Question

Can any one Explain Fourier Series?

Answer 1

fourier series define phenomena in terms of a series of the sums of sines and cosines i believe

Answer 2

It turns out to be that any given periodic function can be well defined / described by summing sin and cos functions.

take for an example a square wave function with 2pi period
It kind of reminds us of a simple sin(x) function

but not exactly, we need to decrese in some places and increase in others
so we add another sin function with a different period sin(3x)

and another one sin(5x)
if we go on forever summing sin functions we can get a very good description of the squre wave
http://en.wikipedia.org/wiki/Fourier_series

why is it important and usefull? because thats how nature works
when we hear and record some sound lets say a piano playing thee notes

the sum of the three notes will form a new periodic funtion, but using fourier serires we can disassemble it back to three original notes

the waveform of an each instrument is different,
http://www.google.co.il/imgres?um=1&hl=iw&rlz=1C1CHEU_iwIL402IL402&authuser=0&tbm=isch&tbnid=IubfSdlMhSqgyM:&imgrefurl=http://swastidesign.blogspot.com/2010/04/physics-of-sound.html&docid=tnOB61VXGeZ6KM&imgurl=http://3.bp.blogspot.com/_mRHX7TPXcLg/S-_v7PfZhzI/AAAAAAAAAu8/JYwmIncvlEI/s1600/Screen%252Bshot%252B2010-05-16%252Bat%252B12.52.27.png&w=416&h=343&ei=UGSsT6q9BoSr0QWbwJjYBA&zoom=1&iact=hc&vpx=539&vpy=241&dur=216&hovh=204&hovw=247&tx=133&ty=150&sig=106460102523178130288&page=1&tbnh=117&tbnw=142&start=0&ndsp=18&ved=1t:429,r:13,s:0,i:96&biw=1066&bih=572

and we can analyze the different overtones of each instument fom the waveform with fourier series

Answer 3

Any periodic function can be expressed as the superposition of sinusoidal waves of different frequencies. Square wave, triangular wave, sawtooth wave can be expressed as the summation of sine and cosine terms of different frequencies. \[f(x)=a0/2+\sum_{n=0}^{?n=infinity}an cosnx+\sum_{n=0}^{?n=infinity}bn sinnx\]

where
\[a0=1/pi*\int\limits_{0}^{pi} f(x) dx\]

\[an=(1/pi)*\int\limits_{0}^{pi}f(x) \cos nx dx\]

\[bn=(1/pi)*\int\limits_{0}^{pi} f(x) \sin nx dx\]