# example of Fourier series

Here we present an example of Fourier series:

Example:

Let $f\colon(-\pi,\pi)\to\mathbb{R}$ 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. Notice that $\cos(nx)$ is an even function, while $f$ and $\sin(nx)$ are odd functions.

 $\displaystyle a_{0}^{f}$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)dx=\frac{1}{2\pi}\int_{-\pi}^{% \pi}xdx=0$ $\displaystyle a_{n}^{f}$ $\displaystyle=$ $\displaystyle\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)dx=\frac{1}{\pi}\int_{-% \pi}^{\pi}x\cos(nx)dx=0$ $\displaystyle b_{n}^{f}$ $\displaystyle=$ $\displaystyle\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(nx)dx=\frac{1}{\pi}\int_{-% \pi}^{\pi}x\sin(nx)dx=$ $\displaystyle=$ $\displaystyle\frac{2}{\pi}\int_{0}^{\pi}x\sin(nx)dx=\frac{2}{\pi}\left(\left[-% \frac{x\cos(nx)}{n}\right]_{0}^{\pi}+\left[\frac{\sin(nx)}{n^{2}}\right]_{0}^{% \pi}=\right)=(-1)^{n+1}\frac{2}{n}$

Notice that $a_{0}^{f},a_{n}^{f}$ are $0$ because $x$ and $x\cos(nx)$ are odd functions. Hence the Fourier series for $f(x)=x$ is:

 $\displaystyle f(x)=x$ $\displaystyle=$ $\displaystyle a_{0}^{f}+\sum_{n=1}^{\infty}(a_{n}^{f}\cos(nx)+b_{n}^{f}\sin(nx% ))=$ $\displaystyle=$ $\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\frac{2}{n}\sin(nx),\quad\forall x% \in(-\pi,\pi)$

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

Title example of Fourier series ExampleOfFourierSeries 2013-03-22 13:57:13 2013-03-22 13:57:13 alozano (2414) alozano (2414) 10 alozano (2414) Example msc 42A16 example of Fourier coefficients ValueOfTheRiemannZetaFunctionAtS2 FourierSineAndCosineSeries