Primeiras somas parciais de

 \dfrac{1}{2}+\dfrac{2}{\pi}\cos x-\dfrac{2}{3\pi}\cos3x+\dfrac{2}{5\pi}\cos5x-\dfrac{2}{7\pi}\cos7x+\cdots

Onda quadrada (a vermelho) no intervalo \lbrack -\pi ,\pi \rbrack

f(x)= \left\{\begin{array}{rl}1&\text{se } -\pi /2\leq x\leq\pi /2\\ 0&\text{se } |x|>\pi /2\end{array}\right.

e as somas parciais dos cinco primeiros termos da série de Fourier

 

f(x)= \dfrac{a_{0}}{2}+\displaystyle\sum_{n=1}^{\infty }\left( a_{n}\cos nx+b_{n}\sin nx\right)

Em virtude de f\left( x\right) ser par b_{n}=0

f\left( x\right) =\dfrac{a_{0}}{2}+\displaystyle\sum_{n=1}^{\infty }a_{n}\cos nx

Os coeficientes a_{n} são

a_{n}=\dfrac{1}{\pi }\displaystyle\int_{-\pi }^{+\pi }f\left( x\right) \cos nx\;dx\qquad n=0,1,2,\cdots

a_{0}=\dfrac{1}{\pi }\displaystyle\int_{-\pi /2}^{+\pi /2}\;dx=1

a_{1}=\dfrac{1}{\pi }\displaystyle\int_{-\pi /2}^{+\pi /2}\cos x\;dx=\dfrac{2}{\pi }

a_{3}=\dfrac{1}{\pi }\displaystyle\int_{-\pi /2}^{+\pi /2}\cos 3x\;dx=-\dfrac{2}{3\pi }

a_{5}=\dfrac{1}{\pi }\displaystyle\int_{-\pi /2}^{+\pi /2}\cos 5x\;dx=-\dfrac{2}{5\pi }

a_{7}=\dfrac{1}{\pi }\displaystyle\int_{-\pi /2}^{+\pi /2}\cos 7x\;dx=-\dfrac{2}{7\pi }

a_{2}=a_{4}=a_{6}=\cdots =a_{2n}=0

NOTA: a série de Fourier nos dois pontos de descontinuidade da função passa a meio do salto dado, isto é, neste caso 1/2.