Laplace Transform of cos(at) * cos(bt)

[seang]

I have to find the laplace transform of cos(at) * cos(bt) and express it as a ratio of two polynomials. I converted both of the cosines into exponentials, and took the laplace transform of those. I think I'm getting confused on the complex side of things.

I get a few things like e^(jt(a+b)), and e^(-jt(a+b)) so I use Euler's formula and get some cosines and jsines. So how do you find the laplace transform of say jsin(t(-a-b)). Shouldn't it just be j*s / (s^2 + (-a-b)^2)?

Except this does not yield the correct answer. Maybe I'm converting all of the transforms to one fraction incorrectly...?

 

[Corneo]

Do you mean to find the Laplace transform of $\cos (at) * \cos (bt)$ ? Where the $*$ denotes convolution? Or did you meant it as multiplication?

 

[leright]

Do you mean to find the Laplace transform of $\cos (at) * \cos (bt)$ ? Where the $*$ denotes convolution? Or did you meant it as multiplication?

I assume he meant it as multiplication, since he is doing all of that work. I thought the same thing myself. If it were convolution it sure would make the problem a lot easier. :p