Sine and cosine series

Summary

The Fourier sine series and the Fourier cosine series expand a function given on the interval $[0, \pi]$ in the functions $\sin nx$ alone or $\cos nx$ alone. They are the Fourier series of the odd and of the even $2\pi$-periodic extension of the function. They appear when separation of variables is applied to the heat or wave equation on an interval. Dirichlet boundary conditions lead to sine series, and Neumann conditions to cosine series.