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!
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.