ilaplace - Maple Help

MTM

 ilaplace
 inverse Laplace integral transform

 Calling Sequence ilaplace(M) ilaplace(M,y) ilaplace(M,y, x)

Parameters

 M - array or expression y - variable expr is transformed with respect to y x - variable in transformed expression

Description

 • The ilaplace(M) calling sequence computes the element-wise inverse Laplace transform of M.  The result, R, is formed as R[i,j] = ilaplace(M[i,j], y, x).
 • ilaplace(L) is the inverse Laplace transform of the scalar L with default independent variable s.  If L is not a function of s, then L is  assumed to be a function of the independent variable returned by findsym(L,1).The default return is a function of t.
 • If L = L(t), then ilaplace returns a function of x.
 • By definition,

$F\left(t\right)={{\int }}_{c-\mathrm{\infty }I}^{c+\mathrm{\infty }I}L\left(s\right){ⅇ}^{st}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}s$,

where c is a real number selected so that all singularities of L(s) are to the left of the line s = c and the integration above proceeds with respect to s.

 • ilaplace(L,y) makes F a function of the variable y instead of the default t.
 • ilaplace(L,y,x) takes L to be a function of x instead of the default t. The integration is then with respect to y.

Examples

 > $\mathrm{with}\left(\mathrm{MTM}\right):$
 > $\mathrm{ilaplace}\left(\frac{1}{{s}^{2}+4}\right)$
 $\frac{{\mathrm{sin}}{}\left({2}{}{t}\right)}{{2}}$ (1)
 > $\mathrm{ilaplace}\left(\frac{1}{{t}^{2}+4}\right)$
 $\frac{{\mathrm{sin}}{}\left({2}{}{x}\right)}{{2}}$ (2)
 > $\mathrm{ilaplace}\left(\frac{1}{{s}^{2}+4},w\right)$
 $\frac{{\mathrm{sin}}{}\left({2}{}{w}\right)}{{2}}$ (3)
 > $\mathrm{ilaplace}\left(\frac{z}{{s}^{2}+4},z,q\right)$
 $\frac{{\mathrm{Dirac}}{}\left({1}{,}{q}\right)}{{{s}}^{{2}}{+}{4}}$ (4)
 > $M≔\mathrm{Matrix}\left(\left[\frac{1}{{s}^{2}+4},\frac{z}{{s}^{2}+4}\right]\right):$
 > $\mathrm{ilaplace}\left(M\right)$
 $\left[\begin{array}{cc}\frac{{\mathrm{sin}}{}\left({2}{}{t}\right)}{{2}}& \frac{{z}{}{\mathrm{sin}}{}\left({2}{}{t}\right)}{{2}}\end{array}\right]$ (5)