Fourier Series

(http://jowett.home.cern.ch/jowett/ComputingNotes/wave.gif)

 
 

Who was Fourier
 

What is a Fourier Series
 

Math of a  Fourier Series
 

Applications of a Fourier Series
 

Bibliography

 

Math of a Fourier Series

 

\begin{displaymath}F_n(x) = a_0 + \sum_{k= 1}^{k=n} \Big(a_k\cos(kx) + b_k\sin(kx)\Big).\end{displaymath}


          When

\begin{displaymath}\left\{\begin{array}{lclr} a_0 &=&\displaystyle \frac{1}{2\pi... ...\pi} F_n(x) \sin(kx)dx,& 1 \leq k \leq n.\\ \end{array}\right.\end{displaymath}

 

 

Above is the mathematical process by which a function from, –Pi to Pi, is transformed in to a trigonometric series. The fist step is to solve for the coefficient a0, ak, and bk. This is the most difficult part. After the coefficients are solved for it is simply a mater of plugging them in to the top equation. The result is a very nice series that can be used to approximate the function. One nice thing about describing a function as a series is that a computer can then be easily programmed to evaluate it.

The following are examples of a Fourier approximation for different values of n. It can easily be seen that the higher the value of n the closer the approximation.

 


http://www.sosmath.com/fourier/pic/pic01.gif