Laplace Transforms

For any absolutely integrable function \( f(t)\) on the interval \ (0,\infty )\), the Laplace transform is the indefinite integral \[ {\cal L} (\lambda ) = f^L (\lambda ) = \int_0^{\infty} f(t)\,e^{-\lambda t} \,{\text d}t , \] subject the the integral converges.

Laplace transforms of piecewise continuous functions

If you have a piecewise-defined polynomial function then there is a “native” command for computing Laplace transforms. This calls Maxima but it’s worth noting that Maxima cannot handle (using the direct interface illustrated in the last few examples) this type of computation.

sage: var('x s')
(x, s)
sage: f1(x) = 1
sage: f2(x) = 1-x
sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
sage: f.laplace(x, s)
-e^(-s)/s + (s + 1)*e^(-2*s)/s^2 + 1/s - e^(-s)/s^2

For other “reasonable” functions, Laplace transforms can be computed using the Maxima interface:

sage: var('k, s, t')
(k, s, t)
sage: f = 1/exp(k*t)
sage: f.laplace(t,s)
1/(k + s)

is one way to compute Laplace transformations and

sage: var('s, t')
(s, t)
sage: f = t^5*exp(t)*sin(t)
sage: L = laplace(f, t, s); L
3840*(s - 1)^5/(s^2 - 2*s + 2)^6 - 3840*(s - 1)^3/(s^2 - 2*s + 2)^5 +
720*(s - 1)/(s^2 - 2*s + 2)^4

is another way.