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I don't understand how using Fourier's trick allows you to calculate ##c_n##. Also, I don't understand how ##\sum_{n=1}^{\infty}c_n\delta_{mn} = c_m##. I get that the Kronecker delta comes out of the fact that ##\psi_m^\ast(x)## and ##\psi_n(x)## are orthogonal. I understand that when ##m = n##, then the Kronecker delta equals 1, but how does this allow us to find ##c_m## and ##c_n## .

$$ \int_{0}^{a} \psi_m^\ast(x) f(x)\mathrm dx = \sum_{n=1}^{\infty}c_n \int_{0}^{a} \psi_m^\ast (x)\psi_n(x)\mathrm dx = \sum_{n=1}^{\infty}c_n\delta_{mn} = c_m $$