The function invlaplace computes the inverse Laplace transform of expr with respect to s.

Expressions which are sums of rational functions of polynomials can be transformed.  Some expressions which involve exponentials, trigonometric, Bessel functions, ln, hypergeometric, gamma, Kelvin functions, and special cases of the Whittaker function can also be transformed.

Both laplace and invlaplace recognize the Dirac-delta (or unit-impulse) function as Dirac(t) and Heaviside's unit step function as Heaviside(t).

The global variables _U1, _U2, _U, ... are used as integration variables if the answer contains convolution integrals.

Users can add their own functions to invlaplace's internal lookup tables with the function addtable.

If the option opt is set to 'NO_INT', then the program will not resort to integration of the original problem if all other methods fail.  This will increase the speed at which the transform will run.

The command with(inttrans,invlaplace) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{inttrans}\right)\:$

$\mathrm{invlaplace}\left(\frac{1}{s-a}+\frac{1}{s^2+b}+1\,s\,t\right)$

${\ⅇ}^{at}+\frac{\sin \left(\sqrt{b}t\right)}{\sqrt{b}}+\mathrm{Dirac}\left(t\right)$

$\mathrm{invlaplace}\left(\frac{1}{s^3+a}+\mathrm{laplace}\left(y\left(t\right)\,t\,s\right)\,s\,t+2\right)$

$\frac{{\ⅇ}^{-a^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(t+2\right)}+{\ⅇ}^{\frac{a^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(t+2\right)}{2}}\left(-\cos \left(\frac{\sqrt{3}{a}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(t+2\right)}{2}\right)+\sqrt{3}\sin \left(\frac{\sqrt{3}{a}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(t+2\right)}{2}\right)\right)}{3a^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}+y\left(t+2\right)$

$\mathrm{invlaplace}\left(\frac{s^2}{{\left(s^2+a^2\right)}^{\frac{3}{2}}}\,s\,t\right)$

$-\mathrm{BesselJ}\left(1\,at\right)ta+\mathrm{BesselJ}\left(0\,at\right)$

$\mathrm{invlaplace}\left(\frac{s}{s-1}\mathrm{laplace}\left(F\left(t\right)\,t\,s\right)\,s\,t\right)$

$F\left(t\right)+{\int }_0^{t}F\left(\mathrm{\_U1}\right){\ⅇ}^{t-\mathrm{\_U1}}\ⅆ\mathrm{\_U1}$

$\mathrm{addtable}\left(\mathrm{invlaplace}\,{\mathrm{myfunc}\left(bt+a\right)}^n\,\frac{\mathrm{Myfunc}\left(s\,b\right)}{n!}+a\cdot 3\,t\,s\,\left\{a\,b\,n\right\}\,n::\mathrm{posint}\right)\:$

$\mathrm{invlaplace}\left(\frac{\exp \left(-3s\right)\mathrm{myfunc}\left(2s\right)}{s}\,s\,t\right)$

$\mathrm{Heaviside}\left(t-3\right)\left({\int }_0^{t-3}\mathrm{Myfunc}\left(\mathrm{\_U1}\,2\right)\ⅆ\mathrm{\_U1}\right)$

$\mathrm{invlaplace}\left({\mathrm{myfunc}\left(s+5\right)}^7\,s\,t\right)$

$\frac{\mathrm{Myfunc}\left(t\,1\right)}{5040}+15$

The inttrans[invlaplace] command was updated in Maple 2019.