Consider the collection $\left\{e^{2\pi ikx/(b-a)}\,:\, k=0,\pm 1, \pm2,\ldots \right\}\,,$ on $(a,b)$. Verify that this system is orthogonal in $L^2$.
I have attempted the two most obvious things: (1) integrating with exponentials, (2) integrating with trig functions. Both are messy, and neither are working out. I don't want to write out a lot of work, because my notes are a tedious nightmarescape, but I basically keep ending up with the following:
Set $\theta=\frac{2\pi}{b-a}$. Take $e^{2\pi ikx/(b-a)}$ and $e^{2\pi ikx/(b-a)}$. Taking their inner product, I get something like $\frac{i}{\theta(k-n)}\left[e^{\theta(k-n)bi}-e^{\theta(k-n)ai}\right]\,.$
I feel like I'm going crazy, but it is not at all apparent that this will end being zero....