1% Copyright The Numerical Algorithms Group Limited 1991.
2% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
3% All rights reserved
4
5% Title: Laplace Transforms
6
7\begin{page}{ExLapSimplePole}{Laplace transform with a single pole}
8\beginscroll
9The Laplace transform of t^n e^(a t) has a pole of order n+1 at x = a
10and no other pole. We divide by n! to get a monic denominator in the
11answer.
12\spadpaste{laplace(t^4 * exp(-a*t) / factorial(4), t, s)}
13\endscroll
14\autobuttons\end{page}
15
16\begin{page}{ExLapTrigTrigh}{Laplace transform of a trigonometric function}
17\beginscroll
18Rather than looking up into a table, we use the normalizer to rewrite
19the trigs and hyperbolic trigs to complex exponentials and
20logarithms.
21\spadpaste{laplace(sin(a*t) * cosh(a*t) - cos(a*t) * sinh(a*t), t, s)}
22\endscroll
23\autobuttons\end{page}
24
25\begin{page}{ExLapDefInt}{Laplace transform requiring a definite integration}
26\beginscroll
27When powers of t appear in the denominator, computing the laplace transform
28requires integrating the result of another laplace transform between a
29symbol and infinity. We use the full power of \Language{}'s integrator
30in such cases.
31\spadpaste{laplace(2/t * (1 - cos(a*t)), t, s)}
32\endscroll
33\autobuttons\end{page}
34
35\begin{page}{ExLapExpExp}{Laplace transform of exponentials}
36\beginscroll
37This is another example where it is necessary to
38integrate the result of another laplace transform.
39\spadpaste{laplace((exp(a*t) - exp(b*t))/t, t, s)}
40\endscroll
41\autobuttons\end{page}
42
43\begin{page}{ExLapSpecial1}{Laplace transform of an exponential integral}
44We can handle some restricted cases of special functions, linear exponential
45integrals among them.
46\beginscroll
47\spadpaste{laplace(exp(a*t+b)*Ei(c*t), t, s)}
48\endscroll
49\autobuttons\end{page}
50
51\begin{page}{ExLapSpecial2}{Laplace transform of special functions}
52\beginscroll
53An example with some interesting special functions.
54\spadpaste{laplace(a*Ci(b*t) + c*Si(d*t), t, s)}
55\endscroll
56\autobuttons\end{page}
57