To Study Fourier Series Representation of a Periodic Function

To Study Fourier Series Representation of a Periodic Function

Fourier series

Objective

To study Fourier series representation of a periodic function

Modules

Module I- Introduction

Module II- Fourier series of a function having period

Module III- Examples

Module I- Introduction

There are many examples in which the periodic input functions are not sinusoidal. The voltage input to a circuit or the force on a spring mass system may be periodic but possess discontinuities. These functions can be expressed as a trigonometric series in terms of sinusoids and cosinusoids within a desired range of variables. This series is known as Fourier series which was developed by the French mathematician Jacques Fourier. Fourier series is a powerful tool for solving some ordinary and partial differential equations.

Orthogonal functions

A set of continuous functions f1(x),f2(x),…,fn(x) which do not vanish identically in [a,b] is said to be orthogonal on the interval [a,b] if

for

andfor m=n.

For example, the set {1, cosx, sinx, cos2x, sin2x,…}is orthogonal in whereas the set {1, cos, sin, cos, sin,…}is orthogonal in [-C,C].

For orthogonal functions, we have the following results:

If m and n are integers then

Similarly,

Periodic Functions

A function f(x) which satisfies the relation f(x+t) = f(x) for all real x and some fixed T is called a periodic function. The smallest positive number T, for which this relation holds, is called a period of f(x).

If T is the period of f(x), then

f(x)=f(x+T)=f(x+2t)=…=f(x+nT)=…

f(x)=f(x-T)=f(x-2t)=…=f(x-nT)=…

Therefore f(x)=f(xnT), where n is a positive integer.

Thus, f(x) repeats itself after periods of T

For example, sinx, cosx, secx and cosecx are periodic functions with periodwhile tanx and cotx are periodic functions with period. The functions sinnx and cosnx are periodic with period.

The sum of a number of periodic functions is also periodic. If T1 and T2 are the periods of f(x) and g(x), then the period of af(x)+bg(x) is the least common multiple of T1 and T2.

For example, cosx, cos2x, cos3x are periodic functions with periodsand respectively.

Therefore, f(x)=cosx +cos2x+cos3x is also periodic with period, since the least common multiple ofand is.

Module II- Fourier series of a function having period

Let us suppose that f(x) is a periodic function with period over the intervalwhich can be represented by a trigonometric series

.

Now the problem is to find the coefficients a0, an and bn. Let us assume that the series on the right hand side can be integrated term by term in the given interval.

To determine a0, integrate both sides between the limitsand.

.

To determine an, multiply both sides by cosnx and then integrate both sides between the limitsand.

Remaining terms vanishes.

.

To determine bn, multiply both sides by sinnx and then integrate both sides between the limitsand.

Remaining terms vanishes.

.

Hence; and.

These values of a0, an and bn are called Euler’s formulae.

Note 1: If , we obtain the Fourier series in (0, ) as

where; and.

Note 2: If , we obtain the Fourier series in as

where; and.

Note 3: If f(x) is an odd function, .

Since cos nx is even, f(x)cos nx is odd. Therefore,

Since sin nx is odd, f(x)sin nx is even.

Therefore, .

Hence, if a periodic function f(x) is odd, its Fourier expansion contains only sine terms,

i.e., , where

Note 4: If f(x) is an even function, .

Since cos nx is even, f(x)cos nx is even.

Therefore,

Since sin nx is odd, f(x)sin nx is odd.

Therefore, .

Hence, if a periodic function f(x) is even, its Fourier expansion contains only cosine terms,

i.e., , where and

Module III- Examples

Example 1. Obtain the Fourier series to represent.

Solution:

Let.

By Euler’s formula, we have

.

.

.

Therefore,

Example 2: Expand as a Fourier series.

Solution:

Let

By Euler’s formula, we have

.

.

When n=1, we have

.

.

When n=1, we have

.

Therefore,

Example 3: Find the Fourier series for in.

Hence show that

Solution:

Here the function f(x) is even, therefore bn=0

Let

Then

.

.

Therefore,

.

Putting , we get

i.e.,.

Therefore,.

Assignment Questions

  1. Obtain the Fourier series for the function f(x)=-5x+2 in .
  2. Find the Fourier series for f(x)=e-x in the interval 0<x<.
  3. Express , , as Fourier series.
  4. If in the interval, show that . Hence obtain the relation .

Quiz questions

  1. The value of the integralwhere is

(a) (b) (c) 0

  1. If in the interval, then the value of a0 is

(a) (b) (c)

  1. If f(x) is an even function then the Fourier series expansion is

(a), where and

(b) , where and

(c) , where and

Answers

  1. (a)2. (b)3. (c)

Glossary

Function: A relation from a set X to another set Y which associates every element of X with a unique element of Y is called a function.

Even function: If f(-x)=f(x), then the function f(x) is said to be an even function.

Odd function: If f(-x)=-f(x), then the function f(x) is said to be an odd function.

Continuous function: A function ƒ : I → D is continuous at c ∈ I if for every ε > 0 there exists a δ > 0 such that for all x ∈ I, .

Series: A series is the sum of the terms of a sequence.

Summary

Orthogonal functions: A set of continuous functions f1(x),f2(x),…,fn(x) which do not vanish identically in [a,b] is said to be orthogonal on the interval [a,b] iffor andfor m=n.

Periodic Functions: A function f(x) which satisfies the relation f(x+t)= f(x) for all real x and some fixed T is called a periodic function. The smallest positive number T, for which this relation holds, is called a period of f(x).

If f(x) is a periodic function with period over the intervalwhich can be represented by a trigonometric series where

; and.

The series on RHS is called Fourier series of f(x). The values of a0, an and bn are called Euler’s formulae.

If , then where ; and.

If a periodic function f(x) is odd, its Fourier expansion contains only sine terms,

i.e., , where

If a periodic function f(x) is even, its Fourier expansion contains only cosine terms,

i.e., , where and

Frequently Asked Questions (FAQs)

  1. Find a Fourier series to represent x-x2 from x= to x=. Hence show that .

Answer:

Let

By Euler’s formula, we have

.

and second integral vanishes

and second integral vanishes

Therefore,

Putting x=0, we get

.

  1. Find the Fourier series for.

Answer:

Let

Now by Euler’s formula, we have

.

Substituting these values, we get

.

  1. Express , as Fourier series.

Hence deduce that.

Answer:

Here f(x) is even. Therefore bn=0.

Let.

Then.

Therefore,

Putting x=0, we get

.

REFERENCE

T.M. Apostol, Mathematical Analysis, Narosa Publishing, New Delhi (1996).
S.C. Malik, Savitha Arora, Mathematical analysis, New Age International Publishers, New Delhi (2003).
C. R. Wylie, L. Barret, Advanced Engineering Mathematics, Mc-Graw Hill, New York (1995).
Shanthi Narayan, A Course of Mathematical Analysis, S. Chand & Company, New Delhi.

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