Topic: What are the Euler’s formulas for the Fourier coefficient s? By what idea did we get them?

CONTENTS

· INTRODUCTION

1. WHO IS FOURIER

2. HISTORY OF FOURIER SERIES

· FOURIER SERIES(THINGS TO LEARN BEFORE FOURIER SERIES)

1. TAYLOR’s EXPANSION

2. PERIODIC FUNCTION

· EULER’s FORMULAE

1. WHAT IS EULER’S FORMULAE

2. HOW DID WE GET EULERS FORMULAE, FROM WHAT IDEA DID WE GET THEM

3. EXAMPLE SOLVING FOURIER SERIES USING EULER’s FORMULAE

· REFERENCE

ü INTRODUCTION

WHO IS FOURIER-

Trigonometry has come a long way since its inception more than

Two thousand years ago. But three developments, more than all

Others stand out as having fundamentally changed the subject:

Ptolemy’s table of chords, which transformed trigonometry into

A practical, computational science; De Moivre’s theorem and

Euler’s formula, which merged trigonometry

With algebra and analysis; and Fourier’s theorem.

Jean Baptiste Joseph Fourier was born in Auxerre in northcentral

France on March 21, 1768. By the age of nine both his

Father and mother had died. Through the influence of some

Friends of the family, Fourier was admitted to a military school

Run by the Benedictine order, where he showed an early talent in

Mathematics. He was particularly interested in the manner in which

Heat flows from a region of high temperature to one of lower

Temperature. Newton had already studied this question and

found that the rate of cooling (drop in temperature) of an object

is proportional to the difference between its temperature

and that of its surroundings. Newton’s law of cooling, however,

governs only the temporal rate of change of temperature, not

its spatial rate of change, or gradient. This latter quantity depends

on many factors: the heat conductivity of the object, its

geometric shape, and the initial temperature distribution on its

boundary. To deal with this problem one must use the analytic

tools of the continuum, in particular partial differential equations

(see p. 53). Fourier showed that to solve such an equation

one must express the initial temperature distribution as a sum

of infinitely many sine and cosine terms—a trigonometric or

Fourier series.

HISTORY OF FOURIER SERIES-

A Fourier series of a periodic function consists of a sum of sine and cosine terms. Sines and cosines are the most fundamental periodic functions.

The Fourier series is named after the French Mathematician and Physicist Josephs Fourier (1768 – 1830). Fourier series has its application in problems pertaining to Heat conduction, acoustics, etc. The subject matter may be divided into the following sub topics.

FORMULA FOR FOURIER SERIES

Consider a real-valued function f(x) which obeys the following conditions called Dirichlet’s conditions :

1. f(x) is defined in an interval (a,a+2l), and f(x+2l) = f(x) so that f(x) is a periodic function of period 2l.

2. f(x) is continuous or has only a finite number of discontinuities in the interval (a,a+2l).

3. f(x) has no or only a finite number of maxima or minima in the interval (a,a+2l).

Also, let

(1)

(2)

(3)

Then, the infinite series

(4)

is called the Fourier series of f(x) in the interval (a,a+2l). Also, the real numbers a₀, a₁, a₂, ….a_n, and b₁, b₂ , ….b_n are called the Fourier coefficients of f(x). The formulae (1), (2) and (3) are called Euler’s formulae.

It can be proved that the sum of the series (4) is f(x) if f(x) is continuous at x. Thus we have

f(x) = ……. (5)

ü FOURIER SERIES

Before we begin the study of Fourier, we will go thru one of the most famous series: Taylor Series, which approximate function by polynomial.

TAYLOR’s EXPANSION

f(x) is Cⁿ⁺¹ over x[a,b]; let x, x₀ [a,b] then

f(x) = P_n(x) + R_n+1(x) where

(n^th Taylor polynomial)

Note:

is called the n^th variation of

for someζ

[x,x₀] (n^th remainder)

Taylor Series:

(expanding at point x₀=a)

Ex.:

(expanding at x₀=0)

(expanding at x₀=0)

PERIODIC FUNCTION

A periodic function has a basic shape which is repeated over and over again. The fundamental range is the time (or sometimes distance) over which the basic shape is defined. The length of the fundamental range is called the period.

A general periodic function f(x) of period T satisfies the condition

f(x+T) = f(x)

Here f(x) is a real-valued function and T is a positive real number.

As a consequence, it follows that

f(x) = f(x+T) = f(x+2T) = f(x+3T) = ….. = f(x+nT)

Thus,

f(x) = f(x+nT), n=1,2,3,…..

The function f(x) = sinx is periodic of period 2p since

Sin(x+2np) = sinx, n=1,2,3,……..

The graph of the function is shown below :

Note that the graph of the function between 0 and 2p is the same as that between 2p and 4p and so on. It may be verified that a linear combination of periodic functions is also periodic.

ü EULER’s FORMULA

WHAT ARE EULER’S FORMULAE

For periodic function f(x), piecewise continuous, with period 2π which can be represented by a trigonometric series: