I am trying to find out expectation of a function of a uniform random variable. I am given a random variable $x$ that is uniformly distributed over the interval $[0, a]$. I want to find out the expectation $E[e^{-i2\pi (m-n)\frac{x}{a}}]$, where $m$, $n$ are integers and $i=\sqrt{-1}$.
I saw derivation at a few places, and it looks like that the expectation evaluates to $\delta_{km}$. It is not clear to me how this derivation is carried out. When I look at the expectation, it appears I am trying to calculate the characteristic function of a uniform random variable. This function is not a delta function. So what am I missing here?