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## 8.7 Laplace Transforms

Axiom can compute some forward Laplace transforms, mostly Laplace transform of elementary function:elementary functions transform:Laplace not involving logarithms, although some cases of special functions are handled.

To compute the forward Laplace transform of F(t) with respect to t and express the result as f(s), issue the command laplace(F(t),t,s).

laplace(sin(a*t)*cosh(a*t)-cos(a*t)*sinh(a*t), t, s)

4 a 3 s 4+4 a 4

Type: Expression Integer

Here are some other non-trivial examples.

laplace((exp(a*t) - exp(b*t))/t, t, s)

-log(s-a)+log(s-b)

Type: Expression Integer
laplace(2/t * (1 - cos(a*t)), t, s)

log(s 2+a 2)-2 log(s)

Type: Expression Integer
laplace(exp(-a*t) * sin(b*t) / b**2, t, s)

1 b s 2+2 a b s+b 3+a 2 b

Type: Expression Integer
laplace((cos(a*t) - cos(b*t))/t, t, s)

log(s 2+b 2)-log(s 2+a 2)2

Type: Expression Integer

Axiom also knows about a few special functions.

laplace(exp(a*t+b)*Ei(c*t), t, s)

e b log(s+c-a c)s-a

Type: Expression Integer
laplace(a*Ci(b*t) + c*Si(d*t), t, s)

a log(s 2+b 2 b 2)+2 c arctan(d s)2 s

Type: Expression Integer

When Axiom does not know about a particular transform, it keeps it as a formal transform in the answer.

laplace(sin(a*t) - a*t*cos(a*t) + exp(t**2), t, s)

(s 4+2 a 2 s 2+a 4)laplace(e t 2,t,s)+2 a 3 s 4+2 a 2 s 2+a 4

Type: Expression Integer
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