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| Title of object: |
example of Fourier series |
| Canonical Name: |
ExampleOfFourierSeries |
| Type: |
Example |
| Created on: |
2003-09-10 16:51:06 |
| Modified on: |
2003-09-10 17:01:29 |
| Classification: |
msc:42A16 |
| Synonyms: |
example of Fourier series=example of Fourier coefficients |
Preamble:
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Content:
Here we present an example of Fourier series:
{\bf Example:}
Let $f\colon \Reals \to \Reals$ be the ``identity'' function,
defined by
$$f(x)=x, \text{ for all }x\in\Reals$$
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.
\begin{eqnarray*}
a_0^f & =&
\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)dx=\frac{1}{2\pi}\int_{-\pi}^{\pi}
x dx= 0
\end{eqnarray*}
\begin{eqnarray*}
a_n^f &=& \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)dx=
\frac{1}{\pi}\int_{-\pi}^{\pi}x \cos(nx)dx = 0
\end{eqnarray*}
\begin{eqnarray*}
b_n^f &=&
\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(nx)dx=\frac{1}{\pi}\int_{-\pi}^{\pi}
x \sin(nx)dx =\\
&=& \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}\right]_0^{\pi}=
\right)=-\frac{2}{n}
\end{eqnarray*}
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:
\begin{eqnarray*}
f(x)=x &=& a_0^f +
\sum_{n=1}^{\infty}(a_n^fcos(nx)+b_n^fsin(nx)) =\\
&=& \sum_{n=1}^{\infty}\frac{-2}{n} \sin(nx)
\end{eqnarray*}
For an application of this Fourier series, see value of the
Riemann zeta function at $s=2$. |
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