Fourier series:

Fourier series: Periodic Functions

A function f(x) is periodic if there is a positive number T such that for every value of x:


?(f(x+T)= f(x))


The smallest such value of T>0 is called the fundamental period or simply the period.


If we know what the graph looks like in an interval of length T, then we can use replication to sketch the entire graph.


Fourier Series

Let f(x) be defined in the interval (-L,L) and outside of this interval by f(x+2L)= f(x)
(i.e assume that f(x) has a period T=2L). Then the trigonometric Fourier series corresponding to f(x)is given by:

?(f(x)=a_0/2+?_(n=1)^??[a_n cos??n?x/L?+b_n sin??n?x/L? ] )

Where a_0,a_n andb_n are constants and usually referred to as Fourier coefficients.
The Fourier coefficients is given by Euler formulas as:

?(a_0=1/L ?_(-L)^L??f(x) dx?)

?(a_n=1/L ?_(-L)^L??f(x) cos??n?x/L? dx ,? for n=1,2,3,?)






Example 1: What is the fundamental period of the following fun

Solution:
a) The period is 3
b) The period is 6
c) The period is 4




Period of Trigonometric Functions