InverseFourierTransform

The inverse Fourier transform under Wolfram's default convention — the two closed forms it documents unconditionally ($\delta(w) \to 1/\sqrt{2\pi}$ and a constant $\to$ a scaled $\delta(t)$), nothing more.

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

InverseFourierTransform(F, w, t)$f(t)$ such that $\mathcal{F}\{f\}(w) = F(w)$, for the two forms below.

Typed signatures

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

Domain: Transforms

Details

• Covered: DiracDelta(w) $\to 1/\sqrt{2\pi}$, and a w-free constant $c \to c\sqrt{2\pi}\,\delta(t)$ (the inverse of FourierTransform's own constant rule).
• Declined: everything else — no rational-function or trig table the way InverseLaplaceTransform has one, since the corresponding FourierTransform images are themselves distributional (a sum of DiracDeltas) rather than an algebraic shape to invert termwise here.

Examples

Basic #

$\mathcal{F}^{-1}\{\delta(w)\} = 1/\sqrt{2\pi}$

live ↗

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

$\mathcal{F}^{-1}\{1\} = \sqrt{2\pi}\,\delta(t)$

live ↗

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

Scope #

A multivariate transform is recombined but stays unevaluated

live ↗

In$\mathrm{InverseFourierTransform}(-\frac{-(uvw)+u+v+w}{(u^2+1)(v^2+1)(w^2+1)}, \bigl\lbrack u, v, w\bigr\rbrack, \bigl\lbrack x, y, z\bigr\rbrack)$Out$\mathrm{InverseFourierTransform}(\frac{uvw-u-v-w}{(u^2+1)(v^2+1)(w^2+1)}, \bigl\lbrack u, v, w\bigr\rbrack, \bigl\lbrack x, y, z\bigr\rbrack)$

live ↗

In$\mathrm{InverseFourierTransform}((uvw)^5\exp(-u^2-v^2-w^2), \bigl\lbrack u, v, w\bigr\rbrack, \bigl\lbrack x, y, z\bigr\rbrack)$Out$\mathrm{InverseFourierTransform}(u^5v^5w^5\exp(-u^2-v^2-w^2), \bigl\lbrack u, v, w\bigr\rbrack, \bigl\lbrack x, y, z\bigr\rbrack)$

About

Examples	4
Checked against	Wolfram Language 2 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/transforms.ts

See also: FourierTransform, InverseLaplaceTransform