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  "body": "\n## Definition\n\nLet $\\mathbb{T} = \\R / 2\\pi\\Z$ be the circle, and write $[x]$ for the class of $x \\in \\R$ in $\\mathbb{T}$.\nFor $n \\in \\Z$, the *Fourier monomial* of index $n$ is the function $e_n \\colon \\mathbb{T} \\to \\C$ given by\n$$e_n([x]) = e^{inx},$$\nwhere $e^{inx}$ is the [[complex exponential|07GJV36896]].\n\n## Remarks\n\nThe function is well defined: if $x' = x + 2\\pi k$ with $k \\in \\Z$, then $e^{inx'} = e^{inx} e^{2\\pi i nk} = e^{inx}$.\nIt is continuous, takes values of absolute value $1$, and satisfies $e_n e_m = e_{n+m}$ and $e_n(s + t) = e_n(s) e_n(t)$.\nSo each $e_n$ is a continuous homomorphism from $\\mathbb{T}$ to the [[circle group|04EAFVFTF7]], and every continuous homomorphism of this kind is one of the $e_n$.\nFor the inner product $\\langle f, g \\rangle = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f(x) \\overline{g(x)} \\, dx$, one has $\\langle e_n, e_m \\rangle = 1$ if $n = m$ and $0$ otherwise.\nExample: $e_3([x]) = e^{3ix} = \\cos 3x + i \\sin 3x$.\nNon-example: $x \\mapsto e^{ix/2}$ is not a function on $\\mathbb{T}$, since $x = 0$ and $x = 2\\pi$ give the same point of $\\mathbb{T}$ but the values $1$ and $-1$.\nOn the circle $\\R / T\\Z$ of length $T \u003e 0$, the Fourier monomials are $x \\mapsto e^{2\\pi i n x / T}$.\n",
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