Laplace Transform of cos(kt) using Power Series expansion

[eyenkay]

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) ->$$\int(e^{-st}f(t))dt$$
Power Series Expansion for cos(x)-> $$\sum\frac{(-1)^{n}}{(2n)!}x^{2n}$$

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$$^{-st}$$ 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 $$\sum\frac{(-1)^{n}}{(2n)!}k^{2n}\int(e^{-st}t^{2n})dt$$..
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?

 

[Dick]

You've got it down to the laplace transform of a power t^(2n). If you don't have a formula that you can use for it, then you derive it just as you say. Use integration by parts and induction to find the formula. Start by doing t, t^2, t^3... It should be pretty obvious what the formula for t^(2n) looks like. Hint: the answer will have a factorial in it.

 

[Count Iblis]

You can also compute the integral of exp(-s t) and then differentiate that 2n times w.r.t. s to bring down a factor t^(2n) in the integrand.

 

[eyenkay]

oh..ok i see, i was forgetting to input the bounds, this causes each term to go to zero except for the last one, where the exponent of t is 0. Thanks.

 

[Mattszo]

You've got it down to the laplace transform of a power t^(2n). If you don't have a formula that you can use for it, then you derive it just as you say. Use integration by parts and induction to find the formula. Start by doing t, t^2, t^3... It should be pretty obvious what the formula for t^(2n) looks like. Hint: the answer will have a factorial in it.

Hi I'm a physics student studying theoretical physics 2, we haven't learned math induction. Is there another way around it?

What about applying the formula

| exp(- sigma t) f(t) | = Mthanks!