In mathematics, a half range Fourier series is a Fourier series defined on an interval instead of the more common , with the implication that the analyzed function should be extended to as either an even (f(-x)=f(x)) or odd function (f(-x)=-f(x)). This allows the expansion of the function in a series solely of sines (odd) or cosines (even). The choice between odd and even is typically motivated by boundary conditions associated with a differential equation satisfied by . Example Calculate the half range Fourier sine series for the function where . Since we are calculating a sine series, Now, When n is odd, When n is even, thus With the special case , hence the required Fourier sine series is Fourier series