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,