This is documentation for Mathematica 3, which was
based on an earlier version of the Wolfram Language.
 Calculus`LaplaceTransform` The term transform usually refers to a mathematical operation that takes a given function and returns a new function. The transformation is often done by means of an integral formula. Commonly used transforms are named after Laplace and Fourier. Transforms are frequently used to change a complicated problem into a simpler one. The simpler problem is then solved, usually using elementary algebraic means. The solution to the simpler problem is taken over to the original problem using the inverse transform. A standard example is the use of the Laplace transform to solve a differential equation. This package implements the Laplace transform. To take a Fourier transform you can use the package Calculus`FourierTransform`. The Laplace transform of a function is the function defined by . The Laplace transform and its inverse. This loads the package. In[1]:= < 3 Sin[2 Pi x]}, U[s][0]==0, U[s][1]==0}, U[s][x], x] Out[15]= The inverse transform gives the solution to the heat equation . In[16]:= (InverseLaplaceTransform[U[s_][x_], s_, t_] := u[x, t];InverseLaplaceTransform[soln, s, t]) Out[16]= Here is the transform of another common partial differential equation, the wave equation. In[17]:= eqn = LaplaceTransform[ D[u[x,t], {t,2}] == 9 D[u[x,t], {x,2}], t, s] Out[17]= DSolve gives the Laplace transform of with respect to , U[s][x]. In[18]:= soln = MapAll[Cancel, PowerExpand[DSolve[{eqn /. {u[x,0] -> 20 Sin[2 Pi x] - 10 Sin[5 Pi x], Derivative[0,1][u][x,0] -> 0}, U[s][0]==0, U[s][1]==0}, U[s][x], x]]] Out[18]= The inverse transform gives the solution to the wave equation . In[19]:= InverseLaplaceTransform[soln, s, t] Out[19]= Options for LaplaceTransform. Many transforms will be performed more quickly with DefiniteIntegral set to False. A transform that does not evaluate with this setting may evaluate if DefiniteIntegral is set to True. Here a condition for the existence of the transform is expressed using If. In[20]:= LaplaceTransform[Cos[d Sqrt[a t]], t, s, DefiniteIntegral->True] Out[20]= The Assumptions option is passed to Integrate which returns a simplified expression. In[21]:= LaplaceTransform[Cos[d Sqrt[a t]], t, s, DefiniteIntegral->True, Assumptions -> {Im[d] == 0 && Re[s/a] > 0}] Out[21]= The default setting of the option Analytic causes unrecognized functions to be treated as analytic. In[22]:= LaplaceTransform[D[g[t, x], t], t, s] Out[22]= Setting Analytic to False forces the transform to be expressed in terms of a directional limit. In[23]:= LaplaceTransform[ D[g[t, x], t], t, s, Analytic -> False] Out[23]=