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[parent] example of Fourier series (Example)

Here we present an example of Fourier series:

Example:

Let $ f\colon (-\pi,\pi) \to \mathbb{R}$ be the “identity” function, defined by

$\displaystyle f(x)=x,$    for all $\displaystyle x\in (-\pi,\pi).$
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.
$\displaystyle a_0^f$ $\displaystyle =$ $\displaystyle \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)dx=\frac{1}{2\pi}\int_{-\pi}^{\pi} x dx= 0$  
$\displaystyle a_n^f$ $\displaystyle =$ $\displaystyle \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)dx= \frac{1}{\pi}\int_{-\pi}^{\pi}x \cos(nx)dx = 0$  
$\displaystyle b_n^f$ $\displaystyle =$ $\displaystyle \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(nx)dx=\frac{1}{\pi}\int_{-\pi}^{\pi} x \sin(nx)dx =$  
  $\displaystyle =$ $\displaystyle \frac{2}{\pi}\int_{0}^{\pi} x\sin(nx) dx= \frac{2}{\pi}\left( \le... ...^{\pi}+\left[\frac{\sin(nx)}{n^2}\right]_0^{\pi}= \right)=(-1)^{n+1}\frac{2}{n}$  

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$.



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See Also: value of the Riemann zeta function at $s=2$, Fourier sine and cosine series

Other names:  example of Fourier coefficients

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Cross-references: Riemann zeta function, Fourier series, odd functions, even function, Fourier coefficients, function
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This is version 7 of example of Fourier series, born on 2003-09-10, modified 2006-02-21.
Object id is 4718, canonical name is ExampleOfFourierSeries.
Accessed 12321 times total.

Classification:
AMS MSC: 42A16 (Fourier analysis :: Fourier analysis in one variable :: Fourier coefficients, Fourier series of functions with special properties, special Fourier series)
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