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  "body": "\n## Summary\n\nThe Clausen function $\\operatorname{Cl}_2$ is the real function $\\theta \\mapsto -\\int_0^\\theta \\log \\abs{2 \\sin(t/2)} \\, dt$, equal to the [[Fourier series|G12KXDHPPS]] $\\sum_{k \\ge 1} \\sin(k\\theta) / k^2$.\nIt is the imaginary part of the [[dilogarithm|G0X0NWSDB7]] on the unit circle.\nIt appears in evaluations of log-sine integrals, in values such as Catalan's constant, and in volumes of hyperbolic polyhedra.\n\n## Explanations\n\n### L1 human\n\nTake the dilogarithm series $\\sum z^k / k^2$ and put $z$ on the unit circle, $z = e^{i\\theta}$.\nThe real part is a quadratic polynomial in $\\theta$ on $[0, 2\\pi]$, but the imaginary part, $\\sum \\sin(k\\theta)/k^2$, is a new function: the Clausen function.\nIts graph looks like a tilted sine wave, odd and periodic with period $2\\pi$, with its maximum at $\\theta = \\pi/3$.\nAt $\\theta = \\pi/2$ the series is $1 - \\frac{1}{9} + \\frac{1}{25} - \\cdots$, which is Catalan's constant $G \\approx 0.91597$.\n",
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