# LaplaceTransform

LaplaceTransform[expr,t,s]

gives the Laplace transform of expr.

LaplaceTransform[expr,{t1,t2,},{s1,s2,}]

gives the multidimensional Laplace transform of expr.

# Details and Options • The Laplace transform of a function is defined to be .
• The multidimensional Laplace transform is given by .
• The lower limit of the integral is effectively taken to be , so that the Laplace transform of the Dirac delta function is equal to 1. »
• Assumptions and other options to Integrate can also be given in LaplaceTransform. »
• In TraditionalForm, LaplaceTransform is output using . »

# Examples

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## Scope(9)

Elementary functions:

Special functions:

Piecewise functions:

In these distributions, the integration region is taken to start at :

Periodic functions:

Multivariate functions:

Integrals:

## Options(2)

### Assumptions(1)

Specify the range for a parameter using Assumptions:

### GenerateConditions(1)

Use to get parameter conditions for when a result is valid:

## Applications(1)

Solve a differential equation using Laplace transforms:

Solve for the Laplace transform:

Find the inverse transform:

Find the solution directly using DSolve:

## Properties & Relations(3)

Use Asymptotic to compute an asymptotic approximation:

LaplaceTransform and InverseLaplaceTransform are mutual inverses:

Use NIntegrate for numerical approximation:

NIntegrate computes the transform for numeric values of the Laplace parameter s:

## Possible Issues(1)

Simplification can be required to get back the original form:

## Neat Examples(1)

LaplaceTransform done in terms of MeijerG:

Introduced in 1999
(4.0)