= Solution
The <complex Fourier series> of a $2L$-periodic function is
$$
\boxed{f(x)\sim\sum_{n\in\mathbb Z}c_ne^{in\pi x/L},\qquad
c_n=\frac1{2L}\int_{-L}^L f(x)e^{-in\pi x/L}\,dx.}
$$
The coefficient formula follows from <orthogonality> of the exponential functions. For an arbitrary <square-integrable function>, this representation means convergence in the $L^2$ <norm>. With appropriate additional regularity it also gives pointwise convergence, to the mean of the one-sided limits at a jump. No pointwise representation is asserted for an unrestricted function merely from periodicity.
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