# 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:
•  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 WorkingPrecision Automatic the precision used in internal computations

# Examples

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## 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:

## 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_2024_inversebilaterallaplacetransform, author="Wolfram Research", title="{InverseBilateralLaplaceTransform}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/InverseBilateralLaplaceTransform.html}", note=[Accessed: 07-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_inversebilaterallaplacetransform, organization={Wolfram Research}, title={InverseBilateralLaplaceTransform}, year={2021}, url={https://reference.wolfram.com/language/ref/InverseBilateralLaplaceTransform.html}, note=[Accessed: 07-September-2024 ]}