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