FourierCoefficient

The $n$-th complex Fourier coefficient of $f$ on $[-\pi, \pi]$, $c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)\, e^{-inx}\,dx$, for a concrete integer $n$ — a curated set of closed forms, not compute-engine's Integrate.

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

FourierCoefficient(f, x, n)$c_n$, for a concrete integer $n$.

Typed signatures

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

Domain: Transforms

Details

• Covered, combined by linearity (a sum, a negation, a constant factor free of $x$):
• - a trig monomial $\cos(x)^p \sin(x)^q$ (nonnegative integer $p, q$ — a plain $\cos(x)$ or $\sin(x)$ is $p=1,q=0$ or $p=0,q=1$), via the exponential binomial expansion $\cos(x) = (E+E^{-1})/2$, $\sin(x) = (E-E^{-1})/(2i)$, $E = e^{ix}$ -- exact, and what covers a "sin/cos polynomial" like $\cos^2(x)$ (checked against Wolfram directly).
• - $\cos(mx)$ / $\sin(mx)$ for a nonzero integer $m$ -- one resonant pair of terms.
• - $x$, $x^2$, $|x|$: the standard sawtooth, parabola, and triangle-wave coefficients.
• - $\sqrt[m]{x}$ for odd $m$ (the real root, an odd function): $c_k$ in $\Gamma$ and the exponential integral $E_{-1/m}(\pm ik\pi)$, checked against Wolfram and numerical quadrature.
• - $e^{ax}$ for a nonzero real constant $a$: $c_n = (-1)^n \sinh(a\pi)/(\pi(a - in))$.
• Declined, on purpose: a symbolic order $n$ (Wolfram answers with a Piecewise closed form there, which is a materially bigger derivation than any single one of the atoms above and not attempted here); $x^k$ for $k > 2$; a mixed polynomial-times-trig product like $x\cos(x)$; and anything else outside the list -- correctness over coverage.
• Deliberately NOT built on compute-engine's own Integrate, which probing found both too weak to reach most of these forms (it declines x^2 * Cos(2x) outright) and, on one it does answer, silently wrong -- Integrate(Abs(x) * Exp(-ix), {x, -pi, pi}) comes back exactly half of what an independent numeric quadrature gives.

Examples

Basic #

$c_2$ of the sawtooth $x$ is $i(-1)^2/2 = i/2$

live ↗

In$\mathrm{FourierCoefficient}(x, x, 2)$Out$\frac{1}{2}i$

Coefficient 5 of $t^2+t$ is complex, $-\frac{2}{25} - \frac{i}{5}$

live ↗

In$\mathrm{FourierCoefficient}(t^2+t, t, 5)$Out$-\frac{2}{25}-\frac{1}{5}i$

Scope #

$c_2$ of $\cos^2(x) = \tfrac12+\tfrac12\cos(2x)$ is $1/4$ -- a sin/cos polynomial, via the binomial expansion

live ↗

In$\mathrm{FourierCoefficient}(\cos(x)^2, x, 2)$Out$\frac{1}{4}$

$c_0$ of $x^2$ is its average value, $\pi^2/3$

live ↗

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

$c_0$ of the triangle wave $|x|$ is $\pi/2$

live ↗

In$\mathrm{FourierCoefficient}(\vert x\vert, x, 0)$Out$\frac{π}{2}$

$c_1$ of $|x|$ is $-2/\pi$ -- checked against an independent numeric quadrature, since Wolfram's own Integrate gets this one wrong by a factor of 2

live ↗

In$\mathrm{FourierCoefficient}(\vert x\vert, x, 1)$Out$-\frac{2}{π}$

$c_1$ of $e^x$, algebraically equal to Wolfram's own $i(-1)^1\sinh(\pi)/((i+1)\pi)$

live ↗

In$\mathrm{FourierCoefficient}(e^{x}, x, 1)$Out$\frac{1}{π}((-\frac{1}{2}-\frac{1}{2}i)\sinh(π))$

live ↗

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

Properties #

Coefficient 3 follows the same $-\frac{2}{n^2} - \frac{i}{n}$ pattern

live ↗

In$\mathrm{FourierCoefficient}(t^2+t, t, 3)$Out$-\frac{2}{9}-\frac{1}{3}i$

live ↗

In$\sum_{n=-2}^2\mathrm{FourierCoefficient}(t^2+t, t, n)\exp(i nt)$Out$(-2-i)\exp(i t)+(\frac{1}{2}-\frac{1}{2}i)\exp(-2i t)+(\frac{1}{2}+\frac{1}{2}i)\exp(2i t)+(-2+i)\exp(-i t)+\frac{π^2}{3}$

Possible issues #

Declines a symbolic order $n$ -- not cheap the way a concrete integer is

live ↗

In$\mathrm{FourierCoefficient}(x, x, n)$Out$\mathrm{FourierCoefficient}(x, x, n)$

Declines $\tan(x)$ -- it has poles inside $[-\pi, \pi]$, so the defining integral doesn't actually converge, even though Wolfram formally returns 0 by symmetry

live ↗

In$\mathrm{FourierCoefficient}(\tan(x), x, 1)$Out$\mathrm{FourierCoefficient}(\tan(x), x, 1)$

About

Examples	12
Checked against	Wolfram Language 10 agree2 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: FourierSeries, Fourier