The Laplace transform is an integral transform widely used to solve various physical problems. It is particular useful to solve linear ordinary differential equations, such as those encountered in the analysis of e electronic circuits.

Denoted as , the Laplace transform transforms a function from time domain to complex frequency domain. A common notation for the Laplace transform is to represent the transformed function as the uppercase version of the original time-domain function:

Where is the original function in the time domain, and is the transformed function in the complex frequency domain, with being a complex variable.

not to be confused with the Lie derivative, which is also commonly denoted

Note that the result of a Laplace transform includes not only the algebraic expression but also the region of convergence (ROC).

Subtopics

Definition

The unilateral Laplace transform is the most commonly used form of the Laplace transform and is typically what people refer to as “the” Laplace transform.

where

The unilateral Laplace transform is only appropriate for analyzing causal signals. There is also a bilateral Laplace transform defined as the following:

Properties

Below are some of the properties of the unilateral Laplace transform. The properties of bilateral Laplace transform are similar, but there are some important differences.

OperationsTime DomainComplex Frequency DomainComments
linear combination
time differentiation
frequency differentiation
time integration
frequency integration
time shifting
frequency shifting
time scaling
time convolution
multiplicationfrequency convolution

Tables

A table of several important unilateral Laplace transforms is given below.

FunctionRegion of convergence
constant1
linear
n-th power
exponential
sine
cosine
unit impulseall s
unit step

Laplace Transform to Solve Differential Equation

The Laplace transform is a powerful tool for solving differential equations. When applied to a differential equation, it typically converts the problem into an algebraic equation that we can readily solve with algebra.

Reference