# BilateralLaplaceTransform

BilateralLaplaceTransform[expr,t,s]

gives the bilateral Laplace transform of expr.

BilateralLaplaceTransform[expr,{t1,t2,,tn},{s1,s2,,sn}]

gives the multidimensional bilateral Laplace transform of expr.

# Details and Options • The bilateral Laplace transform of a function is defined to be .
• The multidimensional bilateral Laplace transform is given by .
• The integral is computed using numerical methods if the third argument, s, is given a numerical value.
• The bilateral Laplace transform of exists only for complex values of such that . In some cases, this strip of definition may extend to a half-plane.
• • The following options can be given:
•  AccuracyGoal Automatic digits of absolute accuracy sought Assumptions \$Assumptions assumptions to make about parameters GenerateConditions True 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

# Examples

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## Basic Examples(2)

Define an exponentially decaying function on the real line:

Compute its bilateral Laplace transform:

Complex plot of the bilateral Laplace transform:

Compute the transform at a single point:

Compute the bilateral Laplace transform of a multivariate function:

## Scope(20)

### Univariate Functions(8)

Bilateral Laplace transform of the UnitStep function:

Bilateral Laplace transform of the UnitBox function:

UnitTriangle function:

Power function:

Exponential function:

Piecewise function:

Product of a cosine and an exponential function:

Complex plot of the bilateral Laplace transform:

### Multivariate Functions(2)

Bilateral Laplace transform of a multivariate function:

Multivariate piecewise function:

### Numerical Evaluation(2)

Calculate the bilateral Laplace transform at a single point:

Alternatively, calculate the Laplace transform symbolically:

Then evaluate it for a specific value of :

For some functions, the bilateral Laplace transform can be evaluated only numerically:

Plot the bilateral Laplace transform using numerical values only:

### Formal Properties(8)

BilateralLaplaceTransform is a linear operator:

Bilateral Laplace transform of is the Laplace transform of evaluated at :

Scaling:

Time shifting:

Convolution property of the bilateral Laplace transform:

Differentiation in the time domain:

Multiplication of the function by t with a positive integer power:

Integration in the time domain:

## Options(4)

### Assumptions(1)

Specify the range for a parameter using Assumptions:

### GenerateConditions(1)

Set GenerateConditions to False to obtain a result without conditions:

### PrincipalValue(1)

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

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

### WorkingPrecision(1)

Use WorkingPrecision to obtain a result with arbitrary precision:

## Properties & Relations(2)

BilateralLaplaceTransform and InverseBilateralLaplaceTransform are mutual inverses:

Use NIntegrate to obtain a numerical approximation:

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

## Neat Examples(1)

Create a table of basic bilateral Laplace transforms: