InverseBilateralLaplaceTransform

InverseBilateralLaplaceTransform[expr,s,t]

gives the inverse bilateral Laplace transform of expr.

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

gives the multidimensional inverse bilateral Laplace transform of expr.

Details and Options

  • The inverse bilateral Laplace transform of a function is defined to be , where the integration is along a vertical line , lying in a strip in which the function is holomorphic. In some cases, the strip of analyticity may extend to a half-plane.
  • The multidimensional inverse bilateral Laplace transform of a function is given by a contour integral of the form .
  • The integral is computed using numerical methods if the third argument, , is given a numerical value.
  • The following options can be given:
  • AccuracyGoalAutomaticdigits of absolute accuracy sought
    Assumptions $Assumptionsassumptions to make about parameters
    GenerateConditionsFalsewhether to generate answers that involve conditions on parameters
    MethodAutomaticmethod to use
    PerformanceGoal$PerformanceGoalaspects of performance to optimize
    PrecisionGoalAutomaticdigits of precision sought
    WorkingPrecisionAutomaticthe precision used in internal computations

Examples

open allclose all

Basic Examples  (2)

Inverse bilateral Laplace transform of a function:

Function with a parameter:

Plot the result:

Scope  (13)

Inverse bilateral Laplace transform of rational function with two real poles:

Rational function with two real and two complex poles:

The following function has two real and four complex poles:

Inverse bilateral Laplace transform of a product of rational and exponential functions:

A rational function with different strips of convergences has different inverse bilateral Laplace transforms:

Rational function whose region of convergence is in the left half-plane:

Function with region of convergence in the right half-plane:

The inverse bilateral Laplace transform of the following rational function is a decaying sinusoidal wave:

Inverse bilateral Laplace transform of a function that is analytic in the whole complex plane:

Inverse bilateral Laplace transform leading to a Gaussian function:

Inverse bilateral Laplace transform of a constant is a Dirac delta function:

Evaluate the inverse bilateral Laplace transform at a single point:

Inverse bilateral Laplace transform at a single point for an analytic function:

Options  (3)

Assumptions  (3)

Specify the range for a parameter using Assumptions:

Use Assumptions to place a pole outside the strip of convergence:

Use Assumptions to restrict the right end of the convergence strip in the left half-plane:

Properties & Relations  (1)

InverseBilateralLaplaceTransform and BilateralLaplaceTransform are mutual inverses:

Neat Examples  (1)

Create a table of basic inverse bilateral Laplace transforms:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_inversebilaterallaplacetransform, author="Wolfram Research", title="{InverseBilateralLaplaceTransform}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/InverseBilateralLaplaceTransform.html}", note=[Accessed: 21-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_inversebilaterallaplacetransform, organization={Wolfram Research}, title={InverseBilateralLaplaceTransform}, year={2021}, url={https://reference.wolfram.com/language/ref/InverseBilateralLaplaceTransform.html}, note=[Accessed: 21-February-2025 ]}