# Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 3 Fourier Series 3.1 Some Properties of Functions Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results will be needed for the remaining sections. We also introduce some notation Preliminary Remarks Joseph Fourier ( ) who gave his name to Fourier series, was not the first to use Fourier series neither did he answer all the questions about them. These series had already been studied by Euler, d Alembert, Bernoulli and others before him. Fourier also thought wrongly that any function could be represented by Fourier series. However, these series bear his name because he studied them extensively. The first concise study of these series appeared in Fourier s publications in 187, 1811 and 1822 in his Théorie analytique de la chaleur. He applied the technique of Fourier series to solve the heat equation. He had the insight to see the power of this new method. His work set the path for techniques that continue to be developed even today. Fourier Series, like Taylor series, are special types of expansion of functions. With Taylor series, we are interested in expanding a function in terms of the special set of functions 1, x, x 2, x 3,... or more generally in terms of 1, (x a), (x a) 2, (x a) 3,... You will remember from calculus that if a function f has a power series representation at a then f (x) = n= f (n) (a) n! (x a) n (3.1) Remember from calculus that a series is an infinite sum. We never use the full se- 117

2 118 CHAPTER 3. FOURIER SERIES N f (n) (a) ries, we usually truncate it. In other words, if we call S N (x) = (x a) n, n! n= then we approximate f (x) by S n (x). S n (x) is called a partial sum. A reason for using Taylor series is that their partial sums are polynomials and polynomials are the easiest functions to work with. With Fourier series, we are interested in expanding a function f in terms of the special set of functions 1, cos πx 2πx 3πx πx 2πx, cos, cos,..., sin, sin, sin 3πx,... Thus, a Fourier series expansion of a function is an expression of the form f (x) = A + n=1 ( A n cos nπx + B n sin nπx ) for some positive constant. In the previous chapters, we saw this was useful in helping us to solve certain PDEs. Another reason for using Fourier series is if f (x) represents some signal (light, sound) since signals are a combination of periodic functions. So it is natural we might want to write f (x) as a Fourier series. However, there are several questions which arise when trying to achieve this. We list them here and will try to answer most of them in this chapter. It is important for the reader to be aware of these questions. 1. Given a function f (x), how do we know if it has a Fourier series representation? Fourier thought every function did. It turns out it is not quite the case, though many functions do. 2. Given a function f (x) which has a Fourier series representation, how do we find the coeffi cients A n and B n? We have already answered this question in the previous chapters. 3. Infinite series do not always converge. Some converge for some values of x and not for others. Does the Fourier series converge and for which values of x? 4. Even if the Fourier series of a function f converges, does it converge to f (x)? We will see that even if a Fourier series converges, it does not always converge to f (x). 5. Since a Fourier series is an infinite sum, the sum rules we know for derivatives and integrals do not apply here. In other words, how do we differentiate and integrate a Fourier series? Can we differentiate and integrate a Fourier series term by term? We assumed we could in the previous chapter, but we do not know that. Though we will see it is true for most functions, it is not always true. 6. Given an initial boundary value problem (IBVP), is the resulting Fourier series really a solution of the IBVP? Keep in mind that to

3 3.1. SOME PROPERTIES OF FUNCTIONS 119 check this, we will have to differentiate the Fourier series, hence the answer to the previous question is relevant. After reviewing some properties of functions, we will review how to represent a function by its Fourier series. We will then answer the question listed above. We will finish these notes by discussing some applications of Fourier series Even, Odd Functions Definition 16 (Even and Odd) et f be a function defined on an interval I (finite or infinite) centered at x =. 1. f is said to be even if f ( x) = f (x) for every x in I. 2. f is said to be odd if f ( x) = f (x) for every x in I. The graph of an even function is symmetric with respect to the y-axis. The graph of an odd function is symmetric with respect to the origin. Example 17 1, x 2, x n (where n is even), and cos x are all even functions Example 18 x, x 3, x n (where n is odd), and sin x are all odd functions. You will recall from calculus the following important theorem about integrating even and odd functions over an interval of the form [ a, a] where a >. Theorem 19 et f be a function which domain includes [ a, a] where a >. 1. If f is even, then a a f (x) dx = 2 a f (x) dx 2. If f is odd, then a f (x) dx = a There are several useful algebraic properties of even and odd functions as shown in the theorem below. Theorem 11 When adding or multiplying even and odd functions, the following is true: even + even = even odd + odd = odd even even = even odd odd = even even odd = odd

4 12 CHAPTER 3. FOURIER SERIES Figure 3.1: Graph of an Even Function Figure 3.2: Graph of an Odd Function

5 3.1. SOME PROPERTIES OF FUNCTIONS Periodic Functions Definition 111 (Periodic) et T >. 1. A function f is called T -periodic or simply periodic if for all x. 2. The number T is called a period of f. f (x + T ) = f (x) (3.2) 3. If f is non-constant, then the smallest positive number T with the above property is called the fundamental period or simply the period of f. Remark 112 et us first remark that if T is a period for f, then nt is also a period for any integer n >. This is easy to see using equation 3.2 repeatedly: f (x) = f (x + T ) = f ((x + T ) + T ) = f (x + 2T ) = f ((x + 2T ) + T ) = f (x + 3T ). = f ((x + (n 1) T ) + T ) = f (x + nt ) Classical examples of periodic functions are sin x, cos x and other trigonometric functions. sin x and cos x have period 2π. tan x has period π. We will see more examples below. Because the values of a periodic function of period T repeat every T units, it is enough to know such a function on any interval of length T. Its graph is obtained by repeating the portion over any interval of length T. Consequently, to define a T -periodic function, it is enough to define it over any interval of length T. Since different intervals may be chosen, the same function may be defined different ways. Example 113 Describe the 2-periodic function shown in figure 3.3 in two different ways: 1. By considering its values on the interval x < 2; 2. By considering its values on the interval 1 x < 1. Solution 1. On the interval x < 2, the function is a portion of the line y = x + 1 thus f (x) = x + 1 if x < 2. The relation f (x + 2) = f (x) describes f for all other values of x. 2. On the interval 1 x < 1, the function consists of two lines. So we have { x 1 if 1 x < f (x) = x + 1 if x, 1 The relation f (x + 2) = f (x) describes f for all other values of x.

6 122 CHAPTER 3. FOURIER SERIES Figure 3.3: A Function of Period 2 Although we have different formulas, they describe the same function. Of course, in practice, we use common sense to select the most appropriate formula. Next, we look at an important theorem concerning integration of periodic functions over one period. Theorem 114 (Integration Over One Period) Suppose that f is T -periodic. Then for any real number a, we have T f (x) dx = a+t a f (x) dx (3.3) Proof. Define F (a) = a+t f (x) dx. By the fundamental theorem of calculus, a F (a) = f (a + T ) f (a) = since f is T -periodic. Hence, F (a) is a constant for all a. In particular, F () = F (a) which implies the theorem. We illustrate this theorem with an example. Example 115 et f be the 2-periodic function shown in figure 3.3. Compute the integrals below: [f (x)]2 dx 2. N N [f (x)]2 dx where N is any positive integer. Solution 116 We answer each part separately. 1. We described this function earlier and noticed that its simplest expression was not over the interval [ 1, 1] but over the interval [, 2]. We should

7 3.1. SOME PROPERTIES OF FUNCTIONS 123 also note that if f is 2-periodic, so is [f (x)] 2 (why?). Using theorem 114, we have 1 1 [f (x)] 2 dx = = 2 2 = 1 3 = 2 3 [f (x)] 2 dx ( x + 1) 2 dx ( x + 1) We break N N [f (x)]2 dx into the sum of N integrals over intervals of length 2. N N [f (x)] 2 dx = N+2 N [f (x)] 2 dx+ By theorem 114, each integral is 2 3. Thus N N N+4 N+2 [f (x)] 2 dx = 2N 3 N [f (x)] 2 dx+...+ [f (x)] 2 dx N 2 The following result about combining periodic functions is important. Theorem 117 When combining periodic functions, the following is true: 1. If f 1, f 2,..., f n are T -periodic, then a 1 f 1 + a 2 f a n f n is also T - periodic. 2. If f and g are two T -periodic functions so is f (x) g (x). 3. If f and g are two T -periodic functions so is f(x) g(x) where g (x). 4. If f has period T and a > then f ( x a ) has period at and f (ax) has period T a. 5. If f has period T and g is any function (not necessarily periodic) then the composition g f has period T. Proof. See problems. We finish this section by looking at another example of a periodic function, which does not involve trigonometric functions but rather the greatest integer function, also known as the floor function, denoted x. x represents the greatest integer not larger than x. For example, 5.2 = 5, 5 = 5, 5.2 = 6, 5 = 5. Its graph is shown in figure 3.4.

8 124 CHAPTER 3. FOURIER SERIES Figure 3.4: Graph of x Figure 3.5: Graph of x x Example 118 et f (x) = x x. This gives the fractional part of x. For x < 1, x =, so f (x) = x. Also, since x + 1 = 1 + x, we get f (x + 1) = x + 1 x + 1 = x x = x x = f (x) So, f is periodic with period 1. Its graph is obtained by repeating the portion of its graph over the interval x < 1. Its graph is shown in figure The practice problems will explore further properties of periodic functions.

9 3.1. SOME PROPERTIES OF FUNCTIONS Orthogonal Family of Functions The functions in the 2-periodic trigonometric system 1, cos πx, cos 2πx, cos 3πx,..., sin πx 2πx 3πx, sin, sin,... are among the most important periodic functions. The reader will verify that they are indeed 2-periodic in the homework. They share another important property. Theorem 119 The family of functions {1, cos nπ x, sin nπ } x : n N forms an orthogonal family on the interval [, ] in other words, if m and n are two nonnegative integers, then 1, cos nπ x = for n = 1, 2,... (3.4) 1, sin nπ x = for n = 1, 2,... sin nπ mπ x, cos x = m, n sin nπ mπ x, sin x = if m n cos nπ mπ x, cos x = if m n Proof. The proof has been done in the previous chapter over several section. We remind the reader of the important trigonometric identities which are used in evaluating these integrals. sin α cos β = 1 [sin (α + β) + sin (α β)] 2 cos α sin β = 1 [sin (α + β) sin (α β)] 2 sin α sin β = 1 [cos (α + β) cos (α β)] 2 cos α cos β = 1 [cos (α + β) + cos (α β)] 2 Remark 12 We also have the useful identities cos 2 mπ xdx = Practice Problems 1. Prove theorem Prove theorem 11. sin 2 mπ xdx = for all m (3.5)

10 126 CHAPTER 3. FOURIER SERIES 3. Sums of periodic functions. Show that if f 1, f 2,..., f n are T -periodic, then a 1 f 1 + a 2 f a n f n is also T -periodic. 4. Sums of periodic functions. et f (x) = cos x + cos πx. (a) Show that the equation f (x) = 2 has a unique solution. (b) Conclude from part a that f is not periodic. Does this contradict the previous problem? 5. Finish proving theorem Operations on periodic functions. (a) Show that if f and g are two T -periodic functions so is f (x) g (x). (b) Show that if f and g are two T -periodic functions so is f(x) g(x) g (x). where (c) Show that if f has period T and a > then f ( x a ) has period at and f (ax) has period T a. (d) Show that if f has period T and g is any function (not necessarily periodic) then the composition g f has period T. 7. Using the previous problem, find the period of the functions below. (a) sin 2x (b) cos 1 2x + 3 sin 2x (c) 1 2+sin x (d) e cos x 8. Show that the functions 1, cos πx are 2-periodic., cos 2πx, cos 3πx,..., sin πx 2πx 3πx, sin, sin, Antiderivative of periodic functions. Suppose that f is 2π-periodic and let a be a fixed real number. Define F (x) = x a f (t) dt for all x Show that F is 2π-periodic if and only if 2π f (t) dt =. (hint: use theorem 114)

### Fourier Series. Some Properties of Functions. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Fourier Series Today 1 / 19

Fourier Series Some Properties of Functions Philippe B. Laval KSU Today Philippe B. Laval (KSU) Fourier Series Today 1 / 19 Introduction We review some results about functions which play an important role

### Lecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x)

ecture VI Abstract Before learning to solve partial differential equations, it is necessary to know how to approximate arbitrary functions by infinite series, using special families of functions This process

### Fourier Series Chapter 3 of Coleman

Fourier Series Chapter 3 of Coleman Dr. Doreen De eon Math 18, Spring 14 1 Introduction Section 3.1 of Coleman The Fourier series takes its name from Joseph Fourier (1768-183), who made important contributions

### MATH 461: Fourier Series and Boundary Value Problems

MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter

### M344 - ADVANCED ENGINEERING MATHEMATICS Lecture 9: Orthogonal Functions and Trigonometric Fourier Series

M344 - ADVANCED ENGINEERING MATHEMATICS ecture 9: Orthogonal Functions and Trigonometric Fourier Series Before learning to solve partial differential equations, it is necessary to know how to approximate

### 6.8 Taylor and Maclaurin s Series

6.8. TAYLOR AND MACLAURIN S SERIES 357 6.8 Taylor and Maclaurin s Series 6.8.1 Introduction The previous section showed us how to find the series representation of some functions by using the series representation

### CHAPTER 2 FOURIER SERIES

CHAPTER 2 FOURIER SERIES PERIODIC FUNCTIONS A function is said to have a period T if for all x,, where T is a positive constant. The least value of T>0 is called the period of. EXAMPLES We know that =

### Introduction to Fourier Series

Introduction to Fourier Series MA 16021 October 15, 2014 Even and odd functions Definition A function f(x) is said to be even if f( x) = f(x). The function f(x) is said to be odd if f( x) = f(x). Graphically,

### Maths 361 Fourier Series Notes 2

Today s topics: Even and odd functions Real trigonometric Fourier series Section 1. : Odd and even functions Consider a function f : [, ] R. Maths 361 Fourier Series Notes f is odd if f( x) = f(x) for

Week 5 Vladimir Dobrushkin 5. Fourier Series In this section, we present a remarkable result that gives the matehmatical explanation for how cellular phones work: Fourier series. It was discovered in the

### 4.3 Limit of a Sequence: Theorems

4.3. LIMIT OF A SEQUENCE: THEOREMS 5 4.3 Limit of a Sequence: Theorems These theorems fall in two categories. The first category deals with ways to combine sequences. Like numbers, sequences can be added,

### Representation of functions as power series

Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

### Recap on Fourier series

Civil Engineering Mathematics Autumn 11 J. Mestel, M. Ottobre, A. Walton Recap on Fourier series A function f(x is called -periodic if f(x = f(x + for all x. A continuous periodic function can be represented

### Chapter Three. Functions. In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics.

Chapter Three Functions 3.1 INTRODUCTION In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics. Definition 3.1: Given sets X and Y, a function from X to

### Engineering Mathematics II

PSUT Engineering Mathematics II Fourier Series and Transforms Dr. Mohammad Sababheh 4/14/2009 11.1 Fourier Series 2 Fourier Series and Transforms Contents 11.1 Fourier Series... 3 Periodic Functions...

### 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is:

CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx

### ANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series

ANALYTICAL MATHEMATICS FOR APPLICATIONS 206 LECTURE NOTES 8 ISSUED 24 APRIL 206 A series is a formal sum. Series a + a 2 + a 3 + + + where { } is a sequence of real numbers. Here formal means that we don

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### Fourier series. Jan Philip Solovej. English summary of notes for Analysis 1. May 8, 2012

Fourier series Jan Philip Solovej English summary of notes for Analysis 1 May 8, 2012 1 JPS, Fourier series 2 Contents 1 Introduction 2 2 Fourier series 3 2.1 Periodic functions, trigonometric polynomials

### M3 PRECALCULUS PACKET 1 FOR UNIT 5 SECTIONS 5.1 TO = to see another form of this identity.

M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 5. USING FUNDAMENTAL IDENTITIES 5. Part : Pythagorean Identities. Recall the Pythagorean Identity sin θ cos θ + =. a. Subtract cos θ from both sides

### Course MA2C02, Hilary Term 2012 Section 8: Periodic Functions and Fourier Series

Course MAC, Hiary Term Section 8: Periodic Functions and Fourier Series David R. Wikins Copyright c David R. Wikins Contents 8 Periodic Functions and Fourier Series 37 8. Fourier Series of Even and Odd

### Fourier Series Expansion

Fourier Series Expansion Deepesh K P There are many types of series expansions for functions. The Maclaurin series, Taylor series, Laurent series are some such expansions. But these expansions become valid

### Taylor and Maclaurin Series

Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

### Lecture5. Fourier Series

Lecture5. Fourier Series In 1807 the French mathematician Joseph Fourier (1768-1830) submitted a paper to the Academy of Sciences in Paris. In it he presented a mathematical treatment of problems involving

### CHAPTER 3. Fourier Series

`A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL

### Fourier Series Representations

Fourier Series Representations Introduction Before we discuss the technical aspects of Fourier series representations, it might be well to discuss the broader question of why they are needed We ll begin

### Introduction to Fourier Series

Introduction to Fourier Series A function f(x) is called periodic with period T if f(x+t)=f(x) for all numbers x. The most familiar examples of periodic functions are the trigonometric functions sin and

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### 1 Mathematical Induction

Extra Credit Homework Problems Note: these problems are of varying difficulty, so you might want to assign different point values for the different problems. I have suggested the point values each problem

### Second Project for Math 377, fall, 2003

Second Project for Math 377, fall, 003 You get to pick your own project. Several possible topics are described below or you can come up with your own topic (subject to my approval). At most two people

### Infinite series, improper integrals, and Taylor series

Chapter Infinite series, improper integrals, and Taylor series. Introduction This chapter has several important and challenging goals. The first of these is to understand how concepts that were discussed

### Fourier Series. 1. Full-range Fourier Series. ) + b n sin L. [ a n cos L )

Fourier Series These summary notes should be used in conjunction with, and should not be a replacement for, your lecture notes. You should be familiar with the following definitions. A function f is periodic

### Fourier Series. A Fourier series is an infinite series of the form. a + b n cos(nωx) +

Fourier Series A Fourier series is an infinite series of the form a b n cos(nωx) c n sin(nωx). Virtually any periodic function that arises in applications can be represented as the sum of a Fourier series.

### Lecture 5 : Continuous Functions Definition 1 We say the function f is continuous at a number a if

Lecture 5 : Continuous Functions Definition We say the function f is continuous at a number a if f(x) = f(a). (i.e. we can make the value of f(x) as close as we like to f(a) by taking x sufficiently close

### ) + ˆf (n) sin( 2πnt. = 2 u x 2, t > 0, 0 < x < 1. u(0, t) = u(1, t) = 0, t 0. (x, 0) = 0 0 < x < 1.

Introduction to Fourier analysis This semester, we re going to study various aspects of Fourier analysis. In particular, we ll spend some time reviewing and strengthening the results from Math 425 on Fourier

### Fourier Series and Sturm-Liouville Eigenvalue Problems

Fourier Series and Sturm-Liouville Eigenvalue Problems 2009 Outline Functions Fourier Series Representation Half-range Expansion Convergence of Fourier Series Parseval s Theorem and Mean Square Error Complex

### 16 Convergence of Fourier Series

16 Convergence of Fourier Series 16.1 Pointwise convergence of Fourier series Definition: Piecewise smooth functions For f defined on interval [a, b], f is piecewise smooth on [a, b] if there is a partition

### Math 1B, lecture 14: Taylor s Theorem

Math B, lecture 4: Taylor s Theorem Nathan Pflueger 7 October 20 Introduction Taylor polynomials give a convenient way to describe the local behavior of a function, by encapsulating its first several derivatives

### Fourier Series, Integrals, and Transforms

Chap. Sec.. Fourier Series, Integrals, and Transforms Fourier Series Content: Fourier series (5) and their coefficients (6) Calculation of Fourier coefficients by integration (Example ) Reason hy (6) gives

### The Geometric Series

The Geometric Series Professor Jeff Stuart Pacific Lutheran University c 8 The Geometric Series Finite Number of Summands The geometric series is a sum in which each summand is obtained as a common multiple

### Trigonometric Functions

Trigonometric Functions MATH 10, Precalculus J. Robert Buchanan Department of Mathematics Fall 011 Objectives In this lesson we will learn to: identify a unit circle and describe its relationship to real

### 1. the function must be periodic; 3. it must have only a finite number of maxima and minima within one periodic;

Fourier Series 1 Dirichlet conditions The particular conditions that a function f(x must fulfil in order that it may be expanded as a Fourier series are known as the Dirichlet conditions, and may be summarized

### Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an

### Applications of Fourier series

Chapter Applications of Fourier series One of the applications of Fourier series is the evaluation of certain infinite sums. For example, n= n,, are computed in Chapter (see for example, Remark.4.). n=

### 0 ( x) 2 = ( x)( x) = (( 1)x)(( 1)x) = ((( 1)x))( 1))x = ((( 1)(x( 1)))x = ((( 1)( 1))x)x = (1x)x = xx = x 2.

SOLUTION SET FOR THE HOMEWORK PROBLEMS Page 5. Problem 8. Prove that if x and y are real numbers, then xy x + y. Proof. First we prove that if x is a real number, then x 0. The product of two positive

### Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

### Continued fractions and good approximations.

Continued fractions and good approximations We will study how to find good approximations for important real life constants A good approximation must be both accurate and easy to use For instance, our

### Sine and Cosine Series; Odd and Even Functions

Sine and Cosine Series; Odd and Even Functions A sine series on the interval [, ] is a trigonometric series of the form k = 1 b k sin πkx. All of the terms in a series of this type have values vanishing

### Lecture 3: Fourier Series: pointwise and uniform convergence.

Lecture 3: Fourier Series: pointwise and uniform convergence. 1. Introduction. At the end of the second lecture we saw that we had for each function f L ([, π]) a Fourier series f a + (a k cos kx + b k

### Lecture Notes for Math 251: ODE and PDE. Lecture 33: 10.4 Even and Odd Functions

Lecture Notes for Math 51: ODE and PDE. Lecture : 1.4 Even and Odd Functions Shawn D. Ryan Spring 1 Last Time: We studied what a given Fourier Series converges to if at a. 1 Even and Odd Functions Before

### Lecture 3. Mathematical Induction

Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

### Advanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Advanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. # 28 Fourier Series (Contd.) Welcome back to the lecture on Fourier

### 5 Indefinite integral

5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse

### 5. ABSOLUTE EXTREMA. Definition, Existence & Calculation

5. ABSOLUTE EXTREMA Definition, Existence & Calculation We assume that the definition of function is known and proceed to define absolute minimum. We also assume that the student is familiar with the terms

### MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### Introduction to Sturm-Liouville Theory

Introduction to Ryan C. Trinity University Partial Differential Equations April 10, 2012 Inner products with weight functions Suppose that w(x) is a nonnegative function on [a,b]. If f (x) and g(x) are

### 36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?

36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this

### Introduction to Series and Sequences Math 121 Calculus II D Joyce, Spring 2013

Introduction to Series and Sequences Math Calculus II D Joyce, Spring 03 The goal. The main purpose of our study of series and sequences is to understand power series. A power series is like a polynomial

### Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series

1 Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series a n n=1 n(x + 2) n 5 n 1. n(x + 2)n Solution: Do the ratio test for the absolute convergence. Let a n =. Then,

### So here s the next version of Homework Help!!!

HOMEWORK HELP FOR MATH 52 So here s the next version of Homework Help!!! I am going to assume that no one had any great difficulties with the problems assigned this quarter from 4.3 and 4.4. However, if

### Test 3 Review. Jiwen He. Department of Mathematics, University of Houston. math.uh.edu/ jiwenhe/math1431

Test 3 Review Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math1431 November 25, 2008 1 / Test 3 Test 3: Dec. 4-6 in CASA Material - Through 6.3. November

### Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

### If f is continuous on [a, b], then the function g defined by. f (t) dt. is continuous on [a, b] and differentiable on (a, b), and g (x) = f (x).

The Fundamental Theorem of Calculus, Part 1 If f is continuous on [a, b], then the function g defined by g(x) = x a f (t) dt a x b is continuous on [a, b] and differentiable on (a, b), and g (x) = f (x).

### Lectures 5-6: Taylor Series

Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

### Sequences and Series

Contents 6 Sequences and Series 6. Sequences and Series 6. Infinite Series 3 6.3 The Binomial Series 6 6.4 Power Series 3 6.5 Maclaurin and Taylor Series 40 Learning outcomes In this Workbook you will

### Analysis for Simple Fourier Series in Sine and Co-sine Form

Analysis for Simple Fourier Series in Sine and Co-sine Form Chol Yoon August 19, 13 Abstract From the history of Fourier series to the application of Fourier series will be introduced and analyzed to understand

### 2. Introduction to Functions. f(a) = b, Discussion

2. INTRODUCTION TO FUNCTIONS 15 2. Introduction to Functions 2.1. Function. Definition 2.1.1. Let A and B be sets. A function f : A B is a rule which assigns to every element in A exactly one element in

### Basic Integration Formulas and the Substitution Rule

Basic Integration Formulas and the Substitution Rule The second fundamental theorem of integral calculus Recall from the last lecture the second fundamental theorem of integral calculus. Theorem Let f(x)

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

### Legendre fractional differential equation and Legender fractional polynomials

International Journal of Applied Mathematical Research, 3 (3) (2014) 214-219 c Science Publishing Corporation www.sciencepubco.com/index.php/ijamr doi: 10.14419/ijamr.v3i3.2747 Research Paper Legendre

### LINEAR RECURSIVE SEQUENCES. The numbers in the sequence are called its terms. The general form of a sequence is. a 1, a 2, a 3,...

LINEAR RECURSIVE SEQUENCES BJORN POONEN 1. Sequences A sequence is an infinite list of numbers, like 1) 1, 2, 4, 8, 16, 32,.... The numbers in the sequence are called its terms. The general form of a sequence

### 1 Review of complex numbers

1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

### MATH : HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS

MATH 16300-33: HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS 25-1 Find the absolute value and argument(s) of each of the following. (ii) (3 + 4i) 1 (iv) 7 3 + 4i (ii) Put z = 3 + 4i. From z 1 z = 1, we have

### Section 1.1 Real numbers. Set Builder notation. Interval notation

Section 1.1 Real numbers Set Builder notation Interval notation Functions a function is the set of all possible points y that are mapped to a single point x. If when x=5 y=4,5 then it is not a function

### Trigonometric Identities

Trigonometric Identities Dr. Philippe B. Laval Kennesaw STate University April 0, 005 Abstract This handout dpresents some of the most useful trigonometric identities. It also explains how to derive new

### Learning Objectives for Math 165

Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given

### Section 4.4. Using the Fundamental Theorem. Difference Equations to Differential Equations

Difference Equations to Differential Equations Section 4.4 Using the Fundamental Theorem As we saw in Section 4.3, using the Fundamental Theorem of Integral Calculus reduces the problem of evaluating a

### MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform

MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish

### This section demonstrates some different techniques of proving some general statements.

Section 4. Number Theory 4.. Introduction This section demonstrates some different techniques of proving some general statements. Examples: Prove that the sum of any two odd numbers is even. Firstly you

### 1. Introduction identity algbriac factoring identities

1. Introduction An identity is an equality relationship between two mathematical expressions. For example, in basic algebra students are expected to master various algbriac factoring identities such as

### Chapter 5: Application: Fourier Series

321 28 9 Chapter 5: Application: Fourier Series For lack of time, this chapter is only an outline of some applications of Functional Analysis and some proofs are not complete. 5.1 Definition. If f L 1

### The Limit of a Sequence of Numbers: Infinite Series

Connexions module: m36135 1 The Limit of a Sequence of Numbers: Infinite Series Lawrence Baggett This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License

### MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 4: Fourier Series and L 2 ([ π, π], µ) ( 1 π

MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 4: Fourier Series and L ([, π], µ) Square Integrable Functions Definition. Let f : [, π] R be measurable. We say that f

### Continuous random variables

Continuous random variables So far we have been concentrating on discrete random variables, whose distributions are not continuous. Now we deal with the so-called continuous random variables. A random

### Sequences of Functions

Sequences of Functions Uniform convergence 9. Assume that f n f uniformly on S and that each f n is bounded on S. Prove that {f n } is uniformly bounded on S. Proof: Since f n f uniformly on S, then given

### Limits and Continuity

Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### k=1 k2, and therefore f(m + 1) = f(m) + (m + 1) 2 =

Math 104: Introduction to Analysis SOLUTIONS Alexander Givental HOMEWORK 1 1.1. Prove that 1 2 +2 2 + +n 2 = 1 n(n+1)(2n+1) for all n N. 6 Put f(n) = n(n + 1)(2n + 1)/6. Then f(1) = 1, i.e the theorem

### Prime Numbers. Chapter Primes and Composites

Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

### HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba

HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain

### Tangent and normal lines to conics

4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

### The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

### 2.3 Bounds of sets of real numbers

2.3 Bounds of sets of real numbers 2.3.1 Upper bounds of a set; the least upper bound (supremum) Consider S a set of real numbers. S is called bounded above if there is a number M so that any x S is less

### Functions: Piecewise, Even and Odd.

Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 21-22, 2015 Tutorials, Online Homework.

### 4.7. Taylor and MacLaurin Series

4.7. TAYLOR AND MACLAURIN SERIES 0 4.7. Taylor and MacLaurin Series 4.7.. Polynomial Approximations. Assume that we have a function f for which we can easily compute its value f(a) at some point a, but