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
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