Fourier Series

Updated 01 Oct 2026

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What it is
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  represents a periodic function using a constant term and sinusoidal harmonics
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Formula
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  with period $T$ and $\omega_0=2\pi/T$ , $f(t)\sim a_0/2+\sum_{k=1}^{\infty}[a_k\cos(k\omega_0t)+b_k\sin(k\omega_0t)]$ .
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  $a_k=(2/T)\int_{t_0}^{t_0+T}f(t)\cos(k\omega_0t),dt$ .
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  $b_k=(2/T)\int_{t_0}^{t_0+T}f(t)\sin(k\omega_0t),dt$ .
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  some materials put the mean directly in a constant called $a_0$ . Keep the coefficient convention consistent.
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Example
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  a zero-mean square wave taking values $\pm1$ has sine coefficients $b_k=4/(\pi k)$ for odd $k$ , and zero for even $k$ , with the corresponding phase/origin choice.
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  “Maximum harmonic $N$ ” and “number of included nonzero terms” are different quantities.
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Numerical computation
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  approximate the integrals using a stated quadrature rule and sufficient sampling resolution.
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  a uniform grid over one period should not accidentally double-count the same periodic endpoint.
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  general harmonic amplitude is $\sqrt{a_k^2+b_k^2}$ ; $|b_k|$ alone applies to a purely sine representation.
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Related
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  Gibbs Phenomenon
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  Root Mean Square Error

Linked references 2