laplace - Maple Help
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inttrans

 laplace
 Laplace transform

 Calling Sequence laplace(expr, t, s)

Parameters

 expr - expression, equation, or set of expressions and/or equations to be transformed t - variable expr is transformed with respect to t s - parameter of transform opt - option to run this under (optional)

Description

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

$F\left(s\right)={\int }_{0}^{\mathrm{\infty }}f\left(t\right){ⅇ}^{-st}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt$

 • 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.

Examples

 > with(inttrans):
 > laplace(t^2+sin(t)=y(t), t, s);
  (1)
 > laplace(t^(3/2)-exp(t)+sinh(a*t), t, s);
 $\frac{{3}{}\sqrt{{\mathrm{\pi }}}}{{4}{}{{s}}^{{5}}{{2}}}}{-}\frac{{1}}{{s}{-}{1}}{+}\frac{{a}}{{-}{{a}}^{{2}}{+}{{s}}^{{2}}}$ (2)
 > laplace(diff(y(t), t\$2)-y(t)=sin(a*t), t, s-2);
 ${\left({s}{-}{2}\right)}^{{2}}{}{?}{-}{\mathrm{D}}{}\left({y}\right){}\left({0}\right){-}\left({s}{-}{2}\right){}{y}{}\left({0}\right){-}{?}{=}\frac{{a}}{{\left({s}{-}{2}\right)}^{{2}}{+}{{a}}^{{2}}}$ (3)
 > laplace(BesselI(0,a*t), t, s);
 $\frac{{1}}{\sqrt{{-}{{a}}^{{2}}{+}{{s}}^{{2}}}}$ (4)
 > laplace(Heaviside(t-c)*f(t),t,s);
 ${ℒ}{}\left({\mathrm{Heaviside}}{}\left({t}{-}{c}\right){}{f}{}\left({t}\right){,}{t}{,}{s}\right)$ (5)
 > assume(c,positive);
 > laplace(Heaviside(t-c)*f(t),t,s);
 ${{ⅇ}}^{{-}{s}{}{\mathrm{c~}}}{}{?}$ (6)
 > addtable(laplace,myfunc(t),Myfunc(s),t,s):
 > laplace(t^3*exp(a*t)*myfunc(4*t),t,w);
 ${-}\frac{{{\mathrm{D}}}^{\left({3}\right)}{}\left({\mathrm{Myfunc}}\right){}\left(\frac{{w}}{{4}}{-}\frac{{a}}{{4}}\right)}{{256}}$ (7)
 > addtable(laplace,myfunc2(t*a)^n,1/((abs(n)+1)/2)!*Myfunc2(s)+a,t,s,{a,n},          n::odd):
 > laplace(myfunc2(4*t)^7,t,w);
 $\frac{{\mathrm{Myfunc2}}{}\left({w}\right)}{{24}}{+}{4}$ (8)

Compatibility

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

 See Also