Lecture 1: Complex Number
|
|
- Kristopher Eaton
- 6 years ago
- Views:
Transcription
1 Lecture 1: Complex Number Key points Definition of complex number, where Modulus and argument Euler's formular Polar expression where if, then and rotates a point on the complex plane by Imaginary unit Rectanular expression or Modulus or Argument or Real part or Imaginary part or Conjugate Polar expression or Conversion functions:,, 1 Imaginary unit Definition: Imaginary Unit: Common sense,,,, where These relations are used in various series expansions appeared in quantum mechanics See Lecture?? See the section of complex plane for graphical representation of these relations In, imaginary unit is I (capital I) I
2 Exercise 11 Compute by hand and confirm the result with Answer 2 Complex number in rectangle expression Complex number in rectangle expression where x and y are real numbers ( ) x real part, y imaginary part If, then is real If, then is pure imaginary Common sense For and, For, is not necessarily the real part of because and can be complex numbers See Exercise 22 Generally, there are multiple expressions to describe a mathematical expression Close to the original mathematical expression 3 4 Close to computer languages (commands or functions) 3 4 Exercise 12 Calculate by hand and confirm the result with
3 Answer Exercise 13 For Answer where and are complex, show that 3 Complex plane, modulus and argument Complex Plain ( plane) Modulus: Argument(Phase): Figure 1 Commen sense For, Popular points on a unit circle: (when, the point on complex plane make a complete rotation along the circle Section 1 )
4 means that when a imaginary unit is multiplied to a complex number, the point rotates by Modulus: 5 or 5 Argument: Use evalf to get the numerical value in decimal form (truncated at some significant figures) Symbolic calculation Exercise 14 Find the modulus and argument of by hand and confirm the result with Answer 4 Complex conjugate
5 Definition: For and its conjugate is See Figure 1 for geometric relation between and In physics, indicates complex conjugate of z In math books, is often used Common sense, If then is real ( If then is pure imaginary ( Complex Conjugate: or or 25 is equivalent to 25 5 Euler's formula and polar expression Euler's formula: Polar Expression: Multipying to a complex number rotates the point by angle Common sense
6 , (In general function evalc converts polar expression to rectangle expression polar expression can be specified by polar function convert changes the expression Exercise 15 Express in polar expression/ Answer 6 Real and Imaginary parts are independent degree of freedome If, then and For, if then, and Warning, solve may be ignoring assumptions on the input variables is not so smart! (1) Warning, solve may be ignoring assumptions on the input
7 variables With a little help by a human, can solve it (2) 7 Examples in physics - Traveling plane wave A plane wave expressed as is unbiqutus in physics (sound wave, electromagnetic wave, a free quantum particle, ) Assuming is real, we can visualize this wave as a circular motion at each position At, and do not change in time The factor makes a circular motion with angular and the starting point of the circular motion is determined by, the rotation speed is the same everywhere On the other hand, the starting point shifts by phase angle See Figure below Now we consider two souond waves A wave traveling in +x and -x direction can be expressed as respectively, where k and are wave vector and frequency A is a complex amplitude
8 Suppose that the wave is coming from x<0 toward x0 and reflected back to x<0, the total wave is Reflection by a soft interface: Reflection by a hard interface: Since, Many physical waves are measured in real number In such cases, we use the real part of complex wave as real wave (Actually, the imaginary part is equally valid as real wave) Writing the amplitrude in polar form Sound wave in an open pipe Sound wave in a closed pipe
9 Homework: Due 9/2 (Tue), 11am 11 Hyperbolic functions Using the Euler's Formula, prove the following relations (Assume that x is real) Quantum wave function The state of a free quantum particle is descrive by a complex function determined by the Schrödinger equation, which is where m is the mass of the particle 1 Show that a traveling wave is a solution to the Schrödinger equation 2 3 where Find the Schrödinger equation for the complex conjugate of In quantum mechanics, a real function plays an important role Show that
Lecture 2: Series Expansion
Lecture 2: Series Expansion Key points Maclaurin series : For, Taylor expansion: Maple commands series convert taylor, Student[NumericalAnalysis][Taylor] 1 Maclaurin series of elementary functions for
More informationcorrelated to the Idaho Content Standards Algebra II
correlated to the Idaho Content Standards Algebra II McDougal Littell Algebra and Trigonometry: Structure and Method, Book 2 2000 correlated to the Idaho Content Standards Algebra 2 STANDARD 1: NUMBER
More informationENGIN 211, Engineering Math. Complex Numbers
ENGIN 211, Engineering Math Complex Numbers 1 Imaginary Number and the Symbol J Consider the solutions for this quadratic equation: x 2 + 1 = 0 x = ± 1 1 is called the imaginary number, and we use the
More informationThe integrating factor method (Sect. 1.1)
The integrating factor method (Sect. 1.1) Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Overview
More informationJUST THE MATHS UNIT NUMBER 6.2. COMPLEX NUMBERS 2 (The Argand Diagram) A.J.Hobson
JUST THE MATHS UNIT NUMBER 6.2 COMPLEX NUMBERS 2 (The Argand Diagram) by A.J.Hobson 6.2.1 Introduction 6.2.2 Graphical addition and subtraction 6.2.3 Multiplication by j 6.2.4 Modulus and argument 6.2.5
More informationAS and A level Further mathematics contents lists
AS and A level Further mathematics contents lists Contents Core Pure Mathematics Book 1/AS... 2 Core Pure Mathematics Book 2... 4 Further Pure Mathematics 1... 6 Further Pure Mathematics 2... 8 Further
More informationComplex Numbers. Integers, Rationals, and Reals. The natural numbers: The integers:
Complex Numbers Integers, Rationals, and Reals The natural numbers: N {... 3, 2,, 0,, 2, 3...} The integers: Z {... 3, 2,, 0,, 2, 3...} Note that any two integers added, subtracted, or multiplied together
More informationChapter 3.4: Complex Zeros of Polynomials
Chapter 3.4: Complex Zeros of Polynomials Imaginary numbers were first encountered in the first century in ancient Greece when Heron of Alexandria came across the square root of a negative number in his
More information18.03 LECTURE NOTES, SPRING 2014
18.03 LECTURE NOTES, SPRING 2014 BJORN POONEN 7. Complex numbers Complex numbers are expressions of the form x + yi, where x and y are real numbers, and i is a new symbol. Multiplication of complex numbers
More informationxvi xxiii xxvi Construction of the Real Line 2 Is Every Real Number Rational? 3 Problems Algebra of the Real Numbers 7
About the Author v Preface to the Instructor xvi WileyPLUS xxii Acknowledgments xxiii Preface to the Student xxvi 1 The Real Numbers 1 1.1 The Real Line 2 Construction of the Real Line 2 Is Every Real
More informationCOMPLEX ANALYSIS-I. DR. P.K. SRIVASTAVA Assistant Professor Department of Mathematics Galgotia s College of Engg. & Technology, Gr.
COMPLEX ANALYSIS-I DR. P.K. SRIVASTAVA Assistant Professor Department of Mathematics Galgotia s College of Engg. & Technology, Gr. Noida An ISO 9001:2008 Certified Company Vayu Education of India 2/25,
More informationLecture 4 Notes: 06 / 30. Energy carried by a wave
Lecture 4 Notes: 06 / 30 Energy carried by a wave We want to find the total energy (kinetic and potential) in a sine wave on a string. A small segment of a string at a fixed point x 0 behaves as a harmonic
More informationQuantum Mysteries. Scott N. Walck. September 2, 2018
Quantum Mysteries Scott N. Walck September 2, 2018 Key events in the development of Quantum Theory 1900 Planck proposes quanta of light 1905 Einstein explains photoelectric effect 1913 Bohr suggests special
More informationMath 4C Fall 2008 Final Exam Study Guide Format 12 questions, some multi-part. Questions will be similar to sample problems in this study guide,
Math 4C Fall 2008 Final Exam Study Guide Format 12 questions, some multi-part. Questions will be similar to sample problems in this study guide, homework problems, lecture examples or examples from the
More informationQuick Overview: Complex Numbers
Quick Overview: Complex Numbers February 23, 2012 1 Initial Definitions Definition 1 The complex number z is defined as: z = a + bi (1) where a, b are real numbers and i = 1. Remarks about the definition:
More informationFLORIDA STANDARDS TO BOOK CORRELATION FOR GRADE 7 ADVANCED
FLORIDA STANDARDS TO BOOK CORRELATION FOR GRADE 7 ADVANCED After a standard is introduced, it is revisited many times in subsequent activities, lessons, and exercises. Domain: The Number System 8.NS.1.1
More informationLecture 6: Vector Spaces II - Matrix Representations
1 Key points Lecture 6: Vector Spaces II - Matrix Representations Linear Operators Matrix representation of vectors and operators Hermite conjugate (adjoint) operator Hermitian operator (self-adjoint)
More informationA video College Algebra course & 6 Enrichment videos
A video College Algebra course & 6 Enrichment videos Recorded at the University of Missouri Kansas City in 1998. All times are approximate. About 43 hours total. Available on YouTube at http://www.youtube.com/user/umkc
More informationComplex Numbers. Basic algebra. Definitions. part of the complex number a+ib. ffl Addition: Notation: We i write for 1; that is we
Complex Numbers Definitions Notation We i write for 1; that is we define p to be p 1 so i 2 = 1. i Basic algebra Equality a + ib = c + id when a = c b = and d. Addition A complex number is any expression
More information01 Harmonic Oscillations
Utah State University DigitalCommons@USU Foundations of Wave Phenomena Library Digital Monographs 8-2014 01 Harmonic Oscillations Charles G. Torre Department of Physics, Utah State University, Charles.Torre@usu.edu
More information1 Complex numbers and the complex plane
L1: Complex numbers and complex-valued functions. Contents: The field of complex numbers. Real and imaginary part. Conjugation and modulus or absolute valued. Inequalities: The triangular and the Cauchy.
More informationChapter 7 PHASORS ALGEBRA
164 Chapter 7 PHASORS ALGEBRA Vectors, in general, may be located anywhere in space. We have restricted ourselves thus for to vectors which are all located in one plane (co planar vectors), but they may
More informationOCR Maths FP1. Topic Questions from Papers. Complex Numbers. Answers
OCR Maths FP1 Topic Questions from Papers Complex Numbers Answers PhysicsAndMathsTutor.com . 1 (i) i Correct real and imaginary parts z* = i 1i Correct conjugate seen or implied Correct real and imaginary
More information(3.1) Module 1 : Atomic Structure Lecture 3 : Angular Momentum. Objectives In this Lecture you will learn the following
Module 1 : Atomic Structure Lecture 3 : Angular Momentum Objectives In this Lecture you will learn the following Define angular momentum and obtain the operators for angular momentum. Solve the problem
More informationLecture 4: Series expansions with Special functions
Lecture 4: Series expansions with Special functions 1. Key points 1. Legedre polynomials (Legendre functions) 2. Hermite polynomials(hermite functions) 3. Laguerre polynomials (Laguerre functions) Other
More information2.0 COMPLEX NUMBER SYSTEM. Bakiss Hiyana bt Abu Bakar JKE, POLISAS BHAB 1
2.0 COMPLEX NUMBER SYSTEM Bakiss Hiyana bt Abu Bakar JKE, POLISAS BHAB 1 COURSE LEARNING OUTCOME 1. Explain AC circuit concept and their analysis using AC circuit law. 2. Apply the knowledge of AC circuit
More informationSECTION 1.4: FUNCTIONS. (See p.40 for definitions of relations and functions and the Technical Note in Notes 1.24.) ( ) = x 2.
SECTION 1.4: FUNCTIONS (Section 1.4: Functions) 1.18 (See p.40 for definitions of relations and functions and the Technical Note in Notes 1.24.) Warning: The word function has different meanings in mathematics
More informationALGEBRA II Aerospace/Engineering
ALGEBRA II Program Goal 5: The student recognizes the importance of mathematics. Number Systems & Their Properties Demonstrate an understanding of the real # system; recognize, apply, & explain its properties;
More informationTennessee s State Mathematics Standards Precalculus
Tennessee s State Mathematics Standards Precalculus Domain Cluster Standard Number Expressions (N-NE) Represent, interpret, compare, and simplify number expressions 1. Use the laws of exponents and logarithms
More informationJUST THE MATHS UNIT NUMBER 1.6. ALGEBRA 6 (Formulae and algebraic equations) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.6 ALGEBRA 6 (Formulae and algebraic equations) by A.J.Hobson 1.6.1 Transposition of formulae 1.6. of linear equations 1.6.3 of quadratic equations 1.6. Exercises 1.6.5 Answers
More informationIntroduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.
Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. Bernoulli-Euler Beams.
More informationTHE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS
The real number SySTeM C O M P E T E N C Y 1 THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS This competency section reviews some of the fundamental
More informationALGEBRA GRADE 7 MINNESOTA ACADEMIC STANDARDS CORRELATED TO MOVING WITH MATH. Part B Student Book Skill Builders (SB)
MINNESOTA ACADEMIC STANDARDS CORRELATED TO MOVING WITH MATH ALGEBRA GRADE 7 NUMBER AND OPERATION Read, write, represent and compare positive and negative rational numbers, expressed as integers, fractions
More informationIntroduction to Complex Numbers Complex Numbers
Introduction to SUGGESTED LEARNING STRATEGIES: Summarize/Paraphrase/ Retell, Activating Prior Knowledge, Create Representations The equation x 2 + 1 = 0 has special historical and mathematical significance.
More informationMeasurement p. 1 What Is Physics? p. 2 Measuring Things p. 2 The International System of Units p. 2 Changing Units p. 3 Length p. 4 Time p. 5 Mass p.
Measurement p. 1 What Is Physics? p. 2 Measuring Things p. 2 The International System of Units p. 2 Changing Units p. 3 Length p. 4 Time p. 5 Mass p. 7 Review & Summary p. 8 Problems p. 8 Motion Along
More informationQuantum Mechanics - I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras. Lecture - 16 The Quantum Beam Splitter
Quantum Mechanics - I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 16 The Quantum Beam Splitter (Refer Slide Time: 00:07) In an earlier lecture, I had
More informationMarch 19 - Solving Linear Systems
March 19 - Solving Linear Systems Welcome to linear algebra! Linear algebra is the study of vectors, vector spaces, and maps between vector spaces. It has applications across data analysis, computer graphics,
More informationMath 220A Fall 2007 Homework #7. Will Garner A
Math A Fall 7 Homework #7 Will Garner Pg 74 #13: Find the power series expansion of about ero and determine its radius of convergence Consider f( ) = and let ak f ( ) = k! k = k be its power series expansion
More informationWave Mechanics in One Dimension
Wave Mechanics in One Dimension Wave-Particle Duality The wave-like nature of light had been experimentally demonstrated by Thomas Young in 1820, by observing interference through both thin slit diffraction
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2006 Christopher J. Cramer. Lecture 5, January 27, 2006
Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2006 Christopher J. Cramer Lecture 5, January 27, 2006 Solved Homework (Homework for grading is also due today) We are told
More informationReady To Go On? Skills Intervention 7-1 Integer Exponents
7A Evaluating Expressions with Zero and Negative Exponents Zero Exponent: Any nonzero number raised to the zero power is. 4 0 Ready To Go On? Skills Intervention 7-1 Integer Exponents Negative Exponent:
More informationApplied Nuclear Physics (Fall 2006) Lecture 2 (9/11/06) Schrödinger Wave Equation
22.101 Applied Nuclear Physics (Fall 2006) Lecture 2 (9/11/06) Schrödinger Wave Equation References -- R. M. Eisberg, Fundamentals of Modern Physics (Wiley & Sons, New York, 1961). R. L. Liboff, Introductory
More informationAP Physics B Syllabus
AP Physics B Syllabus Course Overview Advanced Placement Physics B is a rigorous course designed to be the equivalent of a college introductory Physics course. The focus is to provide students with a broad
More informationDefinition 6.1 (p.277) A positive integer n is prime when n > 1 and the only positive divisors are 1 and n. Alternatively
6 Prime Numbers Part VI of PJE 6.1 Fundamental Results Definition 6.1 (p.277) A positive integer n is prime when n > 1 and the only positive divisors are 1 and n. Alternatively D (p) = { p 1 1 p}. Otherwise
More informationMath Circles Complex Numbers, Lesson 2 Solutions Wednesday, March 28, Rich Dlin. Rich Dlin Math Circles / 24
Math Circles 018 Complex Numbers, Lesson Solutions Wednesday, March 8, 018 Rich Dlin Rich Dlin Math Circles 018 1 / 4 Warmup and Review Here are the key things we discussed last week: The numbers 1 and
More informationOverall Description of Course Trigonometry is a College Preparatory level course.
Radnor High School Course Syllabus Modified 9/1/2011 Trigonometry 444 Credits: 1 Grades: 11-12 Unweighted Prerequisite: Length: Year Algebra 2 Format: Meets Daily or teacher recommendation Overall Description
More informationENGIN 211, Engineering Math. Complex Numbers
ENGIN 211, Engineering Math Complex Numbers 1 Imaginary Number and the Symbol J Consider the solutions for this quadratic equation: x 2 + 1 = 0 x = ± 1 1 is called the imaginary number, and we use the
More informationLecture 14: Ordinary Differential Equation I. First Order
Lecture 14: Ordinary Differential Equation I. First Order 1. Key points Maple commands dsolve 2. Introduction We consider a function of one variable. An ordinary differential equations (ODE) specifies
More informationSection A.7 and A.10
Section A.7 and A.10 nth Roots,,, & Math 1051 - Precalculus I Roots, Exponents, Section A.7 and A.10 A.10 nth Roots & A.7 Solve: 3 5 2x 4 < 7 Roots, Exponents, Section A.7 and A.10 A.10 nth Roots & A.7
More informationSection 1.1 Task List
Summer 017 Math 143 Section 1.1 7 Section 1.1 Task List Section 1.1 Linear Equations Work through Section 1.1 TTK Work through Objective 1 then do problems #1-3 Work through Objective then do problems
More informationGrade Math (HL) Curriculum
Grade 11-12 Math (HL) Curriculum Unit of Study (Core Topic 1 of 7): Algebra Sequences and Series Exponents and Logarithms Counting Principles Binomial Theorem Mathematical Induction Complex Numbers Uses
More informationMATH Spring 2010 Topics per Section
MATH 101 - Spring 2010 Topics per Section Chapter 1 : These are the topics in ALEKS covered by each Section of the book. Section 1.1 : Section 1.2 : Ordering integers Plotting integers on a number line
More informationECE 487 Lecture 6 : Time-Dependent Quantum Mechanics I Class Outline:
ECE 487 Lecture 6 : Time-Dependent Quantum Mechanics I Class Outline: Time-Dependent Schrödinger Equation Solutions to thetime-dependent Schrödinger Equation Expansion of Energy Eigenstates Things you
More informationCM2202: Scientific Computing and Multimedia Applications General Maths: 3. Complex Numbers
CM2202: Scientific Computing and Multimedia Applications General Maths: 3. Complex Numbers Prof. David Marshall School of Computer Science & Informatics A problem when solving some equations There are
More informationHIGH SCHOOL MATH CURRICULUM GRADE ELEVEN ALGEBRA 2 & TRIGONOMETRY N
VALLEY CENTRAL SCHOOL DISTRICT 944 STATE ROUTE 17K MONTGOMERY, NY 12549 Telephone Number: (845) 457-2400 ext. 8121 FAX NUMBER: (845) 457-4254 HIGH SCHOOL MATH CURRICULUM GRADE ELEVEN ALGEBRA 2 & TRIGONOMETRY
More informationMath-2A Lesson 2-1. Number Systems
Math-A Lesson -1 Number Systems Natural Numbers Whole Numbers Lesson 1-1 Vocabulary Integers Rational Numbers Irrational Numbers Real Numbers Imaginary Numbers Complex Numbers Closure Why do we need numbers?
More informationMORE CONSEQUENCES OF CAUCHY S THEOREM
MOE CONSEQUENCES OF CAUCHY S THEOEM Contents. The Mean Value Property and the Maximum-Modulus Principle 2. Morera s Theorem and some applications 3 3. The Schwarz eflection Principle 6 We have stated Cauchy
More informationPre-Calculus & Trigonometry Scope and Sequence
WHCSD Scope and Sequence Pre-Calculus/ 2017-2018 Pre-Calculus & Scope and Sequence Course Overview and Timing This section is to help you see the flow of the unit/topics across the entire school year.
More informationLecture 1 Complex Numbers. 1 The field of complex numbers. 1.1 Arithmetic operations. 1.2 Field structure of C. MATH-GA Complex Variables
Lecture Complex Numbers MATH-GA 245.00 Complex Variables The field of complex numbers. Arithmetic operations The field C of complex numbers is obtained by adjoining the imaginary unit i to the field R
More informationLesson 8: Complex Number Division
Student Outcomes Students determine the modulus and conjugate of a complex number. Students use the concept of conjugate to divide complex numbers. Lesson Notes This is the second day of a two-day lesson
More informationComplex Organisation and Fundamental Physics
Complex Organisation and Fundamental Physics Brian D. Josephson Mind Matter Unification Project Department of Physics/Trinity College University of Cambridge http://www.tcm.phy.cam.ac.uk/~bdj10 Hangzhou
More informationPrecalculus AB Honors Pacing Guide First Nine Weeks Unit 1. Tennessee State Math Standards
Precalculus AB Honors Pacing Guide First Nine Weeks Unit 1 Revisiting Parent Functions and Graphing P.F.BF.A.1 Understand how the algebraic properties of an equation transform the geometric properties
More informationMTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1.
MTH4101 CALCULUS II REVISION NOTES 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) 1.1 Introduction Types of numbers (natural, integers, rationals, reals) The need to solve quadratic equations:
More informationAssignment #1 Chemistry 314 Summer 2008
Assignment #1 Due Thursday, July 17. Hand in for grading, including especially the graphs and tables of values for question 2. 1. This problem develops the classical treatment of the harmonic oscillator.
More informationNatural Numbers Positive Integers. Rational Numbers
Chapter A - - Real Numbers Types of Real Numbers, 2,, 4, Name(s) for the set Natural Numbers Positive Integers Symbol(s) for the set, -, - 2, - Negative integers 0,, 2,, 4, Non- negative integers, -, -
More informationOur goal is to solve a general constant coecient linear second order. this way but that will not always happen). Once we have y 1, it will always
October 5 Relevant reading: Section 2.1, 2.2, 2.3 and 2.4 Our goal is to solve a general constant coecient linear second order ODE a d2 y dt + bdy + cy = g (t) 2 dt where a, b, c are constants and a 0.
More information39.1 Absolute maxima/minima
Module 13 : Maxima, Minima Saddle Points, Constrained maxima minima Lecture 39 : Absolute maxima / minima [Section 39.1] Objectives In this section you will learn the following : The notion of absolute
More informationSection 1.1 Notes. Real Numbers
Section 1.1 Notes Real Numbers 1 Types of Real Numbers The Natural Numbers 1,,, 4, 5, 6,... These are also sometimes called counting numbers. Denoted by the symbol N Integers..., 6, 5, 4,,, 1, 0, 1,,,
More informationMATH Dr. Halimah Alshehri Dr. Halimah Alshehri
MATH 1101 haalshehri@ksu.edu.sa 1 Introduction To Number Systems First Section: Binary System Second Section: Octal Number System Third Section: Hexadecimal System 2 Binary System 3 Binary System The binary
More informationSection 3: Complex numbers
Essentially: Section 3: Complex numbers C (set of complex numbers) up to different notation: the same as R 2 (euclidean plane), (i) Write the real 1 instead of the first elementary unit vector e 1 = (1,
More informationSemiconductor Physics and Devices
Introduction to Quantum Mechanics In order to understand the current-voltage characteristics, we need some knowledge of electron behavior in semiconductor when the electron is subjected to various potential
More informationBREVET BLANC N 1 JANUARY 2012
Exercise. (5 pts) duration: h Presentation and explanations (4 points) Numerical Activities Consider the figure opposite, which is made up of two squares.. a) Calculate the area A of the white part. b)
More informationhp calculators HP 9s Solving Problems Involving Complex Numbers Basic Concepts Practice Solving Problems Involving Complex Numbers
Basic Concepts Practice Solving Problems Involving Complex Numbers Basic concepts There is no real number x such that x + 1 = 0. To solve this kind of equations a new set of numbers must be introduced.
More information10.1 Complex Arithmetic Argand Diagrams and the Polar Form The Exponential Form of a Complex Number De Moivre s Theorem 29
10 Contents Complex Numbers 10.1 Complex Arithmetic 2 10.2 Argand Diagrams and the Polar Form 12 10.3 The Exponential Form of a Complex Number 20 10.4 De Moivre s Theorem 29 Learning outcomes In this Workbook
More informationChapter 4: Radicals and Complex Numbers
Section 4.1: A Review of the Properties of Exponents #1-42: Simplify the expression. 1) x 2 x 3 2) z 4 z 2 3) a 3 a 4) b 2 b 5) 2 3 2 2 6) 3 2 3 7) x 2 x 3 x 8) y 4 y 2 y 9) 10) 11) 12) 13) 14) 15) 16)
More informationCalculus : Summer Study Guide Mr. Kevin Braun Bishop Dunne Catholic School. Calculus Summer Math Study Guide
1 Calculus 2018-2019: Summer Study Guide Mr. Kevin Braun (kbraun@bdcs.org) Bishop Dunne Catholic School Name: Calculus Summer Math Study Guide After you have practiced the skills on Khan Academy (list
More informationThe general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of time-independent Schrödinger equation
Lecture 27st Page 1 Lecture 27 L27.P1 Review Schrödinger equation The general solution of Schrödinger equation in three dimensions (if V does not depend on time) is where functions are solutions of time-independent
More informationComplex numbers, the exponential function, and factorization over C
Complex numbers, the exponential function, and factorization over C 1 Complex Numbers Recall that for every non-zero real number x, its square x 2 = x x is always positive. Consequently, R does not contain
More informationRadnor Middle School Course Overview. Math Introduction to Algebra 1. Prerequisite: Pre-Algebra, Course 3 Grade: 8
Radnor Middle School Course Overview Math Introduction to Algebra 1 General Information Credits: N/A Length: Full Year Weighted: N/A Format: Meets Daily Prerequisite: Pre-Algebra, Course 3 Grade: 8 I.
More informationMATHEMATICS Grade 7 Standard: Number, Number Sense and Operations. Organizing Topic Benchmark Indicator Number and Number Systems
Standard: Number, Number Sense and Operations Number and Number Systems A. Represent and compare numbers less than 0 through familiar applications and extending the number line. 1. Demonstrate an understanding
More informationUnit 1. Revisiting Parent Functions and Graphing
Unit 1 Revisiting Parent Functions and Graphing Revisiting Statistics (Measures of Center and Spread, Standard Deviation, Normal Distribution, and Z-Scores Graphing abs(f(x)) and f(abs(x)) with the Definition
More informationInstituto Superior Técnico. Calculus of variations and Optimal Control. Problem Series nº 2
Instituto Superior Técnico Calculus of variations and Optimal Control Problem Series nº P1. Snell s law of refraction and Fermat s Principle of Least Time Consider the situation shown in figure P1-1 where
More informationSOLUTIONS TO ALL IN-SECTION and END-OF-SECTION EXERCISES
CHAPTER. SECTION. SOLUTIONS TO ALL IN-SECTION and END-OF-SECTION EXERCISES ESSENTIAL PRELIMINARIES The Language of Mathematics Expressions versus Sentences IN-SECTION EXERCISES: EXERCISE. A mathematical
More information19. TAYLOR SERIES AND TECHNIQUES
19. TAYLOR SERIES AND TECHNIQUES Taylor polynomials can be generated for a given function through a certain linear combination of its derivatives. The idea is that we can approximate a function by a polynomial,
More informationChapter 9: Complex Numbers
Chapter 9: Comple Numbers 9.1 Imaginary Number 9. Comple Number - definition - argand diagram - equality of comple number 9.3 Algebraic operations on comple number - addition and subtraction - multiplication
More informationEureka Math Module 4 Topic C Replacing Letters and Numbers
Eureka Math Module 4 Topic C Replacing Letters and Numbers 6.EE.A.2c: Write, read, and evaluate expressions in which letters stand for numbers. 6.EE.A.4: Identify when two expressions are equivalent. Copy
More informationThe Comparison Test & Limit Comparison Test
The Comparison Test & Limit Comparison Test Math4 Department of Mathematics, University of Kentucky February 5, 207 Math4 Lecture 3 / 3 Summary of (some of) what we have learned about series... Math4 Lecture
More informationComplex Variables, Summer 2016 Homework Assignments
Complex Variables, Summer 2016 Homework Assignments Homeworks 1-4, due Thursday July 14th Do twenty-four of the following problems. Question 1 Let a = 2 + i and b = 1 i. Sketch the complex numbers a, b,
More informationMATHEMATICS-I Math 123
Preface We are thankful to Almighty ALLAH Who has given us an opportunity to write the book, named Applied Mathematics-I as Textbook, intending to cover the new syllabus for the first year students of
More informationMATH ALGEBRA AND FUNCTIONS
Students: 1. Use letters, boxes, or other symbols to stand for any number in simple expressions or equations. 1. Students use and interpret variables, mathematical symbols and properties to write and simplify
More informationBound and Scattering Solutions for a Delta Potential
Physics 342 Lecture 11 Bound and Scattering Solutions for a Delta Potential Lecture 11 Physics 342 Quantum Mechanics I Wednesday, February 20th, 2008 We understand that free particle solutions are meant
More informationElectromagnetic Waves
Physics 8 Electromagnetic Waves Overview. The most remarkable conclusion of Maxwell s work on electromagnetism in the 860 s was that waves could exist in the fields themselves, traveling with the speed
More informationComplex Practice Exam 1
Complex Practice Exam This practice exam contains sample questions. The actual exam will have fewer questions, and may contain questions not listed here.. Be prepared to explain the following concepts,
More informationSection 7.1 Rational Functions and Simplifying Rational Expressions
Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.1 Section 7.1 Rational Functions and Simplifying Rational Expressions Complete the outline as you view Video Lecture 7.1. Pause the video
More informationBasics Quantum Mechanics Prof. Ajoy Ghatak Department of physics Indian Institute of Technology, Delhi
Basics Quantum Mechanics Prof. Ajoy Ghatak Department of physics Indian Institute of Technology, Delhi Module No. # 03 Linear Harmonic Oscillator-1 Lecture No. # 04 Linear Harmonic Oscillator (Contd.)
More informationTheory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati
Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 2 Simpul Rotors Lecture - 2 Jeffcott Rotor Model In the
More information3. Infinite Series. The Sum of a Series. A series is an infinite sum of numbers:
3. Infinite Series A series is an infinite sum of numbers: The individual numbers are called the terms of the series. In the above series, the first term is, the second term is, and so on. The th term
More informationYEAR 12 A-LEVEL FURTHER MATHS CONTENT. Matrices: Add/subtract, multiply by scalar, matrix multiplication, zero and identity matrix (CP1 Ch 6)
YEAR 12 A-LEVEL FURTHER MATHS CONTENT Matrices: Add/subtract, multiply by scalar, matrix multiplication, zero and identity matrix (CP1 Ch 6) Complex Numbers: Real and imaginary numbers, adding/subtracting/multiplying
More informationProblem 1. Part a. Part b. Wayne Witzke ProblemSet #4 PHY ! iθ7 7! Complex numbers, circular, and hyperbolic functions. = r (cos θ + i sin θ)
Problem Part a Complex numbers, circular, and hyperbolic functions. Use the power series of e x, sin (θ), cos (θ) to prove Euler s identity e iθ cos θ + i sin θ. The power series of e x, sin (θ), cos (θ)
More informationFOURIER ANALYSIS. (a) Fourier Series
(a) Fourier Series FOURIER ANAYSIS (b) Fourier Transforms Useful books: 1. Advanced Mathematics for Engineers and Scientists, Schaum s Outline Series, M. R. Spiegel - The course text. We follow their notation
More information