# Calculate the Fourier Transform

1. Jan 2, 2012

### rht1369

1. The problem statement, all variables and given/known data

calculate the Fourier Transform of the following function:

2. Relevant equations

x(t) = e-|t| cos(2t)

3. The attempt at a solution

0-∞ et ((e2jt + e-2jt) / 2) e-jωt + ∫0 e-t ((e2jt + e-2jt) / 2) e-jωt

The first integral is easy to calculate and equals: (1/2) * (1/(1+(2-ω)j) + 1/(1+(-2-ω)j))
But how about the second integral?

2. Jan 2, 2012

### AlephZero

You did the first integral OK. Why do you think the second integral is harder than the first one?

But if you sketch a graph of x(t), you might notice something interesting that saves you some work...

3. Jan 2, 2012

### rht1369

My problem is with the infinite boundaries. The first integral is simple since is done from -∞ to 0 and it makes the limit of exponential part will be zero But I am confused about the second one since its boundaries are from 0 to ∞ and I can get it how the limit of exponential part of this function will be zero too!

I sketched the graph. Do you mean that the function is asymmetric? so the doing the integration for one side will be equal to the another?

4. Jan 2, 2012

### AlephZero

The real part of the second integral is $e^{-t}$ which goes to 0 as t goes to $+\infty$. That is similar to the first integral, where $e^{+t}$ goes to 0 as t goes to $-\infty$.

It would be better to call it "an even function" not "asymmetric", but you got the point that the two integrals are equal.