R U S S E L L L. H E R M A N

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "R U S S E L L L. H E R M A N"

Transcription

1 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. L. H E R M A N - V E R S I O N D AT E : J A N U A R Y 1 3,

2 Copyright by Russell L. Herman published by r. l. herman This text has been reformatted from the original using a modification of the Tufte-book documentclass in LATEX. See tufte-latex.googlecode.com. an introduction to fourier and complex analysis with applications to the spectral analysis of signals by Russell Herman is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. These notes have resided at since Spring Sixth printing, January 2016

3

4 Contents Introduction vii 1 Review of Sequences and Infinite Series Sequences of Real Numbers Convergence of Sequences Limit Theorems Infinite Series Geometric Series Convergence Tests Sequences of Functions Infinite Series of Functions Special Series Expansions Power Series Binomial Series The Order of Sequences and Functions Problems Fourier Trigonometric Series Introduction to Fourier Series Fourier Trigonometric Series Fourier Series over Other Intervals Fourier Series on [a, b] Sine and Cosine Series The Gibbs Phenomenon Multiple Fourier Series Problems Generalized Fourier Series and Function Spaces Finite Dimensional Vector Spaces Function Spaces Classical Orthogonal Polynomials Fourier-Legendre Series Properties of Legendre Polynomials The Generating Function for Legendre Polynomials The Differential Equation for Legendre Polynomials Fourier-Legendre Series Examples

5 iv 3.5 Gamma Function Fourier-Bessel Series Appendix: The Least Squares Approximation Appendix: Convergence of Trigonometric Fourier Series Problems Complex Analysis Complex Numbers Complex Valued Functions Complex Domain Coloring Complex Differentiation Complex Integration Complex Path Integrals Cauchy s Theorem Analytic Functions and Cauchy s Integral Formula Laurent Series Singularities and The Residue Theorem Infinite Integrals Integration over Multivalued Functions Appendix: Jordan s Lemma Problems Fourier and Laplace Transforms Introduction Complex Exponential Fourier Series Exponential Fourier Transform The Dirac Delta Function Properties of the Fourier Transform Fourier Transform Examples The Convolution Operation Convolution Theorem for Fourier Transforms Application to Signal Analysis Parseval s Equality The Laplace Transform Properties and Examples of Laplace Transforms Applications of Laplace Transforms Series Summation Using Laplace Transforms Solution of ODEs Using Laplace Transforms Step and Impulse Functions The Convolution Theorem The Inverse Laplace Transform Transforms and Partial Differential Equations Fourier Transform and the Heat Equation Laplace s Equation on the Half Plane Heat Equation on Infinite Interval, Revisited Nonhomogeneous Heat Equation

6 v Problems From Analog to Discrete Signals Analog to Periodic Signals The Dirac Comb Function Discrete Signals Summary The Discrete (Trigonometric) Fourier Transform Discrete Trigonometric Series Discrete Orthogonality The Discrete Fourier Coefficients The Discrete Exponential Transform FFT: The Fast Fourier Transform Applications MATLAB Implementation MATLAB for the Discrete Fourier Transform Matrix Operations for MATLAB MATLAB Implementation of FFT Problems Signal Analysis Introduction Periodogram Examples Effects of Sampling Effect of Finite Record Length Aliasing The Shannon Sampling Theorem Nonstationary Signals Simple examples The Spectrogram Short-Time Fourier Transform Harmonic Analysis Problems Bibliography 317 Index 319

7 vi Dedicated to those students who have endured the various editions of an introduction to fourier and complex analysis with applications to the spectral analysis of signals.

Mean value theorem, Taylors Theorem, Maxima and Minima.

Mean value theorem, Taylors Theorem, Maxima and Minima. MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and express-ions. Permutations and Combinations.

More information

Elementary Differential Equations

Elementary Differential Equations Elementary Differential Equations EIGHTH EDITION Earl D. Rainville Late Professor of Mathematics University of Michigan Phillip E. Bedient Professor Emeritus of Mathematics Franklin and Marshall College

More information

MATHEMATICAL METHODS OF STATISTICS

MATHEMATICAL METHODS OF STATISTICS MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS

More information

MATH Mathematics-Nursing. MATH Remedial Mathematics I-Business & Economics. MATH Remedial Mathematics II-Business and Economics

MATH Mathematics-Nursing. MATH Remedial Mathematics I-Business & Economics. MATH Remedial Mathematics II-Business and Economics MATH 090 - Mathematics-Nursing MATH 091 - Remedial Mathematics I-Business & Economics MATH 094 - Remedial Mathematics II-Business and Economics MATH 095 - Remedial Mathematics I-Science (3 CH) MATH 096

More information

SCHWEITZER ENGINEERING LABORATORIES, COMERCIAL LTDA.

SCHWEITZER ENGINEERING LABORATORIES, COMERCIAL LTDA. Pocket book of Electrical Engineering Formulas Content 1. Elementary Algebra and Geometry 1. Fundamental Properties (real numbers) 1 2. Exponents 2 3. Fractional Exponents 2 4. Irrational Exponents 2 5.

More information

Complex Function Theory. Second Edition. Donald Sarason >AMS AMERICAN MATHEMATICAL SOCIETY

Complex Function Theory. Second Edition. Donald Sarason >AMS AMERICAN MATHEMATICAL SOCIETY Complex Function Theory Second Edition Donald Sarason >AMS AMERICAN MATHEMATICAL SOCIETY Contents Preface to the Second Edition Preface to the First Edition ix xi Chapter I. Complex Numbers 1 1.1. Definition

More information

Precalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES

Precalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 0-1 Sets There are no state-mandated Precalculus 0-2 Operations

More information

FRACTIONAL INTEGRALS AND DERIVATIVES. Theory and Applications

FRACTIONAL INTEGRALS AND DERIVATIVES. Theory and Applications FRACTIONAL INTEGRALS AND DERIVATIVES Theory and Applications Stefan G. Samko Rostov State University, Russia Anatoly A. Kilbas Belorussian State University, Minsk, Belarus Oleg I. Marichev Belorussian

More information

Mathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 Pre-Algebra 4 Hours

Mathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 Pre-Algebra 4 Hours MAT 051 Pre-Algebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT

More information

Fourier Analysis and its applications

Fourier Analysis and its applications Fourier Analysis and its applications Fourier analysis originated from the study of heat conduction: Jean Baptiste Joseph Fourier (1768-1830) Fourier analysis enables a function (signal) to be decomposed

More information

MATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!

MATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing! MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Pre-algebra Algebra Pre-calculus Calculus Statistics

More information

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

More information

SGN-1158 Introduction to Signal Processing Test. Solutions

SGN-1158 Introduction to Signal Processing Test. Solutions SGN-1158 Introduction to Signal Processing Test. Solutions 1. Convolve the function ( ) with itself and show that the Fourier transform of the result is the square of the Fourier transform of ( ). (Hints:

More information

Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks

Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Algebra 2! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

More information

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint

More information

RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A

RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:

More information

Chapter 3: Mathematical Models and Numerical Methods Involving First-Order Differential Equations

Chapter 3: Mathematical Models and Numerical Methods Involving First-Order Differential Equations Massasoit Community College Instructor: Office: Email: Phone: Office Hours: Course: Differential Equations Course Number: MATH230-XX Semester: Classroom: Day and Time: Course Description: This course is

More information

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

More information

On Chebyshev interpolation of analytic functions

On Chebyshev interpolation of analytic functions On Chebyshev interpolation of analytic functions Laurent Demanet Department of Mathematics Massachusetts Institute of Technology Lexing Ying Department of Mathematics University of Texas at Austin March

More information

Math Course Descriptions & Student Learning Outcomes

Math Course Descriptions & Student Learning Outcomes Math Course Descriptions & Student Learning Outcomes Table of Contents MAC 100: Business Math... 1 MAC 101: Technical Math... 3 MA 090: Basic Math... 4 MA 095: Introductory Algebra... 5 MA 098: Intermediate

More information

The continuous and discrete Fourier transforms

The continuous and discrete Fourier transforms FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1

More information

Application of Fourier Transform to PDE (I) Fourier Sine Transform (application to PDEs defined on a semi-infinite domain)

Application of Fourier Transform to PDE (I) Fourier Sine Transform (application to PDEs defined on a semi-infinite domain) Application of Fourier Transform to PDE (I) Fourier Sine Transform (application to PDEs defined on a semi-infinite domain) The Fourier Sine Transform pair are F. T. : U = 2/ u x sin x dx, denoted as U

More information

Purpose of Time Series Analysis. Autocovariance Function. Autocorrelation Function. Part 3: Time Series I

Purpose of Time Series Analysis. Autocovariance Function. Autocorrelation Function. Part 3: Time Series I Part 3: Time Series I Purpose of Time Series Analysis (Figure from Panofsky and Brier 1968) Autocorrelation Function Harmonic Analysis Spectrum Analysis Data Window Significance Tests Some major purposes

More information

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

More information

Elementary Differential Equations and Boundary Value Problems. 10th Edition International Student Version

Elementary Differential Equations and Boundary Value Problems. 10th Edition International Student Version Brochure More information from http://www.researchandmarkets.com/reports/3148843/ Elementary Differential Equations and Boundary Value Problems. 10th Edition International Student Version Description:

More information

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

More information

ORDINARY DIFFERENTIAL EQUATIONS

ORDINARY DIFFERENTIAL EQUATIONS ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.

More information

Credit Number Lecture Lab / Shop Clinic / Co-op Hours. MAC 224 Advanced CNC Milling 1 3 0 2. MAC 229 CNC Programming 2 0 0 2

Credit Number Lecture Lab / Shop Clinic / Co-op Hours. MAC 224 Advanced CNC Milling 1 3 0 2. MAC 229 CNC Programming 2 0 0 2 MAC 224 Advanced CNC Milling 1 3 0 2 This course covers advanced methods in setup and operation of CNC machining centers. Emphasis is placed on programming and production of complex parts. Upon completion,

More information

BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line

BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University

More information

Using the TI-92 Plus: Some Examples

Using the TI-92 Plus: Some Examples Liverpool John Moores University, 1-15 July 000 Using the TI-9 Plus: Some Examples Michel Beaudin École de technologie supérieure,canada mbeaudin@seg.etsmtl.ca 1. Introduction We incorporated the use of

More information

Wavelet analysis. Wavelet requirements. Example signals. Stationary signal 2 Hz + 10 Hz + 20Hz. Zero mean, oscillatory (wave) Fast decay (let)

Wavelet analysis. Wavelet requirements. Example signals. Stationary signal 2 Hz + 10 Hz + 20Hz. Zero mean, oscillatory (wave) Fast decay (let) Wavelet analysis In the case of Fourier series, the orthonormal basis is generated by integral dilation of a single function e jx Every 2π-periodic square-integrable function is generated by a superposition

More information

Introduction to the Finite Element Method

Introduction to the Finite Element Method Introduction to the Finite Element Method 09.06.2009 Outline Motivation Partial Differential Equations (PDEs) Finite Difference Method (FDM) Finite Element Method (FEM) References Motivation Figure: cross

More information

Estimated Pre Calculus Pacing Timeline

Estimated Pre Calculus Pacing Timeline Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to

More information

MATHEMATICS (MATH) 3. Provides experiences that enable graduates to find employment in sciencerelated

MATHEMATICS (MATH) 3. Provides experiences that enable graduates to find employment in sciencerelated 194 / Department of Natural Sciences and Mathematics MATHEMATICS (MATH) The Mathematics Program: 1. Provides challenging experiences in Mathematics, Physics, and Physical Science, which prepare graduates

More information

Option Pricing Formulae using Fourier Transform: Theory and Application

Option Pricing Formulae using Fourier Transform: Theory and Application Option Pricing Formulae using Fourier Transform: Theory and Application Martin Schmelzle * April 2010 Abstract Fourier transform techniques are playing an increasingly important role in Mathematical Finance.

More information

NOV - 30211/II. 1. Let f(z) = sin z, z C. Then f(z) : 3. Let the sequence {a n } be given. (A) is bounded in the complex plane

NOV - 30211/II. 1. Let f(z) = sin z, z C. Then f(z) : 3. Let the sequence {a n } be given. (A) is bounded in the complex plane Mathematical Sciences Paper II Time Allowed : 75 Minutes] [Maximum Marks : 100 Note : This Paper contains Fifty (50) multiple choice questions. Each question carries Two () marks. Attempt All questions.

More information

MATHEMATICS (CLASSES XI XII)

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)

More information

School of Mathematics, Computer Science and Engineering. Mathematics* Associate in Arts Degree COURSES, PROGRAMS AND MAJORS

School of Mathematics, Computer Science and Engineering. Mathematics* Associate in Arts Degree COURSES, PROGRAMS AND MAJORS Mathematics School of Mathematics, Computer Science and Engineering Dean: Lianna Zhao, MD Academic Chair: Miriam Castroconde Faculty: Miriam Castroconde; Terry Cheng; Howard Dachslager, PhD; Ilknur Erbas

More information

The Z transform (3) 1

The Z transform (3) 1 The Z transform (3) 1 Today Analysis of stability and causality of LTI systems in the Z domain The inverse Z Transform Section 3.3 (read class notes first) Examples 3.9, 3.11 Properties of the Z Transform

More information

Frequency Response of FIR Filters

Frequency Response of FIR Filters Frequency Response of FIR Filters Chapter 6 This chapter continues the study of FIR filters from Chapter 5, but the emphasis is frequency response, which relates to how the filter responds to an input

More information

Georgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1

Georgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1 Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse

More information

WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS

WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS Visit this link to read the introductory text for this syllabus. 1. Circular Measure Lengths of Arcs of circles and Radians Perimeters of Sectors and Segments measure in radians 2. Trigonometry (i) Sine,

More information

SIGNAL PROCESSING & SIMULATION NEWSLETTER

SIGNAL PROCESSING & SIMULATION NEWSLETTER 1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty

More information

The Quantum Harmonic Oscillator Stephen Webb

The Quantum Harmonic Oscillator Stephen Webb The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems

More information

Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics

Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics Dimitri Van De Ville Ecole Polytechnique Fédérale de Lausanne Biomedical Imaging Group dimitri.vandeville@epfl.ch

More information

The Convolution Operation

The Convolution Operation The Convolution Operation Convolution is a very natural mathematical operation which occurs in both discrete and continuous modes of various kinds. We often encounter it in the course of doing other operations

More information

Matrices and Polynomials

Matrices and Polynomials APPENDIX 9 Matrices and Polynomials he Multiplication of Polynomials Let α(z) =α 0 +α 1 z+α 2 z 2 + α p z p and y(z) =y 0 +y 1 z+y 2 z 2 + y n z n be two polynomials of degrees p and n respectively. hen,

More information

Introduction to Partial Differential Equations. John Douglas Moore

Introduction to Partial Differential Equations. John Douglas Moore Introduction to Partial Differential Equations John Douglas Moore May 2, 2003 Preface Partial differential equations are often used to construct models of the most basic theories underlying physics and

More information

Numerical Analysis Introduction. Student Audience. Prerequisites. Technology.

Numerical Analysis Introduction. Student Audience. Prerequisites. Technology. Numerical Analysis Douglas Faires, Youngstown State University, (Chair, 2012-2013) Elizabeth Yanik, Emporia State University, (Chair, 2013-2015) Graeme Fairweather, Executive Editor, Mathematical Reviews,

More information

Appendix 3 IB Diploma Programme Course Outlines

Appendix 3 IB Diploma Programme Course Outlines Appendix 3 IB Diploma Programme Course Outlines The following points should be addressed when preparing course outlines for each IB Diploma Programme subject to be taught. Please be sure to use IBO nomenclature

More information

TOMS RIVER REGIONAL SCHOOLS MATHEMATICS CURRICULUM

TOMS RIVER REGIONAL SCHOOLS MATHEMATICS CURRICULUM Content Area: Mathematics Course Title: Precalculus Grade Level: High School Right Triangle Trig and Laws 3-4 weeks Trigonometry 3 weeks Graphs of Trig Functions 3-4 weeks Analytic Trigonometry 5-6 weeks

More information

Algebra and Geometry Review (61 topics, no due date)

Algebra and Geometry Review (61 topics, no due date) Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

More information

DOKUZ EYLUL UNIVERSITY GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES DIRECTORATE COURSE / MODULE / BLOCK DETAILS ACADEMIC YEAR / SEMESTER

DOKUZ EYLUL UNIVERSITY GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES DIRECTORATE COURSE / MODULE / BLOCK DETAILS ACADEMIC YEAR / SEMESTER Offered by: Fen Bilimleri Enstitüsü Course Title: Applied Mathematics Course Org. Title: Applied Mathematics Course Level: Lisansüstü Course Code: MAT 5001 Language of Instruction: İngilizce Form Submitting/Renewal

More information

INTEGRAL METHODS IN LOW-FREQUENCY ELECTROMAGNETICS

INTEGRAL METHODS IN LOW-FREQUENCY ELECTROMAGNETICS INTEGRAL METHODS IN LOW-FREQUENCY ELECTROMAGNETICS I. Dolezel Czech Technical University, Praha, Czech Republic P. Karban University of West Bohemia, Plzeft, Czech Republic P. Solin University of Nevada,

More information

Three Pictures of Quantum Mechanics. Thomas R. Shafer April 17, 2009

Three Pictures of Quantum Mechanics. Thomas R. Shafer April 17, 2009 Three Pictures of Quantum Mechanics Thomas R. Shafer April 17, 2009 Outline of the Talk Brief review of (or introduction to) quantum mechanics. 3 different viewpoints on calculation. Schrödinger, Heisenberg,

More information

Sequence of Mathematics Courses

Sequence of Mathematics Courses Sequence of ematics Courses Where do I begin? Associates Degree and Non-transferable Courses (For math course below pre-algebra, see the Learning Skills section of the catalog) MATH M09 PRE-ALGEBRA 3 UNITS

More information

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

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

More information

AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB.

AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB. AP Calculus BC Course Description: Advanced Placement Calculus BC is primarily concerned with developing the students understanding of the concepts of calculus and providing experiences with its methods

More information

1.5 / 1 -- Communication Networks II (Görg) -- www.comnets.uni-bremen.de. 1.5 Transforms

1.5 / 1 -- Communication Networks II (Görg) -- www.comnets.uni-bremen.de. 1.5 Transforms .5 / -- Communication Networks II (Görg) -- www.comnets.uni-bremen.de.5 Transforms Using different summation and integral transformations pmf, pdf and cdf/ccdf can be transformed in such a way, that even

More information

Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign

More information

Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)

Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary) Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify

More information

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

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

More information

LECTURE 2. Part/ References Topic/Sections Notes/Speaker. 1 Enumeration 1 1.1 Generating functions... 1. Combinatorial. Asst #1 Due FS A.

LECTURE 2. Part/ References Topic/Sections Notes/Speaker. 1 Enumeration 1 1.1 Generating functions... 1. Combinatorial. Asst #1 Due FS A. Wee Date Sections f aculty of science MATH 895-4 Fall 0 Contents 1 Sept 7 I1, I, I3 Symbolic methods I A3/ C Introduction to Prob 4 A sampling 18 IX1 of counting sequences Limit Laws and and generating

More information

Fourier Analysis. u m, a n u n = am um, u m

Fourier Analysis. u m, a n u n = am um, u m Fourier Analysis Fourier series allow you to expand a function on a finite interval as an infinite series of trigonometric functions. What if the interval is infinite? That s the subject of this chapter.

More information

KEANSBURG SCHOOL DISTRICT KEANSBURG HIGH SCHOOL Mathematics Department. HSPA 10 Curriculum. September 2007

KEANSBURG SCHOOL DISTRICT KEANSBURG HIGH SCHOOL Mathematics Department. HSPA 10 Curriculum. September 2007 KEANSBURG HIGH SCHOOL Mathematics Department HSPA 10 Curriculum September 2007 Written by: Karen Egan Mathematics Supervisor: Ann Gagliardi 7 days Sample and Display Data (Chapter 1 pp. 4-47) Surveys and

More information

Final Year Project Progress Report. Frequency-Domain Adaptive Filtering. Myles Friel. Supervisor: Dr.Edward Jones

Final Year Project Progress Report. Frequency-Domain Adaptive Filtering. Myles Friel. Supervisor: Dr.Edward Jones Final Year Project Progress Report Frequency-Domain Adaptive Filtering Myles Friel 01510401 Supervisor: Dr.Edward Jones Abstract The Final Year Project is an important part of the final year of the Electronic

More information

Taylor series. Prof. Enrique Mateus Nieves PhD in Mathematics Education.

Taylor series. Prof. Enrique Mateus Nieves PhD in Mathematics Education. Taylor series As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13. The exponential

More information

AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS

AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc.,

More information

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in

More information

ALGEBRA I / ALGEBRA I SUPPORT

ALGEBRA I / ALGEBRA I SUPPORT Suggested Sequence: CONCEPT MAP ALGEBRA I / ALGEBRA I SUPPORT August 2011 1. Foundations for Algebra 2. Solving Equations 3. Solving Inequalities 4. An Introduction to Functions 5. Linear Functions 6.

More information

NUMERICAL METHODS TOPICS FOR RESEARCH PAPERS

NUMERICAL METHODS TOPICS FOR RESEARCH PAPERS Faculty of Civil Engineering Belgrade Master Study COMPUTATIONAL ENGINEERING Fall semester 2004/2005 NUMERICAL METHODS TOPICS FOR RESEARCH PAPERS 1. NUMERICAL METHODS IN FINITE ELEMENT ANALYSIS - Matrices

More information

AFM Ch.12 - Practice Test

AFM Ch.12 - Practice Test AFM Ch.2 - Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question.. Form a sequence that has two arithmetic means between 3 and 89. a. 3, 33, 43, 89

More information

CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August

More information

COURSE SYLLABUS Pre-Calculus A/B Last Modified: April 2015

COURSE SYLLABUS Pre-Calculus A/B Last Modified: April 2015 COURSE SYLLABUS Pre-Calculus A/B Last Modified: April 2015 Course Description: In this year-long Pre-Calculus course, students will cover topics over a two semester period (as designated by A and B sections).

More information

3. Interpolation. Closing the Gaps of Discretization... Beyond Polynomials

3. Interpolation. Closing the Gaps of Discretization... Beyond Polynomials 3. Interpolation Closing the Gaps of Discretization... Beyond Polynomials Closing the Gaps of Discretization... Beyond Polynomials, December 19, 2012 1 3.3. Polynomial Splines Idea of Polynomial Splines

More information

TMA4213/4215 Matematikk 4M/N Vår 2013

TMA4213/4215 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 information

04 Mathematics CO-SG-FLD004-03. Program for Licensing Assessments for Colorado Educators

04 Mathematics CO-SG-FLD004-03. Program for Licensing Assessments for Colorado Educators 04 Mathematics CO-SG-FLD004-03 Program for Licensing Assessments for Colorado Educators Readers should be advised that this study guide, including many of the excerpts used herein, is protected by federal

More information

Fourier Transform and Image Filtering. CS/BIOEN 6640 Lecture Marcel Prastawa Fall 2010

Fourier Transform and Image Filtering. CS/BIOEN 6640 Lecture Marcel Prastawa Fall 2010 Fourier Transform and Image Filtering CS/BIOEN 6640 Lecture Marcel Prastawa Fall 2010 The Fourier Transform Fourier Transform Forward, mapping to frequency domain: Backward, inverse mapping to time domain:

More information

Analysis/resynthesis with the short time Fourier transform

Analysis/resynthesis with the short time Fourier transform Analysis/resynthesis with the short time Fourier transform summer 2006 lecture on analysis, modeling and transformation of audio signals Axel Röbel Institute of communication science TU-Berlin IRCAM Analysis/Synthesis

More information

Math 1050 Khan Academy Extra Credit Algebra Assignment

Math 1050 Khan Academy Extra Credit Algebra Assignment Math 1050 Khan Academy Extra Credit Algebra Assignment KhanAcademy.org offers over 2,700 instructional videos, including hundreds of videos teaching algebra concepts, and corresponding problem sets. In

More information

Higher Education Math Placement

Higher Education Math Placement Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

More information

Derive 5: The Easiest... Just Got Better!

Derive 5: The Easiest... Just Got Better! Liverpool John Moores University, 1-15 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering

More information

EECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines

EECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation

More information

Prentice Hall Algebra 2 2011 Correlated to: Colorado P-12 Academic Standards for High School Mathematics, Adopted 12/2009

Prentice Hall Algebra 2 2011 Correlated to: Colorado P-12 Academic Standards for High School Mathematics, Adopted 12/2009 Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level

More information

SAN DIEGO COMMUNITY COLLEGE DISTRICT CITY COLLEGE ASSOCIATE DEGREE COURSE OUTLINE

SAN DIEGO COMMUNITY COLLEGE DISTRICT CITY COLLEGE ASSOCIATE DEGREE COURSE OUTLINE MATH 098 CIC Approval: BOT APPROVAL: STATE APPROVAL: EFFECTIVE TERM: SAN DIEGO COMMUNITY COLLEGE DISTRICT CITY COLLEGE ASSOCIATE DEGREE COURSE OUTLINE SECTION I SUBJECT AREA AND COURSE NUMBER: Mathematics

More information

Polynomials and the Fast Fourier Transform (FFT) Battle Plan

Polynomials and the Fast Fourier Transform (FFT) Battle Plan Polynomials and the Fast Fourier Transform (FFT) Algorithm Design and Analysis (Wee 7) 1 Polynomials Battle Plan Algorithms to add, multiply and evaluate polynomials Coefficient and point-value representation

More information

Lecture 8 ELE 301: Signals and Systems

Lecture 8 ELE 301: Signals and Systems Lecture 8 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 2-2 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 2-2 / 37 Properties of the Fourier Transform Properties of the Fourier

More information

DRAFT. Further mathematics. GCE AS and A level subject content

DRAFT. Further mathematics. GCE AS and A level subject content Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed

More information

PYKC Jan-7-10. Lecture 1 Slide 1

PYKC Jan-7-10. Lecture 1 Slide 1 Aims and Objectives E 2.5 Signals & Linear Systems Peter Cheung Department of Electrical & Electronic Engineering Imperial College London! By the end of the course, you would have understood: Basic signal

More information

6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.

6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w. hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,

More information

Big Ideas in Mathematics

Big Ideas in Mathematics Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards

More information

ENCOURAGING THE INTEGRATION OF COMPLEX NUMBERS IN UNDERGRADUATE ORDINARY DIFFERENTIAL EQUATIONS

ENCOURAGING THE INTEGRATION OF COMPLEX NUMBERS IN UNDERGRADUATE ORDINARY DIFFERENTIAL EQUATIONS Texas College Mathematics Journal Volume 6, Number 2, Pages 18 24 S applied for(xx)0000-0 Article electronically published on September 23, 2009 ENCOURAGING THE INTEGRATION OF COMPLEX NUMBERS IN UNDERGRADUATE

More information

THE FAST FOURIER TRANSFORM

THE FAST FOURIER TRANSFORM THE FAST FOURIER TRASFORM LOG CHE ABSTRACT. Fast Fourier transform (FFT) is a fast algorithm to compute the discrete Fourier transform in O( log ) operations for an array of size 2 J. It is based on the

More information

Univariate and Multivariate Methods PEARSON. Addison Wesley

Univariate and Multivariate Methods PEARSON. Addison Wesley Time Series Analysis Univariate and Multivariate Methods SECOND EDITION William W. S. Wei Department of Statistics The Fox School of Business and Management Temple University PEARSON Addison Wesley Boston

More information

Basic Math Course Map through algebra and calculus

Basic Math Course Map through algebra and calculus Basic Math Course Map through algebra and calculus This map shows the most common and recommended transitions between courses. A grade of C or higher is required to move from one course to the next. For

More information

Applied Linear Algebra I Review page 1

Applied Linear Algebra I Review page 1 Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties

More information

Design of FIR Filters

Design of FIR Filters Design of FIR Filters Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205 Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet, Dr. Malcolm Macleod and Prof. Peter Rayner 68 FIR as

More information

The Fourth International DERIVE-TI92/89 Conference Liverpool, U.K., 12-15 July 2000. Derive 5: The Easiest... Just Got Better!

The Fourth International DERIVE-TI92/89 Conference Liverpool, U.K., 12-15 July 2000. Derive 5: The Easiest... Just Got Better! The Fourth International DERIVE-TI9/89 Conference Liverpool, U.K., -5 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de technologie supérieure 00, rue Notre-Dame Ouest Montréal

More information

Advanced Signal Processing and Digital Noise Reduction

Advanced Signal Processing and Digital Noise Reduction Advanced Signal Processing and Digital Noise Reduction Saeed V. Vaseghi Queen's University of Belfast UK WILEY HTEUBNER A Partnership between John Wiley & Sons and B. G. Teubner Publishers Chichester New

More information

The Heat Equation. Lectures INF2320 p. 1/88

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)

More information