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  "body": "\n## Definition\n\nThe *Clausen function* is the function $\\operatorname{Cl}_2 \\colon \\R \\to \\R$ given by\n$$\\operatorname{Cl}_2(\\theta) = -\\int_0^{\\theta} \\log \\abs{2 \\sin \\tfrac{t}{2}} \\, dt,$$\nwhere $\\log$ is the [[natural logarithm|04DXAFN2ES]].\nThe integral converges for every real $\\theta$, because the logarithmic singularities of the integrand at the points of $2\\pi\\Z$ are integrable.\n\n## Remarks\n\nEquivalently, $\\operatorname{Cl}_2$ is the sum of the uniformly convergent [[Fourier series|G12KXDHPPS]]\n$$\\operatorname{Cl}_2(\\theta) = \\sum_{k=1}^{\\infty} \\frac{\\sin k\\theta}{k^2},$$\nand $\\operatorname{Cl}_2(\\theta) = \\operatorname{Im} \\operatorname{Li}_2(e^{i\\theta})$, where $\\operatorname{Li}_2$ is the [[dilogarithm|G0X0NWSDB7]].\nThe function is odd and $2\\pi$-periodic, and it vanishes at every multiple of $\\pi$.\n\nExample: at $\\theta = \\pi/2$, $\\sin(k\\pi/2)$ is $0$ for even $k$ and $(-1)^{m}$ for $k = 2m + 1$, so $\\operatorname{Cl}_2(\\pi/2) = \\sum_{m \\ge 0} (-1)^m / (2m+1)^2 = G$, Catalan's constant.\nIts maximum on $\\R$ is $\\operatorname{Cl}_2(\\pi/3) \\approx 1.01494$.\n\nNon-example: the series $\\sum_{k \\ge 1} \\cos(k\\theta)/k^2$ is not the Clausen function; it is the real part of $\\operatorname{Li}_2(e^{i\\theta})$, which equals $\\frac{\\pi^2}{6} - \\frac{\\pi\\theta}{2} + \\frac{\\theta^2}{4}$ for $0 \\le \\theta \\le 2\\pi$.\n\nHigher Clausen functions $\\operatorname{Cl}_n$ are defined from $\\operatorname{Li}_n(e^{i\\theta})$ in a similar way.\n",
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