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LaplaceTransform [f[t],t,s]

gives the symbolic Laplace transform of f[t] in the variable t as F[s] in the variable s.

LaplaceTransform [f[t],t,]

gives the numeric Laplace transform at the numerical value .

LaplaceTransform [f[t1,…,tn],{t1,…,tn},{s1,…,sn}]

gives the multidimensional Laplace transform of f[t1,…,tn].

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Basic Uses  
Elementary Functions  
Special Functions  
Show More Show More
Piecewise Functions  
Periodic Functions  
Generalized Functions  
Multivariate Functions  
Formal Properties  
Numerical Evaluation  
Fractional Calculus  
Options  
Assumptions  
GenerateConditions  
Principal Value  
Working Precision  
Applications  
Ordinary Differential Equations  
Fractional Differential Equations  
Evaluation of Integrals  
Other Applications  
Properties & Relations  
Possible Issues  
Neat Examples  
See Also
Tech Notes
Related Guides
Related Links
History
Cite this Page

LaplaceTransform

LaplaceTransform [f[t],t,s]

gives the symbolic Laplace transform of f[t] in the variable t as F[s] in the variable s.

LaplaceTransform [f[t],t,]

gives the numeric Laplace transform at the numerical value .

LaplaceTransform [f[t1,…,tn],{t1,…,tn},{s1,…,sn}]

gives the multidimensional Laplace transform of f[t1,…,tn].

Details and Options

  • Laplace transforms are typically used to transform differential and partial differential equations to algebraic equations, solve and then inverse transform back to a solution.
  • Laplace transforms are also extensively used in control theory and signal processing as a way to represent and manipulate linear systems in the form of transfer functions and transfer matrices. The Laplace transform and its inverse are then a way to transform between the time domain and frequency domain.
  • The Laplace transform of a function is defined to be .
  • The multidimensional Laplace transform is given by .
  • The integral is computed using numerical methods if the third argument, s, is given a numerical value.
  • The asymptotic Laplace transform can be computed using Asymptotic .
  • The Laplace transform of exists only for complex values of s in a half-plane .
  • 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. »
  • The following options can be given:
  • AccuracyGoal Automatic digits of absolute accuracy sought
    Assumptions $Assumptions assumptions to make about parameters
    GenerateConditions False whether to generate answers that involve conditions on parameters
    Method Automatic method to use
    PerformanceGoal $PerformanceGoal aspects of performance to optimize
    PrecisionGoal Automatic digits of precision sought
    PrincipalValue False whether to find Cauchy principal value
    WorkingPrecision Automatic the precision used in internal computations
  • Use GenerateConditions "ConvergenceRegion" to obtain the region of convergence for the Laplace transform.
  • In TraditionalForm , LaplaceTransform is output using . »

Examples

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Basic Examples  (4)

Compute the Laplace transform of a function:

Define a piecewise function:

Compute its Laplace transform:

Compute the transform at a single point:

Compute the Laplace transform of a multivariate function:

Define a multivariate piecewise function:

Compute its Laplace transform:

Scope  (67)

Basic Uses  (4)

Laplace transform of a function for a symbolic parameter s:

Laplace transforms of trigonometric functions:

Evaluate the Laplace transform for a numerical value of the parameter s:

TraditionalForm formatting:

Elementary Functions  (13)

Laplace transform of a power function:

Square root function:

Laplace transforms of polynomials:

Exponential function:

Product of an exponential and a linear function:

Expressions involving trigonometric functions:

Expressions involving hyperbolic functions:

Ratio of an exponential and a linear function:

Ratio of sine and linear functions:

Composition of elementary functions:

Logarithmic function:

Product of logarithmic and power functions:

Square of a logarithmic function:

Special Functions  (10)

Laplace transform of error and square root functions composition:

Bessel functions:

Products involving Bessel functions:

Sine integral function:

Laguerre polynomials:

Airy function:

Chebyshev polynomial:

Struve function:

Fresnel function:

Gamma function:

Hypergeometric function:

Piecewise Functions  (9)

Laplace transform of a piecewise function:

Restriction of a sine function to a half-period:

Exponential function with a left cutoff:

Triangular function:

Polynomial function with a left cutoff:

Ramp:

UnitStep :

Product of UnitStep and cosine functions:

Laplace transform of Floor :

Periodic Functions  (5)

Laplace transform of SquareWave :

TriangleWave :

SawtoothWave :

Full-wave-rectified function with period :

Rectified wave:

Generalized Functions  (5)

Laplace transform of HeavisideTheta :

DiracDelta :

Derivative of DiracDelta :

HeavisideLambda :

HeavisidePi :

Multivariate Functions  (9)

Bivariate Laplace transform of a constant:

Exponential function:

Power function:

BesselJ :

Square root:

Composition of cosine and square root:

Laplace transform of a multivariate power function:

Cosine:

Logarithm:

Formal Properties  (6)

The Laplace transform is a linear operator:

Laplace transform of is the Laplace transform of evaluated at :

Laplace transform of a first-order derivative:

Laplace transform of a second-order derivative:

Laplace transform of a product with monomials:

Laplace transform threads itself over equations:

Numerical Evaluation  (3)

Calculate the Laplace transform at a single point:

Alternatively, calculate the Laplace transform symbolically:

Then evaluate it for specific value of :

Plot the Laplace transform using numerical values only:

For some functions, the Laplace transform cannot be evaluated symbolically:

Evaluate the Laplace transform numerically and plot it:

Calculate a multivariate Laplace transform at a single point in the plane:

Fractional Calculus  (3)

Laplace transform of the MittagLefflerE functions:

ComplexPlot in the -domain:

Inverse Laplace transform to the time domain:

Laplace transform of the MittagLefflerE functions involving parameters:

Inverse Laplace transform to the time domain:

Laplace transform of the CaputoD fractional derivative:

Apply to sine function:

Compare this with the LaplaceTransform of the CaputoD derivative of the sine function:

Options  (4)

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:

Principal Value  (1)

The Laplace transform of the following function is not defined due to the singularity at :

Use PrincipalValue to obtain the Cauchy principal value for the integral:

Working Precision  (1)

Use WorkingPrecision to obtain a result with arbitrary precision:

Applications  (12)

Ordinary Differential Equations  (5)

Solve a differential equation using Laplace transforms:

Solve for the Laplace transform:

Find the inverse transform:

Plot the solution:

Find the solution directly using DSolve :

Solve the following differential equation:

Solve for the Laplace transform:

Find the inverse transform:

Plot the solution:

Solve an RL circuit to find the current :

Verify with DSolveValue :

Green's function for an RL circuit:

Use the Green's function to solve the RL circuit:

Solve a system of ODEs:

Fractional Differential Equations  (3)

Solve a fractional-order differential equation using Laplace transforms:

Solve for the Laplace transform:

Find the inverse transform:

Plot the solution:

Find the solution directly using DSolve :

Solve the following fractional integro-differential equation:

Solve for the Laplace transform:

Find the inverse transform:

Find the solution directly using DSolve :

The following equation describes a fractional harmonic oscillator of order 1.9:

Solve for the Laplace transform:

Find the inverse transform:

Plot the solution:

Find the solution directly using DSolve :

Evaluation of Integrals  (2)

Calculate the following integral:

Compute the Laplace transform and interchange the order of Laplace transform and integration:

Perform the integration over :

Use InverseLaplaceTransform to obtain the original integral:

Verify the result:

Integral involving the Bessel function:

Perform a change of variables and introduce an auxiliary variable :

Apply the Laplace transform and interchange the order of Laplace transform and integration:

Perform the integration over :

Use InverseLaplaceTransform to obtain :

The original integral equals :

Verify the result:

Other Applications  (2)

Compute a Laplace transform using a series expansion:

The odd coefficients vanish:

The transformed series can be summed using Regularization :

Verify the result directly using LaplaceTransform :

Laplace transform of Sinc using series expansions:

Odd coefficients vanish:

Verify the result:

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  (2)

LaplaceTransform done in terms of MeijerG :

Create a table of basic Laplace transforms:

History

Introduced in 1999 (4.0) | Updated in 2020 (12.2) ▪ 2023 (13.3)

Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplaceTransform.html (updated 2023).

Text

Wolfram Research (1999), LaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/LaplaceTransform.html (updated 2023).

CMS

Wolfram Language. 1999. "LaplaceTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/LaplaceTransform.html.

APA

Wolfram Language. (1999). LaplaceTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LaplaceTransform.html

BibTeX

@misc{reference.wolfram_2025_laplacetransform, author="Wolfram Research", title="{LaplaceTransform}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/LaplaceTransform.html}", note=[Accessed: 16-November-2025]}

BibLaTeX

@online{reference.wolfram_2025_laplacetransform, organization={Wolfram Research}, title={LaplaceTransform}, year={2023}, url={https://reference.wolfram.com/language/ref/LaplaceTransform.html}, note=[Accessed: 16-November-2025]}
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