F(s)=\int_0^{\infty}f(t)e^{-st}\,dtVariables
- f(t): time-domain function
- s: complex frequency
How to use this formula
Maps a time-domain function to a complex-frequency function useful for initial-value problems.
Important notes
- Convergence requires a suitable region in the complex plane.
Quick example
The transform of 1 is 1/s for Re(s)>0.
Applicability, worked calculation, and verification
Domain and applicability
Apply Laplace 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
- Convergence requires a suitable region in the complex plane.
- Integration limits, the differential, and any constant of integration must be included where required.
- State the domain together with any initial or boundary conditions, and confirm the regularity assumptions required by the method.
Do not use this formula when
- Do not use Laplace Transform when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Laplace Transform, check zero, negative, and extreme input values before relying on the result.
- When using Laplace Transform, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form F(s)=\int_0^{\infty}f(t)e^{-st}\,dt for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For f(t)=1 and Re(s)>0, F(s)=∫₀∞e^(-st)dt=[-e^(-st)/s]₀∞=1/s.
- Require Re(s)>0 so e^(-st) tends to zero.
- Integrate e^(-st) to -e^(-st)/s.
- Evaluate the endpoints to obtain 1/s.
Independent verification
Differentiating the antiderivative and checking the endpoint limit confirms the result.
Common mistakes
- Keep the integration bounds and differential attached to Laplace Transform, and include a constant of integration for an indefinite integral.
- Do not present a general solution as the requested solution before applying all initial or boundary conditions.
Continue the workflow
Use Laplace 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(s)=\int_0^{\infty}f(t)e^{-st}\,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 Laplace Transform.
Verified against: NIST Digital Library of Mathematical Functions
Automated quality check: Passed the core formula indexing gate.
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 Laplace Transform used for?
Maps a time-domain function to a complex-frequency function useful for initial-value problems.
Can I copy this formula as LaTeX?
Yes. Copy F(s)=\int_0^{\infty}f(t)e^{-st}\,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.