PSP-RH-2-2010

A expansão em série de Fourier da função real f(x)= \begin{cases}0, & - \pi < x < 0 \\ {\large{1 \over 2}}, & 0< x< \pi \end{cases}, f(x+2 \pi)=f(x), para todo x ≠ k \pi, k ∈ \mathbb{Z}, é

{\large{1 \over 2}}+{\large{2 \over \pi}} \left[ \sin(x)+{\large{\cos(3x) \over 3}}+{\large{ \sin(5x) \over 5}}+{\large{ \cos(7x) \over 7}}+... \right]
{\large{1 \over 2}}+{\large{2 \over \pi}} \left[ \sin(x)-{\large{\sin(3x) \over 3}}+{\large{ \sin(5x) \over 5}}+{\large{ \sin(7x) \over 7}}+... \right]
{\large{1 \over 2}}+{\large{1 \over \pi}} \left[ \cos(x)-{\large{\cos(3x) \over 3}}+{\large{ \cos(5x) \over 5}}-{\large{ \cos(7x) \over 7}}+... \right]
{\large{1 \over 4}}+{\large{1 \over \pi}}\left[ \cos(x)+{\large{\cos(3x) \over 3}}+{\large{\cos(5x) \over 5}}+{\large{\cos(7x) \over 7}}+ ... \right]
{\large{1 \over 4}}+{\large{1 \over \pi}}\left[\sin(x)+{\large{\sin(3x) \over 3}}+{\large{\sin(5x) \over 5}}+{\large{\sin(7x) \over 7}}+... \right]