Fourier Series

Fourier Series
Definition
Fourier series are infinite series that represent periodic functions in terms of cosines and sines. As such, Fourier series are of greatest importance to the engineer and applied mathematician. To define Fourier series, we first need some background material.
A function f(x) is called a periodic function if f(x) is defined for all real x, except possibly at some points, and if there is some positive number 2L, called a period of , such that for all x

f(x+2L)=f(x)

The graph of a periodic function has the characteristic that it can be obtained by periodic repetition of its graph (e.g. Fig. 1) in any interval of length 2L. The smallest positive period is often called the fundamental period. If f(x) has period 2L, it also has the period 4L because
f(x+4L)=f(?(x+2L)?f(x) +2L)=f(x+2L)=f(x)

The functions in the figure above have the same 2? period.

The Fourier Series Expansion Equation

The Fourier Expansion for a function f(x) defined in the interval [-L, L] is:
f(x)=a_0/2+?_(n=1)^??(a_n cos??n?/L? x+b_n sin??n?/L? x)
The constants a_0,a_n and b_n in the equation above can be calculated as:
a_0=1/L ?_(-L)^L??f(x) dx?
a_n=1/L ?_(-L)^L??f(x) cos??n?/L? x dx?
b_n=1/L ?_(-L)^L??f(x) sin??n?/L? x dx?
If the function f(x) (with 2L period) defined in an interval different from [-L, L]; the Fourier series will be:
f(x)=a_0/2+?_(n=1)^??(a_n cos??n?/L? x+b_n sin??n?/L? x)