 If f(x) is an odd function, then the Fourier Half Range sine series of f is defined to be

which is just a form of complete Fourier series with the only difference that formula_3 and formula_4 is zero, and the series is defined for half of the interval.In the formula we have ...If f(x) is an even function, then the Fourier cosine series is defined to be

where

This notion can be generalized to functions which are not even or odd, but then the above formulas will look different.