Fourier Series/Square Wave/Special Cases/Half Interval Pi

Special Case of Fourier Series for Square Wave
Let $\map S x$ be the square wave defined on the real numbers $\R$ as:


 * $\forall x \in \R: \map S x = \begin {cases}

1 & : x \in \openint 0 \pi \\ -1 & : x \in \openint {-\pi} 0 \\ \map S {x + 2 \pi} & : x < -\pi \\ \map S {x - 2 \pi} & : x > +\pi \end {cases}$

Then its Fourier series can be expressed as:

Proof
From Fourier Series for Square Wave, the real function $\map f x$ defined on the open interval $\openint {-l} l$ as:


 * $\map f x = \begin{cases} -1 & : x \in \openint {-l} 0 \\ 1 & : x \in \openint 0 l \end {cases}$

has a Fourier series which can be expressed as:

The result follows by setting $l = \pi$.