Fourier Series

340

College Mathematics

5.

FOURIER SERIES

5.1 Introduction In various engineering problems it will be necessary to express a function in a series of sines and cosines which are periodic functions. Most of the single valued functions which are used in applied mathematics can be expressed in the form. 1 a0 + a1 cos x + a 2 cos2x +KK 2 + b1 sin x + b2 sin2x +KK within a desired range of values of x. Such a series is called a Fourier Series in the name of the French mathematician Jacques Foureier (1768 - 1830) 5.2 Periodic Functions Definition : If at equal intervals of the abscissa ‘x’ the value of each ordinate f(x) repeats itself then f(x) is called a periodic function. i.e., A function f(x) is said to be a periodic function if there exists a real number α such that f(x + α ) = f(x) for all x. The number α is called the period of f(x). ∴ we have f(x) = f(x + α ) = f(x + 2α ) = f(x + 3α ) = …………………..= f(x + n α ) = …………. Ex : (i) sin x = sin (x + 2π ) = sin (x +4π ) = …………… .......….= sin (x + 2n π ) = …………… Hence sin x is a periodic function of the period 2 π . (ii) cos x = cos(x + 2π ) = cos (x + 4π ) = ……… ……….. = cos (x + 2n π ) = ……………. Hence cos x is a periodic function of the period 2 π . We define the Fourier series in terms of these two periodic functions.

Fourier Series 341 5.3 Fourier Series Definition : A series of the form ∞ ∞ a nπ x nπ x f ( x) = 0 + ∑ a n cos( ) + ∑ bn sin( ) 2 n =1 l l n =1 is called a Fourier series of f(x) with period 2l in the interval ( c, c +2l ) where l is any positive real number and a 0, a n, bn are given by the formulae called Euler’s Formulae :

1 a0 = l

c+ 2 l

∫

c

f (x ) dx ,

an = bn =

1 l 1 l

c+ 2 l

∫

c

f ( x )cos( f ( x)sin(

nπ x )dx l

nπ x )dx l c These coefficients a 0, a n, bn are known as Fourier coefficients.

c+ 2l

∫

In particular if l = π , the Fourier series of f(x) with period 2π in the interval (c, c+2 π ) is given by ∞ ∞ a f ( x) = 0 + ∑ an cos nx + ∑bn sin...