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

Special Case of Fourier Series for Triangle Wave
Let $\map T x$ be the triangle wave defined on the real numbers $\R$ as:


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

\size x & : x \in \closedint {-\pi} \pi \\ \map T {x + 2 \pi} & : x < -\pi \\ \map T {x - 2 \pi} & : x > +\pi \end {cases}$ where $\size x$ denotes the absolute value of $x$.

Then its Fourier series can be expressed as:

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


 * $\map T x = \size x$

has a Fourier series which can be expressed as:

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