Math Guide

Fourier series VI

In mathematics we come across a number of series’ most of which have practical implications. Explicitly a series is described by mentioning all the terms or a few terms to indicate the pattern. But they can be more conveniently expressed in a recursive form.

Forexample, the series Sn= 1 + 2 + 3 + 4 + ….+ n, indicates the sum of natural numbers up to‘n’ terms and also tells us the pattern that the series is anarithmetic series of natural numbers for ‘n. terms. The recursiveformula to represent the same meaning is, Sn= (1 + n)(n/2).

Inexact analogy to what we described, periodic functions can beexpressed as sum (may even be sum of infinite terms) of simplefunctions. That is in a form of series. Here again the series isexpressed in recursive form. These types of series are called Fourierseries or Fourier Series Transform or Generalized Fourier Series.

Normallyin Fourier-Series Expansion Examples, we prefer to select the ‘simplefunctions’ as trigonometric functions. Suppose f(x) is a continuousfunction in the interval [-π, π], then f(x) can be expressed inseries form as,

f(x)= (a0/2)+, where N isgreater than 0 and an and bn means anand bn.

Theconstants a0,anand bnare called as Fourier coefficients. They are determined from thefollowing formulas,

an= where n ≥0, and bn=where n ≥ 1.

Theabove statement is a recursive formula for Fourier-series. FourierSeries Proof can be shown in many ways but all of them are cumbersomeand beyond the scope of this article.

Letus illustrate an example as one of the Fourier-series problems. 

Let afunction be defined as f(x) = x/(2π), a typical harmonic function.Now the Fourier coefficients are,

a0= = 0

an= = 0

bn= = [(-1)n+1]/(nπ)

Therefore,the series in recursive form is,

f(x)= (1/π)*

Thisis an example of a function expressed as Fourier’s series. But whatis the advantage we get out of this. The explanation is, the computerprogramming accepts such functions for a limited domain. In practicalapplications, functions of these types are required to be studied inlimited domains. 

Therefore, one can be benefited by this derivation.

Thus,we had seen the Fourier series as a practical tool that helps inpractical situations.