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  "body": "\n## Summary\n\nA Fourier–Legendre series expands a function on $[-1, 1]$ as a sum of [[Legendre polynomials|QFYGJVA996]].\nIt is the [[generalized Fourier series|SNJC87TBHD]] for the orthogonal system of Legendre polynomials in $L^2([-1, 1])$.\nSince this system is complete, the series of every square-integrable function converges to it in the mean-square sense.\nSuch series appear when one solves the Laplace equation in spherical coordinates.\n\n## Explanations\n\n### L1 human\n\nThe Legendre polynomials $P_0, P_1, P_2, \\dots$ play on the interval $[-1, 1]$ the role that sines and cosines play on a circle.\nEach one is orthogonal to all the others, so one can read off how much of each $P_n$ a function contains by one integral.\nSumming these pieces rebuilds the function.\nFor example, $x^2 = \\tfrac13 P_0(x) + \\tfrac23 P_2(x)$, where $P_0(x) = 1$ and $P_2(x) = (3x^2 - 1)/2$.\n",
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