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  "body": "\n## Definition\n\nLet $f \\colon [0, \\pi] \\to \\R$ be Lebesgue [[integrable|X7GTF3D98F]].\nFor $n \\ge 0$ put\n$$a_n = \\frac{2}{\\pi} \\int_0^\\pi f(x) \\cos nx \\, dx, \\qquad b_n = \\frac{2}{\\pi} \\int_0^\\pi f(x) \\sin nx \\, dx.$$\nThe *Fourier cosine series* of $f$ is the [[trigonometric series|FZK2M8KGXQ]] $\\frac{a_0}{2} + \\sum_{n=1}^{\\infty} a_n \\cos nx$.\nThe *Fourier sine series* of $f$ is the trigonometric series $\\sum_{n=1}^{\\infty} b_n \\sin nx$.\n\n## Remarks\n\nThe cosine series is the [[Fourier series|G12KXDHPPS]] of the even $2\\pi$-periodic function equal to $f$ on $[0, \\pi]$.\nThe sine series is the Fourier series of the odd $2\\pi$-periodic function equal to $f$ on $(0, \\pi)$.\nExample: for $f(x) = x$ on $[0, \\pi]$, integration by parts gives $b_n = \\frac{2(-1)^{n+1}}{n}$, so the sine series is $2 \\sum_{n \\ge 1} \\frac{(-1)^{n+1}}{n} \\sin nx$.\nFor the same $f$, $a_0 = \\pi$, $a_n = 0$ for even $n \\ge 2$ and $a_n = -\\frac{4}{\\pi n^2}$ for odd $n$.\nNon-example: $f(x) = 1/x$ on $(0, \\pi]$ is not integrable on $[0, \\pi]$, so its sine and cosine coefficients are not defined.\n",
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