---
title: "The Laplace Transform"
description: "An improper integral converts a function of time into a function of a new variable, turning differentiation into multiplication. Its usefulness rests on the transform being reversible."
canonical: https://lightmysky.com/learn/mathematics/the-laplace-transform-mt_8j8GXzITN0
source: https://lightmysky.com/learn/mathematics/the-laplace-transform-mt_8j8GXzITN0.md
retrieved: 2026-09-12
---

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# The Laplace Transform

An improper integral converts a function of time into a function of a new variable, turning differentiation into multiplication. Its usefulness rests on the transform being reversible.

Subject: Mathematics · Area: Differential Equations · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/the-laplace-transform-mt_8j8GXzITN0

## Ready when they can

- Compute the transform of a simple function from the defining integral
- Use the derivative rule to see why the transform turns calculus into algebra
- Read an inverse transform off a table after partial fractions

## Lesson: Turning calculus into algebra

You start with a function of time, f(t). Multiply it by e to the power minus s t. Then integrate that product from zero to infinity. The result is a new function of s, written F(s). This recipe is the Laplace transform.

**Example.** Take f(t) equal to e to the power a t. The integral becomes e to the power (a minus s) t from zero to infinity. That works out to 1 over (s minus a). It converges only when s is bigger than a. For e to the power 3t, the transform is 1 over (s minus 3), valid for s above 3.

Differentiation in the t world becomes multiplication in the s world. The rule is that L of f prime equals s times F(s) minus f(0). Because of it, a differential equation turns into an algebra equation. You solve for F(s) with ordinary algebra, using the starting values directly.

**Example.** To go back, split F(s) into simple pieces with partial fractions. Then match each piece to a table entry. For example, 1 over ((s minus 1)(s plus 2)) splits into one third over (s minus 1) minus one third over (s plus 2). Reading the table backward gives one third e to the t minus one third e to the minus 2t.

**Recap.** The Laplace transform trades a time function for an s function, swaps derivatives for multiplication, and inverts through partial fractions and a table.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Series Solutions About an Ordinary Point](https://lightmysky.com/learn/mathematics/series-solutions-about-an-ordinary-point-mt_E6NiSd7MED)
- [Improper Integrals and Their Convergence](https://lightmysky.com/learn/mathematics/improper-integrals-and-their-convergence-mt_gvtZPFRA6Z)
- [Partial Fractions for Rational Integrands](https://lightmysky.com/learn/mathematics/partial-fractions-for-rational-integrands-mt_PA8qxs424c)

## Opens up

- [Solving Initial Value Problems with Laplace Transforms](https://lightmysky.com/learn/mathematics/solving-initial-value-problems-with-laplace-transforms-mt_0BhjPVNUwl)