Use, Advantages, Periodic Function of Fourier Series

UNIT ‒ III

FOURIER SERIES

INTRODUCTION

Fourier series, is named after the French Mathematician cum physicist Jean‒Baptiste Joseph Fourier (1768 ‒ 1830). He introduced Fourier Series in 1822, while he was investigating the problem of heat conduction. The series of sines and cosines are known after him.

Fourier Series are series of cosine and sine terms and arise in the important practical task of representing general periodic functions. They constitute a very important tool in solving problems that involve ordinary and partial differential equations.

Use of Fourier Series:

Fourier series are particularly suitable for expansion of periodic functions. We come across many periodic functions in voltage, current, flux, density, applied force, potential and electromagnetic force in electricity. Hence, Fourier Series are very useful in electrical engineering problems.

PERIODIC FUNCTION:

Definition : Periodic Function

A function f(x) is said to be periodic, if and only if f(x + p) = f(x) is true for some value of p and every value of x. The smallest value of P for which this equation is true for every value of x will be called the period of the function f(x).

Example:

1. sin x = sin (x + 2π) = sin (x+4π) = ….

So sinx is a periodic function with the period 2π. This is also called Sinusoidal periodic function.

2. The trignometric functions sinx and cosx are periodic functions with functions fundamental (primitive) period 2π.

3. sin 2x and cos 2v are also periodic functions with fundamental period π.

4. tanx is a periodic function with period π.

5. Find the period of sinnx where n is a positive integer

Solution

Let f(x) = sin nx = sin (nx + 2π)

 = sin n ( x + 2π/n ) = f(x + 2π/n)

Therefore, 2π/n is the period of sin nx, 2π/n is the period of cos nx

 π/n is the period of tan nx

6. Show that a constant has any positive number as period.

Solution:

Let f(x) = c, where c is a constant.

then f(x+k) = c, k being any positive number

that is f(x + k) = f(x)

So f(x) is periodic with period k.

Note: Since, there is no least value of k, we say that f(x) = c has no fundamental period.

7. Let f : R → R be the function defined by

Let p be any rational number. If x is rational, then x + p is also rational and if x is irrational, then x + p is also irrational.

Hence, every rational number is a period of f and f has no fundamental period.

8. Let f and g be periodic functions with period p and let a and b be real numbers. Prove that af + bg is also a periodic function with period p.

Solution: Since, f and g are periodic with period p

 f(x+p) = f(x)            ……..(1),

  g (x + p) = g(x)          ……….(2)

Now (af+bg) (x +p) = af(x+p) + bg(x +p)

= af(x) + bg(x) by (1) and (2)

= (af + bg) (x)

Hence, af + bg is periodic with period p.

9. If p is a period of f(x), show that np is also a period where n is any integer (positive or negative)

Solution:

Since

 f(x) = f(x+p) = f (x + 2p) = ….  = f[x + (n‒1)p] = f (x + np) it follows that np is period of f.

10. Draw the graph of y = |x| in ‒1≤x≤1

Note: |x| = ‒x, x ≤ 0

= x, x≥0

CONTINUITY OF A FUNCTION :

The left hand limit of f(x) at x = a is defined as x approaches a from the left and is denoted by f(a −).

f(a ‒) = Lt_h→0 f(a − h) as h→0 through positive values.

The right hand limit of f(x) at x = a is defined as x approaches a from the right and is denoted by f(a +)

 f(a +) = Lt_h→0 f (a + h)

A function f(x) is said to be continuous at x=a if f(a+) = f(a) = f(a−)

Note:

1. f(a) is different from f(a +) and f(a −), f(a) means the value of f(x) at x = a.

2. If there is a finite jump in the graph of y = f(x) at x=a, the function is not continuous at x = a. (i.e.,) The function is not defined at x = a.

In such cases, both right hand and left hand limits are not equal. The function f(x) is piecewise continuous in an interval (a, b) means that f(x) is continuous at all, but a finite number of points in (a, b).

Example: f(x) = 

Here x = 1 is a point of finite discontinuity for f(x).

 [ f (1−) ≠ f(1+) since f(1‒) = 1, f(1+) = ½ ]

ADVANTAGES OF FOURIER SERIES:

1. Discontinuous function can be represented by Fourier series, although derivatives of the discontinuous functions do not exist. (This is not true for Taylor's series).

2. The Fourier series is useful in expanding the periodic functions, since outside the closed interval, there exists a periodic extension of the function.

3. Expansion of an oscillating function by Fourier series gives all modes of oscillation (fundamental and all overtones) which is extremely useful in physics.

4. Fourier series of a discontinuous function is not uniformly convergent at all points.

5. Term by term integration of a convergent Fourier series is always valid, and it may not be valid if the series is not convergent. However, term by term, differentiation of a Fourier series is not valid if the series is not convergent.