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  "body": "\n## Definition\n\nLet $P_n$ be the [[Legendre polynomial|QFYGJVA996]] of degree $n$, for $n \\ge 0$, normalized by $P_n(1) = 1$.\nLet $f \\colon [-1, 1] \\to \\C$ be a [[Lebesgue integrable|DWJF7ABPR1]] function.\nThe *Fourier–Legendre coefficients* of $f$ are\n$$a_n = \\frac{2n + 1}{2} \\int_{-1}^{1} f(x) P_n(x) \\, dx, \\qquad n \\ge 0.$$\nThe *Fourier–Legendre series* of $f$ is the formal series\n$$\\sum_{n=0}^{\\infty} a_n P_n(x).$$\n\n## Remarks\n\nThe factor $(2n+1)/2$ comes from the orthogonality relations $\\int_{-1}^{1} P_m P_n \\, dx = \\frac{2}{2n+1} \\delta_{mn}$.\nFor $f$ in the [[space|SASP14MA1Z]] $L^2([-1, 1])$, the series converges to $f$ in $L^2$, because the Legendre polynomials form a complete orthogonal system there.\n\nExample: let $f(x) = x^2$.\nThen $a_0 = \\frac12 \\int_{-1}^1 x^2 \\, dx = \\frac13$, $a_1 = \\frac32 \\int_{-1}^1 x^3 \\, dx = 0$, and $a_2 = \\frac52 \\int_{-1}^1 x^2 \\cdot \\frac{3x^2 - 1}{2} \\, dx = \\frac52 \\cdot \\frac{4}{15} = \\frac23$.\nAll $a_n$ with $n \\ge 3$ vanish, since $x^2$ is a combination of $P_0$ and $P_2$ and is orthogonal to $P_n$ for $n \\ge 3$.\nSo the series is $\\frac13 P_0 + \\frac23 P_2$, which equals $x^2$.\n\nNon-example: $f(x) = 1/(1 - x)$ on $[-1, 1)$ is not integrable near $x = 1$.\nAlready $a_0 = \\frac12 \\int_{-1}^{1} \\frac{dx}{1 - x}$ diverges, so $f$ has no Fourier–Legendre series.\n",
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