8.7 Laplace Transforms¶

FriCAS 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)


 4a3s4+4a4

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(s2+a2)-2log(s)

Type: Expression Integer

laplace(exp(-a*t) * sin(b*t) / b^2, t, s)


 1bs2+2abs+b3+a2b

Type: Expression Integer

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


 log(s2+b2)-log(s2+a2)2

Type: Expression Integer

FriCAS also knows about a few special functions.

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


 eblog(s+c-ac)s-a

Type: Expression Integer

laplace(a*Ci(b*t) + c*Si(d*t), t, s)


 alog(s2+b2b2)+2carctan(ds)2s

Type: Expression Integer

When FriCAS 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)


 (s4+2a2s2+a4)laplace(et2,t,s)+2a3s4+2a2s2+a4

Type: Expression Integer