full Fourier series

The full Fourier series of f on [-L,L] is f(x)\sim\dfrac{a_0}2+\sum_{n=1}^\infty\left(a_n\cos\dfrac{n\pi x}L+b_n\sin\dfrac{n\pi x}L\right), with a_n=\dfrac1L\displaystyle\int_{-L}^Lf(x)\cos\dfrac{n\pi x}L\,dx and b_n=\dfrac1L\displaystyle\int_{-L}^Lf(x)\sin\dfrac{n\pi x}L\,dx; restricting to an even f collapses every b_n to zero and recovers the Fourier cosine series, and restricting to an odd f collapses every a_n to zero and recovers the Fourier sine series.