InverseLaplaceTransform
InverseLaplaceTransform[expr,s,t]
gives the inverse Laplace transform of expr.
InverseLaplaceTransform[expr,{s1,s2,…},{t1,t2,…}]
gives the multidimensional inverse Laplace transform of expr.
Details and Options

- The inverse Laplace transform of a function
is defined to be
, where γ is an arbitrary positive constant chosen so that the contour of integration lies to the right of all singularities in
.
- The multidimensional inverse 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 - In TraditionalForm, InverseLaplaceTransform is output using ℒ-1. »

Examples
open allclose allBasic Examples (4)
Scope (46)
Basic Uses (3)
Compute the inverse Laplace transform of a function for a symbolic parameter t:
Use a numerical value for the parameter:
TraditionalForm formatting:
Rational Functions (5)
Elementary Functions (5)
Logarithmic Functions (4)
Special Functions (5)
Function involving BesselK:
Ratio of an incomplete gamma function and a power function:
Inverse of a polygamma function:
Piecewise Functions (5)
Periodic Functions (4)
Generalized Functions (3)
Multivariate Functions (8)
Inverse Laplace transform of a bivariate rational function:
Rational function that is separable with respect to p and q:
Rational function that is not separable with respect to p and q:
Rational function whose inverse is related to BesselJ:
Inverse Laplace transform of a bivariate function involving a radical:
Inverse Laplace transform of a multivariate function in three variables:
Numerical Inversion (4)
Calculate the inverse Laplace transform at a single point:
Alternatively, calculate the inverse Laplace transform symbolically:
Then evaluate it for a specific value of t:
Plot the inverse Laplace transform numerically and compare it with the exact result:
Function whose inverse is a piecewise function with respect to t:
Function whose inverse is a periodic function with respect to t:
Options (3)
GenerateConditions (1)
By default, InverseLaplaceTransform assumes that the result is defined for non-negative t:
Use GenerateConditions to obtain the range of validity for the result:
Method (1)
Use the default method for numerical evaluation:
Use Method to obtain the result from all six different methods:
Working Precision (1)
Use WorkingPrecision to obtain a result with arbitrary precision:
Applications (2)
Compute the step response to the linear system with transfer function :
Solve a differential equation using Laplace transforms:
Solve for the Laplace transform:
Find the solution directly using DSolve:
Properties & Relations (2)
Use Asymptotic to compute an asymptotic approximation:
InverseLaplaceTransform and LaplaceTransform are mutual inverses:
Neat Examples (2)
Text
Wolfram Research (1999), InverseLaplaceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseLaplaceTransform.html (updated 2020).
BibTeX
BibLaTeX
CMS
Wolfram Language. 1999. "InverseLaplaceTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/InverseLaplaceTransform.html.
APA
Wolfram Language. (1999). InverseLaplaceTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseLaplaceTransform.html