complex Fourier series

The full Fourier series rewritten with complex exponentials in place of separate sine and cosine terms.

The complex Fourier series of $f$ on $[-L,L]$ is $f(x)\sim\sum_{n=-\infty}^\infty c_ne^{in\pi x/L}$ , with $c_n=\dfrac1{2L}\displaystyle\int_{-L}^Lf(x)e^{-in\pi x/L}\,dx$ ; Euler's formula converts this term by term into the same $a_n,b_n$ coefficients as the full Fourier series, with $c_n=\tfrac12(a_n-ib_n)$ for $n>0$ .