Proving Continuous Function: Expectation Values of Periodic Functions Over Time

[thegaussian]

Hi. I'm looking to at how expectation values of periodic functions evolve in time, and i need to prove that $\exp ( i \theta )$ is continuous in time (this is the expectation of the exponential of the angle).
My formula is:

$\exp( i \theta) = \exp ( i t /2) \sum_{n=-\infty}^{\infty} a_n a^*_{n-1} \exp (- i n t )$

where $a_n$ are the Fourier coefficients of the initial function, $*$ represents the complex conjugate. Now how do I go about proving it's continuous? We have basically a complex exponential factor (that's obviously continuous) multiplied by a Fourier series, but I just have no idea really where to go from there.

Any help would be much appreciated.

Thanks!

 

[fresh_42]

An ordinary $\varepsilon-\delta$ prove should do. You can even take $\exp(- i t)$ out of the sum.