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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

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

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

 See Also