(0)}$, where $M(r)$ is the maximum of $\\abs{f}$ on $\\abs{z} = r$.\nThis bounds the number of zeros in terms of the growth, which is the first step of the [[Hadamard factorization theorem|KYSR7NHNFQ]].\nThe sum can be written as $\\int_0^r n(t)/t \\, dt$, where $n(t)$ counts the zeros in $\\abs{z} \\le t$.\nIn this form the formula is the starting point of [[Nevanlinna theory|1EHH63VNAD]] and of its [[first main theorem|QJWCFAK4AK]].\n",
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