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I'm stuck with a general thing that's been bothering me for a while, I'm working with PDE's, mostly S-L problems. The solutions to my problem are basically sums, normally I'd solve the eigenvalue problem, then use fourier to get the coefficents in the sum, but I'm wondering, if my eigenfunction isn't normalized, is this still okay to just do?

Been stuck with the issue of computing the coefficients for a long time now, hopefully someone can make this understandable, thanks in advance!

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If you don't normalize your eigenfunctions, then the normalization constant is simply absorbed into that mode's coefficient, so you can still expand your original function as a sum of those non-normalized eigenfunctions. However, often times, it's easier to work with normalized eigenfunctions, because orthonormal basis have nice properties that can be easier to work with.

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    So if they are not actually asking for the coefficient infront of the eigenfunction, you would say that it's most likely better to normalize it first, and then calculate the coefficient for the fourier series? – 2017-02-11
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    Let's fix a mode to illustrate the idea. Say the non-normalized eigenfunction is $f$ and it has norm 5, so the normalized eigenfunction is $f/5$. Then that term in the sum will look like $c_f \frac{f}{5}$ or $c_f' f$ where $c_f'= \frac{c_f}{5}$. Either way, these are equivalent, its just a matter of what you call the coefficient and what you call the eigenfuction. – 2017-02-11
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    So if you're just looking for the expansion, it doesn't really matter. But say you want to do other things it may be easier to normalize. For instance, if you want to take the dot product of eigenfunctions, if they are normalized, the dot product will simply be the kronecker delta, if theyre not normalized, the dot product will be a scaled kronecker that you would have to computed for each eigenfunction (although you would probably have the norm lying around anyways). – 2017-02-11