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:
  • AccuracyGoalAutomaticdigits of absolute accuracy sought
    Assumptions $Assumptionsassumptions to make about parameters
    GenerateConditions Truewhether to generate answers that involve conditions on parameters
    MethodAutomaticmethod to use
    PerformanceGoal$PerformanceGoalaspects of performance to optimize
    PrecisionGoalAutomaticdigits of precision sought
    PrincipalValue Falsewhether to find Cauchy principal value
    WorkingPrecision Automaticthe 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:

DiracDelta:

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:

Wolfram Research (2021), BilateralLaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.html.

Text

Wolfram Research (2021), BilateralLaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.html.

CMS

Wolfram Language. 2021. "BilateralLaplaceTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.html.

APA

Wolfram Language. (2021). BilateralLaplaceTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.html

BibTeX

@misc{reference.wolfram_2025_bilaterallaplacetransform, author="Wolfram Research", title="{BilateralLaplaceTransform}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.html}", note=[Accessed: 21-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_bilaterallaplacetransform, organization={Wolfram Research}, title={BilateralLaplaceTransform}, year={2021}, url={https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.html}, note=[Accessed: 21-February-2025 ]}