# Illustration Sawtooth Function Approximated by Two Different Fourier Series

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A sawtooth function in the interval of length $$L=1$$\begin{align} f(x) ~=~ \begin{cases} -x, &\mbox{} \left(0,~ \frac{1}{2}\right) \\ 1-x, &\mbox{} \left(\frac{1}{2},~ 1\right) \end{cases} \end{align}is approximated here by two different Fourier series:\begin{align} f(x) ~=~ \underset{n}{\boxed{+}} ~ \frac{1}{ \mathrm{i}\,2\pi\,n } \, \text{e}^{\mathrm{i}\,2\pi\,n\, (x-1/2)} \end{align}

The not so good approximation (red) is the Fourier series terminated at $$n_{\text{max}}=1$$. The better approximation (blue) is the Fourier series terminated at $$n_{\text{max}} = 20$$.