# Fourier Series

Most of the phenomena studied in the domain of engineering and science are periodic in nature> For instance, current and voltage in an alternating current circuit. These periodic functions could be analyzed into their constituent components (fundamentals and harmonics) by a process called Fourier analysis.

Periodic functions occur frequently in the problems studied during engineering education. Their representation in terms of simple periodic functions such as sine and cosine, which leads to Fourier series(FS). Fourier series is a very powerful tool in connection with various problems involving partial differential equations.

The Fourier series representation of analytic functions is derived from Laurent expansions. The elementary complex analysis is used to derive additional fundamental results in the harmonic analysis including the representation of C∞ periodic functions by Fourier series, the representation of rapidly decreasing functions by Fourier integrals, and Shannon’s sampling theorem. The ideas are classical and of transcendent beauty.

Laurent series yield Fourier series
A difficult thing to understand and/or motivate is the fact that arbitrary periodic functions have Fourier series representations. In this section, we prove that periodic analytic functions have such a representation using Laurent expansions.

(Periodic function)
A function is periodic of period L if f(x+L) = f(x) for all x in the domain of f. The smallest positive value of L is called the fundamental period.

The trigonometric functions sin x and cos x are examples of periodic functions with fundamental period 2π and tan x is periodic with fundamental period π. A constant function is a periodic function with arbitrary period L.

It is easy to verify that if the functions f1, . . . , fn are periodic of period L, then any linear combination

c1f1(x)+…+cnfn(x)
is also periodic. Furthermore, if the infinite series

12ao+∑∞n=1ancosnπxL+bnsinnπxL
consisting of 2L-periodic functions converges for all x, then the function to which it converges will be periodic of period 2L. There are two symmetry properties of functions that will be useful in the study of Fourier series.

Even and Odd Function
A function f(x) is said to be even if f(−x) = f(x).
The function f(x) is said to be odd if f(−x) = −f(x). Graphically, even functions have symmetry about the y-axis, whereas odd functions have symmetry around the origin.