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  "body": "\n## Definition\n\nLet $f \\colon \\R \\to \\C$ be a [[periodic function|089KWVYD69]] of period $2\\pi$ that is Lebesgue [[integrable|X7GTF3D98F]] on $[-\\pi, \\pi]$, and let $n \\in \\Z$.\nThe $n$-th *Fourier coefficient* of $f$ is the complex number\n$$\\hat f(n) = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f(x) e^{-inx} \\, dx,$$\nwhere the integral is the [[Lebesgue integral|DWJF7ABPR1]].\n\n## Remarks\n\nIn terms of the [[Fourier monomials|TZYSW0975T]] $e_n$, $\\hat f(n) = \\langle f, e_n \\rangle$ for the inner product $\\langle f, g \\rangle = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f \\overline{g}$.\nBy periodicity the integral may be taken over any interval of length $2\\pi$.\nThe bound $\\abs{\\hat f(n)} \\le \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\abs{f}$ holds for every $n$.\nExample: for $f(x) = \\cos x$, the formula $\\cos x = \\frac{1}{2}(e^{ix} + e^{-ix})$ and the orthogonality of the $e_n$ give $\\hat f(1) = \\hat f(-1) = \\frac{1}{2}$ and $\\hat f(n) = 0$ for $\\abs{n} \\ne 1$.\nNon-example: the $2\\pi$-periodic function equal to $1/x$ on $(0, 2\\pi]$ is not integrable on a period, so its Fourier coefficients are not defined.\nThe real coefficients $a_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x) \\cos nx \\, dx$ and $b_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x) \\sin nx \\, dx$ satisfy $a_n = \\hat f(n) + \\hat f(-n)$ and $b_n = i(\\hat f(n) - \\hat f(-n))$.\nOther notations are $c_n$ and $\\mathcal{F}f(n)$; for period $T$ the coefficient is $\\frac{1}{T} \\int_0^T f(x) e^{-2\\pi i n x / T} \\, dx$.\n",
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