# Find the exponential Fourier transform of the given f(x)

I attempted this problem, but my final answer does not seem simplified enough. The problem is that I don't know how to simplify it more and I lack the experience to be able to tell when it's ok to stop simplifying.

My attempt:

$f(x)=|x|, |x|<1 \quad ; \quad =0 \quad$everywhere else

$$g(a)=\frac{1}{2\pi}\int_{-1}^1 |x|e^{-iax}dx$$

$$=\frac{1}{\pi}\int_0^1 xe^{-iax}dx$$

$$=\frac{1}{\pi} \left [\frac{xe^{-iax}}{-ia}|_0^1 -\int_0^1 \frac{e^{-iax}}{-ia} \right ]$$

$$=-\frac{1}{i\pi a} \left [e^{-ia} + \frac{e^{-ia}-1}{ia} \right ]$$

I'm new to Fourier transforming, so I'm not sure what a useable final answer looks like. Can this be simplified further?

• You should have left it as an integral from $-1$ to $1$ and simple your answer into trig functions. Commented Oct 20, 2016 at 3:30
• @JackyChong Why? This is an even function across the y-axis, so can't I just integrate from $0$ to $1$ and multiply by $2$? Commented Oct 20, 2016 at 3:31
• Sure, but you could simplify your expression if you just stick to integration from $-1$ to $1$. Commented Oct 20, 2016 at 3:32
• But then how do I integrate the absolute value of $x$? Commented Oct 20, 2016 at 3:33
• Let me write a short note in the answer section Commented Oct 20, 2016 at 3:33

You can simplify it further by using trig functions instead of the Euler notation. I would not recommend integrating from -1 to 1 since (you already seem to know) that integrating $|x|$ would have you dealing with the sgn(x) function and that's over-complicating the problem.