\hat f(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dtVariables
- f(t): input function
- ω: angular frequency
- i: imaginary unit
How to use this formula
Transforms a time- or space-domain function into frequency components.
Important notes
- Normalization conventions differ by field.
Quick example
A Gaussian transforms to another Gaussian under common conventions.
Applicability, worked calculation, and verification
Domain and applicability
Apply Fourier Transform only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- Normalization conventions differ by field.
- 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.
Do not use this formula when
- Do not use Fourier Transform when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Fourier Transform, check zero, negative, and extreme input values before relying on the result.
- When using Fourier Transform, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form \hat f(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
Using F(ω)=∫f(t)e^(-iωt)dt, the transform of f(t)=e^(-t²/2) is √(2π)e^(-ω²/2).
- Complete the square in -t²/2-iωt.
- Shift the Gaussian integral; the oscillatory term contributes e^(-ω²/2).
- Use ∫e^(-u²/2)du=√(2π).
Independent verification
At ω=0, the transform equals the area under the Gaussian, √(2π), as the formula predicts.
Common mistakes
- Keep the integration bounds and differential attached to Fourier Transform, and include a constant of integration for an indefinite integral.
- Verify the result of Fourier Transform with a known case, inverse operation, dimensional check, or independent calculation before publishing it.
Continue the workflow
Use 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
\hat f(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Fourier Transform.
Verified against: NIST Digital Library of Mathematical Functions
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 Fourier Transform used for?
Transforms a time- or space-domain function into frequency components.
Can I copy this formula as LaTeX?
Yes. Copy \hat f(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt 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.