Evaluate the following code to see how to compute Laplace transforms with Sage:
var('t, a, k, s') print("Elementary transform:") f(t) = sin(k * t) F(s) = f.laplace(t, s) F.show() print("Inverse transform:") G(s) = 1 / ((s - 1) * (s + 2) * (s + 4)) G.show() g(t) = G.inverse_laplace(s, t) g.show()
The results are shown in the following screenshot:
We used Sage to compute the Laplace transform and the inverse Laplace transform. Normally, finding the Laplace transform would have required looking up the elementary forms in a table. Computing the inverse transform requires performing a partial fraction expansion and then finding the resulting terms in a table. The process is much faster with Sage, and Sage can compute transforms that cannot practically be computed by hand. We used the
laplace
method to compute the forward transform, and inverse_laplace
to compute the reverse transform. Each of these methods computes the transform...