# LaplaceTransform

LaplaceTransform[f[t],t,s]

gives the symbolic Laplace transform of f[t] in the variable t and returns a transform 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(62)

### 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:

### 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(5)

Laplace transform of error and square root functions composition:

Bessel functions:

Products involving Bessel functions:

Sine integral function:

Laguerre polynomials:

Airy 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:

Product of UnitStep and cosine functions:

Laplace transform of Floor:

### Periodic Functions(5)

Laplace transform of SquareWave:

Full-wave-rectified function with period :

Rectified wave:

### Generalized Functions(5)

Laplace transform of HeavisideTheta:

Derivative of DiracDelta:

### Multivariate Functions(9)

Bivariate Laplace transform of a constant:

Exponential function:

Power function:

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 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: