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  "body": "\n## Statement\n\nLet $f$ be continuous on the closed half-plane $\\set{z \\in \\C : \\operatorname{Re} z \\ge 0}$ and [[holomorphic|0NAHXR1FRH]] on the open half-plane $\\operatorname{Re} z \u003e 0$.\nAssume there are constants $C \u003e 0$ and $\\varepsilon \u003e 0$ with $\\abs{f(z)} \\le C / \\abs{z}^{1 + \\varepsilon}$ for all $z \\ne 0$ with $\\operatorname{Re} z \\ge 0$.\nThen\n$$\\sum_{n=0}^{\\infty} f(n) = \\int_0^\\infty f(x)\\,dx + \\frac{f(0)}{2} + i \\int_0^\\infty \\frac{f(it) - f(-it)}{e^{2\\pi t} - 1}\\,dt.$$\n\n## Remarks\n\nProof idea: the kernels $1/(e^{\\mp 2\\pi i z} - 1)$ have simple poles at the integers.\nApply the residue theorem to $f$ times such a kernel on large rectangles in the upper and the lower quarter planes, with one side on the imaginary axis.\nThe residues at the integers produce the sum, the decay of $f$ kills the far sides, and the sides on the imaginary axis give the last integral.\n\nExample: $f(z) = 1/(z + 1)^2$ satisfies the hypotheses with $\\varepsilon = 1$, since $\\abs{z + 1} \\ge \\abs{z}$ when $\\operatorname{Re} z \\ge 0$.\nHere $f(it) - f(-it) = -4it/(1 + t^2)^2$, so the formula reads\n$$\\frac{\\pi^2}{6} = 1 + \\frac12 + 4 \\int_0^\\infty \\frac{t}{(1 + t^2)^2 (e^{2\\pi t} - 1)}\\,dt.$$\nApplied to $f(z) = (z + a)^{-s}$ it gives an integral formula for the [[Hurwitz zeta function|G4SW7HJG88]].\n\nSome decay or growth condition is needed for both sides to make sense: for $f = 1$ the series and the integral $\\int_0^\\infty f$ both diverge.\n\n## Explanations\n\n### L1 human\n\nThe formula compares a sum $f(0) + f(1) + f(2) + \\cdots$ with the integral $\\int_0^\\infty f$.\nThe difference is half of the first term plus a correction that only uses the values of $f$ on the imaginary axis.\nThis correction is often small and easy to estimate, because of the rapidly growing factor $e^{2\\pi t}$.\nSo the formula turns hard sums into integrals.\n\n### L2 human\n\nThe Abel–Plana formula is an exact alternative to the [[Euler–Maclaurin formula|9P839VB8S7]] for functions that extend holomorphically to the right half-plane.\nEuler–Maclaurin gives an asymptotic series with Bernoulli numbers, while Abel–Plana gives one convergent integral.\nExpanding $f(it) - f(-it)$ in powers of $t$ and integrating term by term recovers the Euler–Maclaurin terms, since $\\int_0^\\infty t^{2k-1} / (e^{2\\pi t} - 1)\\,dt$ is a multiple of a Bernoulli number.\nIt is a standard tool for the zeta functions and for the analytic continuation of series in physics, such as the computation of Casimir energies.\n",
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