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