LaTeX
f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat f(\omega)e^{i\omega t}\,d\omegaVariables
- f̂(ω): Fourier transform
- t: original-domain variable
How to use this formula
Reconstructs a function from its angular-frequency representation.
Important notes
- The 2π factor depends on the transform convention.
Quick example
Apply the inverse transform to recover a signal after frequency-domain filtering.
Applicability, worked calculation, and verification
Assumptions and domain checks
- The 2π factor depends on the transform convention.
- For the Inverse Fourier Transform, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- Integration limits, the differential, and any constant of integration must be included where required.
- Check the domain and the convergence, continuity, differentiability, or measurability conditions used by the result.
Worked example
Apply the inverse transform to recover a signal after frequency-domain filtering.
Common mistakes
- When copying Inverse Fourier Transform, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Verify the result of Inverse Fourier Transform with a known case, inverse operation, dimensional check, or independent calculation before publishing it.
Continue the workflow
Use Inverse Fourier Transform in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat f(\omega)e^{i\omega t}\,d\omega. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Inverse Fourier Transform used for?
Reconstructs a function from its angular-frequency representation.
Can I copy this formula as LaTeX?
Yes. Copy f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat f(\omega)e^{i\omega t}\,d\omega or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.