- #1
eyenkay
- 7
- 0
Homework Statement
The problem just states to find the Laplace Transform of cos(kt) from its power series expansion, instead of using the formula for the transform of a periodic function.
Homework Equations
Equation for Laplace transform of a function f(t) ->[tex]\int(e^{-st}f(t))dt[/tex]
Power Series Expansion for cos(x)-> [tex]\sum\frac{(-1)^{n}}{(2n)!}x^{2n}[/tex]
The Attempt at a Solution
I've been trying to apply the formula for the Laplace Transform directly to the expansion of cos, but I get stuck in the integration.. Once you apply the formula, I figured you can bring the e[tex]^{-st}[/tex] inside the sum since it doesn't depend on n, and therefore you treat it like a constant wrt the sum. Then interchange the order of the sum and the integral, and end up with [tex]\sum\frac{(-1)^{n}}{(2n)!}k^{2n}\int(e^{-st}t^{2n})dt[/tex]..
This is what I can't figure out how to integrate, if you try it by parts you just get t to the 2n-1, then 2n-2... etc.
Any ideas?