FourierSeries

The order-$n$ complex exponential Fourier series of $f$ on $[-\pi, \pi]$, $\sum_{k=-n}^{n} c_k\, e^{ikx}$, each $c_k$ from FourierCoefficient.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageFourierSeries!↗

FourierSeries(f, x, n)the order-$n$ Fourier series of $f(x)$ on $[-\pi, \pi]$, for a nonnegative integer $n$.

Typed signatures

Type	Package	Applies
(expression, expression, integer<0..>) -> expression	analytic	always

Domain: Transforms

Details

• Built entirely from closed forms, not a call through compute-engine's own Integrate — probing found Integrate both too weak to reach most of these forms (it declines x^2 * Cos(2x) outright) and, worse, silently wrong on one it does answer (Integrate(Abs(x) * Exp(-ix), {x, -pi, pi}) comes back exactly half of what an independent numeric quadrature gives). See FourierCoefficient for exactly what's covered.
• Always period $2\pi$ on $[-\pi, \pi]$, matching Wolfram's own default — there's no period argument here.

Examples

Basic #

The order-2 series of the sawtooth $x$ — every $c_k$ nonzero, none of it collapses

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In$\mathrm{FourierSeries}(x, x, 2)$Out$-i\exp(i x)+(-\frac{1}{2}i)\exp(-2i x)+\frac{1}{2}i\exp(2i x)+i\exp(-i x)$

A truncated series of $t/2$ with a few exponential terms

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In$\mathrm{FourierSeries}(\frac{t}{2}, t, 3)$Out$(-\frac{1}{2}i)\exp(i t)+(-\frac{1}{4}i)\exp(-2i t)+(-\frac{1}{6}i)\exp(3i t)+\frac{1}{6}i\exp(-3i t)+\frac{1}{4}i\exp(2i t)+\frac{1}{2}i\exp(-i t)$

Scope #

$\sin(x)$ is already band-limited to $k = \pm1$, so raising the order past 1 adds nothing

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In$\mathrm{FourierSeries}(\sin(x), x, 3)$Out$(-\frac{1}{2}i)\exp(i x)+\frac{1}{2}i\exp(-i x)$

$\cos^2(x) = \tfrac12 + \tfrac14 e^{2ix} + \tfrac14 e^{-2ix}$, via the binomial expansion, not integration

live ↗

In$\mathrm{FourierSeries}(\cos(x)^2, x, 2)$Out$\frac{1}{4}(\exp(-2i x))+\frac{1}{4}(\exp(2i x))+\frac{1}{2}$

Order 0 is just $c_0$, the average value $\pi^2/3$

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In$\mathrm{FourierSeries}(x^2, x, 0)$Out$\frac{π^2}{3}$

The series of $|t|$ has a constant and $1/k^2$ odd terms

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In$\mathrm{FourierSeries}(\vert t\vert, t, 2)$Out$-\frac{1}{π}(2\exp(-i t))-\frac{1}{π}(2\exp(i t))+\frac{π}{2}$

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In$\mathrm{FourierSeries}(\exp(-t), t, 3)$Out$\frac{1}{π}((-\frac{1}{2}-\frac{1}{2}i)\sinh(π)\exp(-i t))+\frac{1}{π}((\frac{1}{5}-\frac{2}{5}i)\sinh(π)\exp(2i t))+\frac{1}{π}((-\frac{1}{10}-\frac{3}{10}i)\sinh(π)\exp(-3i t))+\frac{1}{π}((-\frac{1}{10}+\frac{3}{10}i)\sinh(π)\exp(3i t))+\frac{1}{π}((\frac{1}{5}+\frac{2}{5}i)\sinh(π)\exp(-2i t))+\frac{1}{π}((-\frac{1}{2}+\frac{1}{2}i)\sinh(π)\exp(i t))+\frac{\sinh(π)}{π}$

Possible issues #

Declines a polynomial-times-trig product like $x\cos(x)$ -- Wolfram has a closed form here, but it falls outside the atoms this covers (see FourierCoefficient)

live ↗

In$\mathrm{FourierSeries}(x\cos(x), x, 1)$Out$\mathrm{FourierSeries}(x\cos(x), x, 1)$

About

Examples	8
Checked against	Wolfram Language 7 agree1 differ
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/fourier-series.ts

See also: FourierCoefficient, Fourier