Fourier Series for $|x|$

Question

How can I calculate the Fourier series for |x| (where $x\in[−\pi,\pi]$) in the complex form? Thanks.

Answer 1

As the comment suggests:

$$\int\limits_{-\pi}^\pi |x|e^{inx}dx=-\int\limits_{-\pi}^0 xe^{inx}dx+\int\limits_0^\pi x e^{inx}dx$$

In both cases you can do integration by parts:

$$u=x\;,\;\;u'=1\\v'=e^{inx}\;,\;\;v=-\frac ine^{inx}$$

so for example

$$\int\limits_{-\pi}^0 xe^{inx}dx=\left.-\frac inxe^{inx}\right|_{-\pi}^0+\frac in\int\limits_{-\pi}^0e^{inx}dx=-\frac in\left(0-(-\pi)e^{-in\pi}\right)+\left.\frac1{n^2}e^{inx}\right|_{-\pi}^0=$$

$$=\frac{\pi i}n(-1)^{n+1}+\frac1{n^2}\left(1-(-1)^n\right)=\begin{cases}-\frac{\pi i}n&,\;\;n\;\text{is even}\\{}\\\frac{\pi i}n+\frac2{n^2}&,\;\;n\;\text{is odd}\end{cases}$$

and etc.