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  "body": "\n## Summary\n\nThe Legendre polynomials $P_0, P_1, P_2, \\dots$ are the polynomials obtained by making $1, x, x^2, \\dots$ orthogonal on $[-1, 1]$ with the constant weight, normalised by $P_n(1) = 1$.\nThey are the simplest [[classical orthogonal polynomials|47WJ34NC9K]], the Jacobi case $\\alpha = \\beta = 0$.\nThey solve [[Legendre's equation|W6VE30V10Y]] and appear in the Laplace equation in spherical coordinates, in multipole expansions, and in Gauss–Legendre quadrature.\n\n## Explanations\n\n### L1 human\n\nOn the interval from $-1$ to $1$, two functions are called orthogonal when the integral of their product is zero.\nStart with $1$, $x$, $x^2$, and so on, and adjust each one so that it is orthogonal to all the earlier ones.\nThe results, scaled to equal $1$ at $x = 1$, are $P_0 = 1$, $P_1 = x$, $P_2 = \\frac{3x^2 - 1}{2}$, and so on.\nThey play the role on $[-1, 1]$ that sines and cosines play on a circle.\n",
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