Half Range Fourier Series

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, b_n= \frac{2}{\pi} \int_0^\pi \cos(x)\sin(nx)\,\mathrm{d}x = \frac{2n((-1)^n+1)}{\pi(n^2-1)}\quad \forall n\ge 2

When n is odd, When n is even, thus

With the special case, hence the required Fourier sine series is

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