{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $f \\colon \\R \\to \\C$ be a [[periodic function|089KWVYD69]] of period $2\\pi$ whose real and imaginary parts are [[Riemann integrable|0N1JQXZXHC]] on $[-\\pi, \\pi]$.\nFor each integer $n$, the $n$-th *Fourier coefficient* of $f$ is\n$$\\hat f(n) = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f(x) e^{-inx} \\, dx,$$\nwhere $e^{-inx}$ is the [[complex exponential|07GJV36896]].\nThe *Fourier series* of $f$ is the formal [[series|0NM626FH6Q]]\n$$\\sum_{n = -\\infty}^{\\infty} \\hat f(n) e^{inx},$$\nwhose $N$-th partial sum is $S_N f(x) = \\sum_{n=-N}^{N} \\hat f(n) e^{inx}$.\n\n## Remarks\n\nFor real $f$ the series is often written $\\frac{a_0}{2} + \\sum_{n \\ge 1} (a_n \\cos nx + b_n \\sin nx)$, with $a_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x) \\cos nx \\, dx$ and $b_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x) \\sin nx \\, dx$.\nExample: let $f(x) = x$ for $-\\pi \u003c x \\le \\pi$, extended with period $2\\pi$.\nThen $a_n = 0$ since $f$ is odd, and integration by parts gives $b_n = \\frac{2}{\\pi} \\int_0^\\pi x \\sin nx \\, dx = \\frac{2 (-1)^{n+1}}{n}$.\nSo the Fourier series is $2 \\sum_{n \\ge 1} \\frac{(-1)^{n+1}}{n} \\sin nx$; at $x = \\pi/2$ the series converges to $\\pi/2$, which is the Leibniz formula $1 - \\frac{1}{3} + \\frac{1}{5} - \\cdots = \\frac{\\pi}{4}$.\nNon-example: the $2\\pi$-periodic function equal to $1/\\abs{x}$ for $0 \u003c \\abs{x} \\le \\pi$ is not integrable on $[-\\pi, \\pi]$, so its Fourier coefficients are not defined.\nThe series is formal: the definition says nothing about convergence, and there are continuous functions whose Fourier series diverge at some points.\nIn Lebesgue theory the same formulas define the Fourier series of every integrable $2\\pi$-periodic function.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "G12KXDHPPS",
    "kind": "definition",
    "msc": [
      "42A16",
      "42A20"
    ],
    "name": "Fourier series",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fourier_series"
      }
    ],
    "tag": "S36XX2PVNH",
    "uses": [
      "089KWVYD69",
      "0N1JQXZXHC",
      "07GJV36896",
      "0NM626FH6Q"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/S36XX2PVNH/",
  "links": [
    {
      "field": "defines",
      "target": "G12KXDHPPS"
    },
    {
      "field": "uses",
      "target": "089KWVYD69"
    },
    {
      "field": "uses",
      "target": "0N1JQXZXHC"
    },
    {
      "field": "uses",
      "target": "07GJV36896"
    },
    {
      "field": "uses",
      "target": "0NM626FH6Q"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/S36XX2PVNH.md",
  "msc": [
    "42A16",
    "42A20"
  ],
  "name": "Fourier series",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/S36/S36XX2PVNH.md",
  "status": null,
  "tag": "S36XX2PVNH",
  "uri": "m2://S36XX2PVNH",
  "url": "https://math2.tamnd.com/wiki/fourier-series-definition/",
  "writing": {
    "tag": "S36XX2PVNH",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}