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  "body": "\n## Summary\n\nThe Fourier monomials are the functions $e_n(x) = e^{inx}$, $n \\in \\Z$, viewed as functions on the circle $\\mathbb{T} = \\R / 2\\pi\\Z$.\nThey are the continuous homomorphisms from the [[circle group|04EAFVFTF7]] to the unit circle of $\\C$, that is, the characters of the circle.\nThey form an [[orthonormal basis|0S9GGTR0EX]] of $L^2(\\mathbb{T})$, and the coefficients of a function in this basis are its [[Fourier coefficients|5NN2YZZR3V]].\n\n## Explanations\n\n### L1 human\n\nThe monomial $e_n$ runs around the unit circle of the complex plane $n$ times while $x$ runs once from $0$ to $2\\pi$.\nFor $n$ negative it runs in the opposite direction, and $e_0$ is the constant $1$.\nIts real and imaginary parts are the pure waves $\\cos nx$ and $\\sin nx$.\nEvery [[trigonometric polynomial|J3NJQ5FZJ5]] is a finite combination of these monomials, as an ordinary polynomial is a combination of powers.\nFor example, $e_2 \\cdot e_{-3} = e_{-1}$, because exponents add.\n",
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