example of Fourier series

Here we present an example of Fourier series:

Example:

Let $f:(-\pi ,\pi )\to ℝ$ be the “identity” function,defined by

$$f(x)=x,\text{ for all }x\in (-\pi ,\pi ).$$

We will compute the Fourier coefficients for this function. Noticethat $\cos (nx)$ is an even function, while $f$ and $\sin (nx)$ areodd functions.

$a_0^f$$=$${\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}f(x)𝑑x={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}x𝑑x=0$

$a_n^f$$=$${\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}f(x)\cos (nx)𝑑x={\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}x\cos (nx)𝑑x=0$

$b_n^f$$=$${\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}f(x)\sin (nx)𝑑x={\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}x\sin (nx)𝑑x=$

$=$${\displaystyle \frac{2}{\pi }}{\displaystyle {\int }_0^{\pi }}x\sin (nx)dx={\displaystyle \frac{2}{\pi }}({[-{\displaystyle \frac{x\cos (nx)}{n}}]}_0^{\pi }+{\left[{\displaystyle \frac{\sin (nx)}{n^2}}\right]}_0^{\pi }=)={(-1)}^{n+1}{\displaystyle \frac{2}{n}}$

Notice that $a_0^f,a_n^f$ are $0$ because $x$ and $x\cos (nx)$ areodd functions. Hence the Fourier series for $f(x)=x$ is:

$f(x)=x$$=$$a_0^f+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}(a_n^f\cos (nx)+b_n^f\sin (nx))=$

$=$${\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{(-1)}^{n+1}{\displaystyle \frac{2}{n}}\sin (nx),\forall x\in (-\pi ,\pi )$

For an application of this Fourier series, see value of theRiemann zeta function at $s=2$.