PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (205)
Orphanage
Unclass'd
Unproven (490)
Corrections (8)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
[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$.



"example of Fourier series" is owned by alozano.
(view preamble)

View style:

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

Other names:  example of Fourier coefficients

This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: Riemann zeta function, Fourier series, odd functions, even function, Fourier coefficients, function
There is 1 reference to this entry.

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 11862 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)
Pending Errata and Addenda
None.
[ View all 5 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)