Clarification on "Next frequencies to add"

So far, your visualization of the Laplace transform is the best on the internet. Nicely done!

I'm trying to figure out how to come up with the functions to be added to the final solution. These functions are annotated as F(s)e^st/2pij, where s = 1+0.1j.

As the animation plays out, the variable that is changing is r, and since your notebook explains that the main exponential equation evaluates to 2e^ct*cos(rt) and that c is visibly equal to 1 in the code, this part is easy to figure out.

However, what I can't figure out is F(s), what does the Laplace transform of the equation, which is F(s), evaluates to if we only have r? What does F(c+rj) evaluates to given that we have c=1 and r is a changing constant?