The laplace function computes the Laplace transform (F(s)) of expr (f(t)) with respect to t, using the definition:

Expressions involving a wide variety of functions including exponentials, trigonometrics, Bessel functions, error functions, and many others can be transformed.

The laplace function also recognizes derivatives (diff or Diff) and integrals (int or Int).

When transforming expressions like diff(y(t), t, s), laplace will insert the initial values $y\left(0\right)$, $\mathrm{D}\left(y\right)\left(0\right)$, etc.  $\mathrm{D}\left(y\right)\left(0\right)$ is the value of the first derivative at 0; $\mathrm{D}\left(\mathrm{D}\left(y\right)\right)\left(0\right)$ is the second derivative at 0, and so on.

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

Users can add their own functions to laplace's internal lookup table by using the addtable function.

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,laplace) allows the use of the abbreviated form of this command.

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

$\mathrm{laplace}\left(t^2+\sin \left(t\right)=y\left(t\right)\,t\,s\right)$

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

$\mathrm{laplace}\left(t^{\frac{3}{2}}-\exp \left(t\right)+\sinh \left(at\right)\,t\,s\right)$

$\frac{3\sqrt{\mathrm{\pi }}}{4s^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}-\frac{1}{s-1}+\frac{a}{-a^2+s^2}$

$\mathrm{laplace}\left(\mathrm{diff}\left(y\left(t\right)\,\mathrm{`\$`}\left(t\,2\right)\right)-y\left(t\right)=\sin \left(at\right)\,t\,s-2\right)$

${\left(s-2\right)}^2\mathrm{laplace}\left(y\left(t\right)\,t\,s-2\right)-\mathrm{D}\left(y\right)\left(0\right)-\left(s-2\right)y\left(0\right)-\mathrm{laplace}\left(y\left(t\right)\,t\,s-2\right)=\frac{a}{{\left(s-2\right)}^2+a^2}$

$\mathrm{laplace}\left(\mathrm{BesselI}\left(0\,at\right)\,t\,s\right)$

$\frac{1}{\sqrt{-a^2+s^2}}$

$\mathrm{laplace}\left(\mathrm{Heaviside}\left(t-c\right)f\left(t\right)\,t\,s\right)$

$\mathrm{laplace}\left(\mathrm{Heaviside}\left(t-c\right)f\left(t\right)\,t\,s\right)$

$\mathrm{assume}\left(c\,\mathrm{positive}\right)$

$\mathrm{laplace}\left(\mathrm{Heaviside}\left(t-c\right)f\left(t\right)\,t\,s\right)$

${\ⅇ}^{-s\mathrm{c~}}\mathrm{laplace}\left(f\left(t+\mathrm{c~}\right)\,t\,s\right)$

$\mathrm{addtable}\left(\mathrm{laplace}\,\mathrm{myfunc}\left(t\right)\,\mathrm{Myfunc}\left(s\right)\,t\,s\right)\:$

$\mathrm{laplace}\left(t^3\exp \left(at\right)\mathrm{myfunc}\left(4t\right)\,t\,w\right)$

$-\frac{{\mathrm{D}}^{\left(3\right)}\left(\mathrm{Myfunc}\right)\left(\frac{w}{4}-\frac{a}{4}\right)}{256}$

$\mathrm{addtable}\left(\mathrm{laplace}\,{\mathrm{myfunc2}\left(ta\right)}^n\,\frac{1}{\left(\frac{\mathrm{abs}\left(n\right)+1}{2}\right)!}\mathrm{Myfunc2}\left(s\right)+a\,t\,s\,\left\{a\,n\right\}\,n::\mathrm{odd}\right)\:$

$\mathrm{laplace}\left({\mathrm{myfunc2}\left(4t\right)}^7\,t\,w\right)$

$\frac{\mathrm{Myfunc2}\left(w\right)}{24}+4$

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