Laplace Transform of t*e^(it): Real Part

[cragar]

Homework Statement

t*e^(it) how do we take the laplace transform of this .

would it be 1/((s-i)^2) then how would we get the real part of that .

 

[cipher42]

I'm not quite sure how the Laplace transform of a complex-valued function such as e^(it) works, but sense you want to take the real part of that is it the case that what you're really looking for the Laplace transform of is t*cos(t)? Because that isn't so hard and introducing complex numbers seems like the long way around.

 

[cragar]

then how would i take the laplace transform of t*cos(t) based on if i knew the
lapace tranform of t and cos(t) how does this product work out.

 

[cipher42]

A handy result in the theory of Laplace transforms says that:
$$L\{t^nf(t)\}(s)=(-1)^n\frac{d^n}{ds^n}L\{f\}(s)$$
which we can use to compute:
$$L\{t\cos{t}\}=-\frac{d}{ds}L\{\cos{t}\}$$
Now all you need to do is look up the transform for $\cos{t}$, differentiate and you're on your way home!

 

[cragar]

thanks , what is this rule called .

 

[cipher42]

That's a good question; I'm not sure if this identity has a name or not. Perhaps someone else will know.

It's not too hard too prove. All you do is write out the Laplace transform for t^n * f(t) and use integration by parts n-times (which you could also do to get the answer with any ol' f(t), but if you remember it, the identity is pretty quick).

 

[cragar]

the only problem is my computer won't let me see the latex black boxes so i am having to look at it how you typed it in so i am having trouble reading it .