gives the Laplace transform of expr.


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 . »


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Basic Examples  (3)

Scope  (9)

Elementary functions:

Special functions:

Piecewise functions:

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

Periodic functions:

Multivariate functions:


LaplaceTransform threads itself over equations:

TraditionalForm formatting:

Options  (2)

Assumptions  (1)

Specify the range for a parameter using Assumptions:

GenerateConditions  (1)

Use GenerateConditions->True 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