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Laplace Transform

\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} f(t) e^{-st}\, dt

Converts differential equations into algebraic equations in the s-domain, simplifying analysis of dynamic systems.

By Pierre-Simon Laplace

Formal Sciences

Laplace Transform

1785 · Pierre-Simon Laplace

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Story PortalWho discovered this?Discover how Pierre-Simon Laplace and others shaped this equation.Visual PortalWhat does it look like?See the equation come alive in the Visual Studio.Machine PortalWhere is it used?Explore machines powered by this equation — control systems.Math PortalWhat does it derive from?Trace the mathematical lineage from fourier transform.Future PortalWhere is it going?Autonomous vehicle control

Why it matters: Enabled control theory, circuit analysis, and solving complex differential equations systematically.

Discoverers: Pierre-Simon Laplace (1785)

What does it mean?

Converts differential equations into algebraic equations in the s-domain, simplifying analysis of dynamic systems.

Why should I care?

Enabled control theory, circuit analysis, and solving complex differential equations systematically.

Equation Compass

North — Prerequisites

West — History

Pierre-Simon Laplace

East — Applications

South — Derivations

Derivation

Variables & Units

Symbol	Name	Unit	Meaning
$f(t)$	Time function	—	Input in time domain
$F(s)$	Laplace transform	—	Output in s-domain
$s$	Complex frequency	—	σ + iω

Worked Example

L{e^(at)} = 1/(s-a) for Re(s) > a.

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Sources & further reading

Search Wikipedia Search Wolfram MathWorld Open in Wolfram|Alpha Watch explainers on YouTube

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