Maths 361 Fourier Series Notes 2


 Shannon Floyd
 1 years ago
 Views:
Transcription
1 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 all x [, ]. f is even if f( x) = f(x) for all x [, ]. Some useful properties of odd and even functions : 1. If f is even If f is odd f(x) dx = f(x) dx 0 f(x) dx = 0. If f, g are even functions and q, r are odd functions then fg and qr are even functions, fq is an odd function. A function defined on the interval [0, ] can be extended to [, ] as an even function or an odd function: The odd extension of f is defined by f odd (x) = The even extension of f is defined by f even (x) = { f(x), x [0, ], f( x), x [, 0) { f(x), x [0, ], f( x), x [, 0) 1
2 Section 1.3 : Real trig Fourier series Recall from last lecture : If {φ n } is a complete orthogonal set in P S[a, b] then f P S[a, b] can be represented as a sum of φ n : f(x), φ n (x) f(x) = c n φ n = φ φ n n (x) Equality will hold except possibly for a finite number of points x [a, b]. We would like to express a function f(x) as a sum of cosine and sine terms, i.e., would like for some a 0, a n, b n, n = 1,... f(x) = a 0 + a n cos + b n sin The right side of equation (1) is the called the real trig Fourier representation of f(x). Assuming (for now) no problems with convergence of the infinite series, we can calculate a 0, a n, b n using earlier results. In particular, { ( ) ( )} S = 1, cos, sin is an orthogonal set in the IPS P S[, ] with inner product f, g, = f(x)g(x) dx, and is in fact complete (for pointwise convergence). So if f P S[, ] then (1) a 0 = f, 1 1 for n = 1,.... a n = f, cos ( ) cos ( ) b n = f, sin ( ) sin ( ) We can calculate 1 =, cos ( ) = and sin ( ) = for n = 1,,.... Summary : A function f P S[, ] has the (classical or real trig) Fourier representation where f(x) a 0 + a n cos + b n sin a 0 = 1 f(x) dx a n = 1 f(x) cos dx
3 for n = 1,... b n = 1 f(x) sin dx Note that in () means has the Fourier representation. We do not (yet) know whether f is equal to its Fourier representation at any particular point x although we expect this to be the case at all but finitely many points of [, ]. Example : Calculate the real trig Fourier representation of f(x) = 1 + x for x [, ] We find that i.e., a 0 = 1, all other a n = 0, b n = nπ ( 1)n+1 ( ) f(x) 1 + nπ ( 1)n+1 sin We can use Matlab to plot the first few terms of the Fourier series: 3
4 Example : Calculate the real trig Fourier representation of f(x) = { 1, x < 0 1, 0 x We find that f(x),3,5... ( ) 4 nπ sin i.e., all a n = 0, b n = 4 nπ for odd n and b n = 0 for even n. We can use Matlab to plot the first few terms of the Fourier series: 4
5 Today s topics: Convergence of Fourier series Sketching Fourier series Fourier cosine and sine series Recommended reading: Haberman 3. and 3.3 (excluding Gibb s phenomenon) Recommended exercises: Haberman 3..1(b),(d),(f); 3..(a),(c),(f); 3.3.1(d); 3.3.(d); 3.3.4; Section 1.4 Convergence of Fourier series Consider a function f : [a, b] R. We define f per, the periodic extension of f, by f per (x + n(b a)) = f(x) for each integer n and each x [a, b). Theorem (Dirichlet) : et f P S[, ]. et for n = 1,.... a n = 1 b n = 1 a 0 = 1 f(x) dx f(x) cos dx f(x) sin dx Define N N S N (x) = a 0 + a n cos + b n sin, i.e., S N is the sum of the first N + 1 terms in the real trig Fourier representation of f for x [, ]. Then for x R. lim S N(x) = f per(x + ) + f per (x ) N 5
6 Dirichlet s theorem says that for f P S[, ], the Fourier series of f converges to f per whenever f per is continuous; the average of the right and left limits of f per whenever f per has a jump discontinuity. See section 5.5 of PDE s : an introduction by W. A. Strauss for a proof of this theorem. Sketching Fourier series We can now sketch the Fourier representation of a function without first calculating a 0, a n, b n. For f P S[a, b], let S (x) = lim N S N be the real trig Fourier representation of f in [, ]. To sketch S (x): 1. Sketch f per (x) without marking value at points of discontinuity.. Mark at points of discontinuity. f per (x + ) + f per (x ) 6
7 1.5 Fourier sine and cosine series The sets { ( )} S 1 = sin and { ( )} S = 1, cos are orthogonal and complete for pointwise convergence in P S[0, ] with inner product f, g = Define the Fourier sine series of f to be 0 f(x)g(x) dx b n sin where b n = 0 f(x) sin dx Define the Fourier cosine series of f to be a 0 + a n cos where a 0 = 1 f(x) dx 0 a n = f(x) cos dx 0 Theorem : If f P S[0, ], then at each x R the Fourier sine series of f converges to f per (x + ) + f per (x ) where f per is the periodic extension of f odd and f odd is the odd extension of f to [, ] as defined in the last lecture. Similarly, if f P S[0, ], then at each x R the Fourier cosine series of f converges to f per (x + ) + f per (x ) where f per is the periodic extension of f even and f even is the even extension of f to [, ] as defined in the last lecture. 7
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 (1768183), who made important contributions
More informationMATH 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
More informationLecture 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
More informationWeek 15. Vladimir Dobrushkin
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
More informationCHAPTER 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 =
More informationCHAPTER 3. Fourier Series
`A SERIES OF CLASS NOTES FOR 20052006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL
More informationFourier 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
More information16 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
More informationFourier Series. 1. Fullrange 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
More information1. 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
More informationEngineering 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...
More informationRecap 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
More informationFourier 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
More informationIntroduction 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,
More informationFourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks
Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results
More informationFourier Series and SturmLiouville Eigenvalue Problems
Fourier Series and SturmLiouville Eigenvalue Problems 2009 Outline Functions Fourier Series Representation Halfrange Expansion Convergence of Fourier Series Parseval s Theorem and Mean Square Error Complex
More informationMath 201 Lecture 30: Fourier Cosine and Sine Series
Math ecture 3: Fourier Cosine and Sine Series Mar. 3, Many examples here are taken from the textbook. The first number in ( refers to the problem number in the UA Custom edition, the second number in (
More informationM344  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
More informationChapt.12: Orthogonal Functions and Fourier series
Chat.12: Orthogonal Functions and Fourier series J.P. Gabardo gabardo@mcmaster.ca Deartment of Mathematics & Statistics McMaster University Hamilton, ON, Canada Lecture: January 10, 2011. 1/3 12.1:Orthogonal
More informationAn Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig
An Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig Abstract: This paper aims to give students who have not yet taken a course in partial
More information1. Periodic Fourier series. The Fourier expansion of a 2πperiodic function f is:
CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2πperiodic 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
More informationLecture 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
More informationEven, odd functions. Main properties of even, odd functions. Sine and cosine series. Evenperiodic, oddperiodic extensions of functions.
Sine and Cosine Series (Sect..4. Even, odd functions. Main properties of even, odd functions. Sine and cosine series. Evenperiodic, oddperiodic etensions of functions. Even, odd functions. Definition
More informationIntroduction 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
More informationFourier 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.
More informationChapter 14: Fourier Transforms and Boundary Value Problems in an Unbounded Region
Chapter 14: Fourier Transforms and Boundary Value Problems in an Unbounded Region 王奕翔 Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw December 25, 213 1 / 27 王奕翔 DE Lecture
More informationFourier Series & Fourier Transforms
Fourier Series & Fourier Transforms nicholas.harrison@imperial.ac.uk 19th October 003 Synopsis ecture 1 : Review of trigonometric identities Fourier Series Analysing the square wave ecture : The Fourier
More informationIntroduction 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
More informationAdvanced 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
More informationSine 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
More information) + 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
More informationMATH31011/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
More informationLecture 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
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationCourse 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
More informationLecture 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
More informationFourier 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
More informationLearning 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
More informationJuan Ruiz Álvarez. Calculus I
1 1 Departamento de Matemáticas. Universidad de Alcalá de Henares. Outline Introduction and definitions 1 Introduction and definitions 3 4 5 Outline Introduction and definitions 1 Introduction and definitions
More information3 Trigonometric Fourier Series
3 Trigonometric Fourier Series Ordinary language is totally unsuited for epressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical
More informationAn introduction to generalized vector spaces and Fourier analysis. by M. Croft
1 An introduction to generalized vector spaces and Fourier analysis. by M. Croft FOURIER ANALYSIS : Introduction Reading: Brophy p. 5863 This lab is u lab on Fourier analysis and consists of VI parts.
More informationFourier Series for Periodic Functions. Lecture #8 5CT3,4,6,7. BME 333 Biomedical Signals and Systems  J.Schesser
Fourier Series for Periodic Functions Lecture #8 5C3,4,6,7 Fourier Series for Periodic Functions Up to now we have solved the problem of approximating a function f(t) by f a (t) within an interval. However,
More informationFourier 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
More information6.6 The Inverse Trigonometric Functions. Outline
6.6 The Inverse Trigonometric Functions Tom Lewis Fall Semester 2015 Outline The inverse sine function The inverse cosine function The inverse tangent function The other inverse trig functions Miscellaneous
More informationChapter 11 Fourier Analysis
Chapter 11 Fourier Analysis Advanced Engineering Mathematics WeiTa Chu National Chung Cheng University wtchu@cs.ccu.edu.tw 1 2 11.1 Fourier Series Fourier Series Fourier series are infinite series that
More informationMA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity
MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x
More informationChapter 5: Trigonometric Functions of Real Numbers
Chapter 5: Trigonometric Functions of Real Numbers 5.1 The Unit Circle The unit circle is the circle of radius 1 centered at the origin. Its equation is x + y = 1 Example: The point P (x, 1 ) is on the
More informationTrigonometric 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
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationFourier cosine and sine series even odd
Fourier cosine and sine series Evaluation of the coefficients in the Fourier series of a function f is considerably simler is the function is even or odd. A function is even if f ( x) = f (x) Examle: x
More informationSection 0.4 Inverse Trigonometric Functions
Section 0.4 Inverse Trigonometric Functions Short Recall from Trigonometry Definition: A function f is periodic of period T if f(x + T ) = f(x) for all x such that x and x+t are in the domain of f. The
More informationIn this chapter, we define continuous functions and study their properties.
Chapter 7 Continuous Functions In this chapter, we define continuous functions and study their properties. 7.1. Continuity Continuous functions are functions that take nearby values at nearby points. Definition
More informationFourier 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
More informationx a x 2 (1 + x 2 ) n.
Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number
More informationSolutions to Linear Algebra Practice Problems
Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the
More informationf x a 0 n 1 a 0 a 1 cos x a 2 cos 2x a 3 cos 3x b 1 sin x b 2 sin 2x b 3 sin 3x a n cos nx b n sin nx n 1 f x dx y
Fourier Series When the French mathematician Joseph Fourier (768 83) was tring to solve a problem in heat conduction, he needed to epress a function f as an infinite series of sine and cosine functions:
More informationTMA4213/4215 Matematikk 4M/N Vår 2013
Norges teknisk naturvitenskapelige universitet Institutt for matematiske fag TMA43/45 Matematikk 4M/N Vår 3 Løsningsforslag Øving a) The Fourier series of the signal is f(x) =.4 cos ( 4 L x) +cos ( 5 L
More information2 n = n=1 a n is convergent and we let. i=1
Lecture 4 : Series So far our definition of a sum of numbers applies only to adding a finite set of numbers. We can extend this to a definition of a sum of an infinite set of numbers in much the same way
More informationSolutions to Sample Midterm 2 Math 121, Fall 2004
Solutions to Sample Midterm Math, Fall 4. Use Fourier series to find the solution u(x, y) of the following boundary value problem for Laplace s equation in the semiinfinite strip < x : u x + u
More information3. 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.
More informationR U S S E L L L. H E R M A N
R U S S E L L L. H E R M A N A N I N T R O D U C T I O N T O F O U R I E R A N D C O M P L E X A N A LY S I S W I T H A P P L I C AT I O N S T O T H E S P E C T R A L A N A LY S I S O F S I G N A L S R.
More informationFourier Series. 5.1 Introduction
5 Fourier Series 5. Introduction In this chapter we will look at trigonometric series. Previously, we saw that such series epansion occurred naturally in the solution of the heat equation and other boundary
More informationThe Fourier Transform
The Fourier Transorm Fourier Series Fourier Transorm The Basic Theorems and Applications Sampling Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGrawHill, 2. Eric W. Weisstein.
More informationLecture5. Fourier Series
Lecture5. Fourier Series In 1807 the French mathematician Joseph Fourier (17681830) submitted a paper to the Academy of Sciences in Paris. In it he presented a mathematical treatment of problems involving
More informationMath 241: More heat equation/laplace equation
Math 241: More heat equation/aplace equation D. DeTurck University of Pennsylvania September 27, 2012 D. DeTurck Math 241 002 2012C: Heat/aplace equations 1 / 13 Another example Another heat equation problem:
More informationHarmonic Formulas for Filtering Applications
Harmonic Formulas for Filtering Applications Trigonometric Series Harmonic Analysis The history of the trigonometric series, for all practical purposes, came of age in 1822 with Joseph De Fourier s book
More informationM3 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
More informationDerivatives: rules and applications (Stewart Ch. 3/4) The derivative f (x) of the function f(x):
Derivatives: rules and applications (Stewart Ch. 3/4) The derivative f (x) of the function f(x): f f(x + h) f(x) (x) = lim h 0 h (for all x for which f is differentiable/ the limit exists) Property:if
More informationMath 462: HW5 Solutions
Mth 6: HW5 Soutions Due on August 5, Jcky Chong Jcky Chong Remrk: We re working in the context of Riemnn Integrs. Probem 5.. Find the Fourier cosine series of the functions sin x in the interv (, ). Use
More informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from
More informationThe Alternating Series Test
The Alternating Series Test So far we have considered mostly series all of whose terms are positive. If the signs of the terms alternate, then testing convergence is a much simpler matter. On the other
More informationFourier Series. 1. Introduction. 2. Square Wave
531 Fourier Series Tools Used in Lab 31 Fourier Series: Square Wave Fourier Series: Triangle Wave Fourier Series: Gibbs Effect Fourier Series: Coefficients How does a Fourier series converge to a function?
More information1 Inner Products and Norms on Real Vector Spaces
Math 373: Principles Techniques of Applied Mathematics Spring 29 The 2 Inner Product 1 Inner Products Norms on Real Vector Spaces Recall that an inner product on a real vector space V is a function from
More informationApplications 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=
More information36 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
More informationSolvability of Laplace s Equation Kurt Bryan MA 436
1 Introduction Solvability of Laplace s Equation Kurt Bryan MA 436 Let D be a bounded region in lr n, with x = (x 1,..., x n ). We see a function u(x) which satisfies u = in D, (1) u n u = h on D (2) OR
More informationME231 Measurements Laboratory Spring Fourier Series. Edmundo Corona c
ME23 Measurements Laboratory Spring 23 Fourier Series Edmundo Corona c If you listen to music you may have noticed that you can tell what instruments are used in a given song or symphony. In some cases,
More informationTaylor 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
More informationLecture 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
More informationFourier Analysis. Tom Leinster
Fourier Analysis om Leinster 2013 14 Contents A Preparing the ground 2 A1 he algebraist s dream........................ 2 A2 Pseudohistorical overview...................... 5 A3 Integration..............................
More informationMath 5311 Gateaux differentials and Frechet derivatives
Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.
More information10.2 Fourier Series PROBLEMS
1. Fourier Series 547 PROBEMS In each of Problems 1 through 1 either solve the given boundar value problem or else show that it has no solution. 1. + =, () =, (π) = 1. + =, () = 1, (π) = 3. + =, () =,
More informationHow to roughly sketch a sinusoidal graph
34 CHAPTER 17. SINUSOIDAL FUNCTIONS Definition 17.1.1 (The Sinusoidal Function). Let A,, C and D be fixed constants, where A and are both positive. Then we can form the new function ( ) π y = A sin (x
More informationGRAPHING IN POLAR COORDINATES SYMMETRY
GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry  yaxis,
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationThe 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)
More information1 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
More informationFOURIER SERIES. 1. Periodic Functions. Recall that f has period T if f(x + T ) = f(x) for all x. If f and g are periodic
FOURIER SERIES. Periodic Functions Reca that f has period T if f(x + T ) f(x) for a x. If f and g are periodic with period T, then if h(x) af(x) + bg(x), a and b are constants, and thus h aso has period
More informationInner product. Definition of inner product
Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product
More informationMath 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
More informationChapter 2. Fourier Analysis
Chapter 2. Fourier Analysis Reading: Kreyszig, Advanced Engineering Mathematics, 0th Ed., 20 Selection from chapter Prerequisites: Kreyszig, Advanced Engineering Mathematics, 0th Ed., 20 Complex numbers:
More informationUnit 8 Inverse Trig & Polar Form of Complex Nums.
HARTFIELD PRECALCULUS UNIT 8 NOTES PAGE 1 Unit 8 Inverse Trig & Polar Form of Complex Nums. This is a SCIENTIFIC OR GRAPHING CALCULATORS ALLOWED unit. () Inverse Functions (3) Invertibility of Trigonometric
More informationCorrelation and Convolution Class Notes for CMSC 426, Fall 2005 David Jacobs
Correlation and Convolution Class otes for CMSC 46, Fall 5 David Jacobs Introduction Correlation and Convolution are basic operations that we will perform to extract information from images. They are in
More informationMath 201 Lecture 23: Power Series Method for Equations with Polynomial
Math 201 Lecture 23: Power Series Method for Equations with Polynomial Coefficients Mar. 07, 2012 Many examples here are taken from the textbook. The first number in () refers to the problem number in
More informationWEEK #5: Trig Functions, Optimization
WEEK #5: Trig Functions, Optimization Goals: Trigonometric functions and their derivatives Optimization Textbook reading for Week #5: Read Sections 1.8, 2.10, 3.3 Trigonometric Functions From Section 1.8
More informationMATH 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,
More informationAbsolute Maxima and Minima
Absolute Maxima and Minima Definition. A function f is said to have an absolute maximum on an interval I at the point x 0 if it is the largest value of f on that interval; that is if f( x ) f() x for all
More informationRepresentation 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
More informationI. Pointwise convergence
MATH 40  NOTES Sequences of functions Pointwise and Uniform Convergence Fall 2005 Previously, we have studied sequences of real numbers. Now we discuss the topic of sequences of real valued functions.
More informationSeries FOURIER SERIES. Graham S McDonald. A selfcontained Tutorial Module for learning the technique of Fourier series analysis
Series FOURIER SERIES Graham S McDonald A selfcontained Tutorial Module for learning the technique of Fourier series analysis Table of contents Begin Tutorial c 004 g.s.mcdonald@salford.ac.uk 1. Theory.
More information