FourierTransform

The Fourier transform under Wolfram's default convention ($\mathrm{FourierParameters} \to \{0, 1\}$) — $F(w) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t) e^{iwt}\,dt$ — as a rule table over the standard pairs plus linearity and the modulation (shift) theorem.

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

FourierTransform(f, t, w)$F(w) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t) e^{iwt}\,dt$, Wolfram's default FourierParameters -> {0, 1}.

Typed signatures

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

Domain: Transforms

Details

• Covered: DiracDelta(t - a), UnitStep(t), $\sin(at)$, $\cos(at)$, the Gaussian $e^{-at^2}$ ($\mathrm{Re}(a) > 0$, checked via isPositive), $e^{-a|t|}$ ($a > 0$), a constant, sums (linearity), a constant factor, and the modulation theorem — an $e^{iat}$ factor shifts $w \to w + a$ in the rest's transform, covering $e^{iat}$ alone ($\to \sqrt{2\pi}\,\delta(w+a)$) and further products with the table above.
• Declined: an opaque function; UnitStep(t - a) for $a \neq 0$ (only the unshifted UnitStep(t) is covered — DiracDelta(t - a) for any a is covered, since its transform is just a phase); a product of two independently-transformable pieces outside the modulation shape; any FourierParameters other than the default (a 4th argument is declined outright).

Examples

Basic #

$\mathcal{F}\{\delta(t)\} = 1/\sqrt{2\pi}$

live ↗

In$\mathrm{FourierTransform}(\mathrm{DiracDelta}(t), t, w)$Out$\sqrt{\frac{1}{2π}}$

$\mathcal{F}\{e^{-t^2}\} = \frac{1}{\sqrt{2}}e^{-w^2/4}$ ($a = 1 > 0$, checked via isPositive)

live ↗

In$\mathrm{FourierTransform}(\exp(-t^2), t, w)$Out$\frac{1}{2}(\sqrt{2}\exp(-\frac{w^2}{4}))$

$\mathcal{F}\{\cos(at)\} = \sqrt{\pi/2}\,(\delta(w-a)+\delta(w+a))$

live ↗

In$\mathrm{FourierTransform}(\cos(at), t, w)$Out$\frac{1}{2}(\sqrt{2}(\mathrm{DiracDelta}(a+w)+\mathrm{DiracDelta}(w-a))\sqrt{π})$

Scope #

live ↗

In$\mathrm{FourierTransform}((xyz)^3\exp(-x^2-y^2-z^2), \bigl\lbrack x, y, z\bigr\rbrack, \bigl\lbrack u, v, w\bigr\rbrack)$Out$\mathrm{FourierTransform}(x^3y^3z^3\exp(-x^2-y^2-z^2), \bigl\lbrack x, y, z\bigr\rbrack, \bigl\lbrack u, v, w\bigr\rbrack)$

live ↗

In$\mathrm{FourierTransform}(\frac{\mathrm{SquareWave}(\frac{y}{2π})}{e^{\vert x\vert}}, \bigl\lbrack x, y\bigr\rbrack, \bigl\lbrack u, v\bigr\rbrack)$Out$\mathrm{FourierTransform}(\mathrm{SquareWave}(\frac{y}{2π})\exp(-\vert x\vert), \bigl\lbrack x, y\bigr\rbrack, \bigl\lbrack u, v\bigr\rbrack)$

About

Examples	5
Checked against	Wolfram Language 2 agree2 differ1 unevaluated
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/transforms.ts

See also: InverseFourierTransform, LaplaceTransform