# Section 3: Complex numbers

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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, 0), and (consequently) write the real α instead of αe 1 = (α, 0) ; (ii) write i (the imaginary unit) instead of the second elementary unit vector e 2 = (0, 1), and (consequently) for β R : write βi instead of βe 2 = (0, β) ; (iii) for α, β R : write α + βi instead of αe 1 + βe 2 = (α, β). So: C = { z = α + βi : α, β R } R ; for z = α + βi one calls α the real part of z and β the imaginary part of z, for short: α = Re(z) and β = Im(z). New: A further algebraic structure of C (see next) Section 3: Complex numbers 1

2 The field of complex numbers: algebraic structure Besides the vector space structure, (α 1 + β 1 i) + (α 2 + β 2 i) = (α 1 α 2 ) + (β 2 + β 2 )i (addition) λ (α + βi) = (α + βi) λ = (λα) + (λβ)i (multipl. by a scalar), one defines: Multiplication in C; reciprocal and division (α 1 + β 1 i) (α 2 + β 2 i) = (α 1 α 2 β 1 β 2 ) + (α 1 β 2 + α 2 β 1 )i Comment: On the left hand side we have formally multiplied out and used i 2 = 1 to obtain the right hand side. If z = α + βi 0 (i.e., not α = β = 0), then 1 z = ( ) 1 α 2 +β α βi, i.e., with this 1 2 z one has 1 z z = 1. So the division of any two complex numbers (but, of course, the denominator must be nonzero) is given by: z 1 1 z 2 = z 1 z 2, if z 2 0. Section 3: Complex numbers / 3.1 Algebraic structure 2

3 Multiplication in C extends the well-known multiplication in R preserving its properties (associativity, commutativity, distributivity): (z 1 z 2 )z 3 = z 1 (z 2 z 3 ), z 1 z 2 = z 2 z 1, (z 1 +z 2 )z 3 = z 1 z 3 +z 2 z 3 Recall: Absolute value z = α 2 + β 2 for z = α + βi. Properties: z 1 + z 2 z 1 + z 2, z 1 z 2 = z 1 z 2. Conjugation: A useful automorphism in C For z = α + βi one defines z = α βi, and calls z the conjugated complex of z. The function z z from C to C is an automorphism, i.e.: z 1 + z 2 = z 1 + z 2 and z 1 z 2 = z 1 z 2. Interesting to note: Re(z) = 1 2 (z + z), Im(z) = 1 2i (z z), zz = z 2 and 1 z = 1 z. z 2 Section 3: Complex numbers / 3.1 Algebraic structure 3

4 The complex exponential function Continuation of the real exponential function exp(x) = e x, x R, to a complex-valued function of a complex variable, exp(z) = e z, z C, by: e z = e α ( cos(β) + sin(β)i ), for all z = α + βi. Using well-known properties of the real exponential function and of the sine and cosine functions we see: Basic property of exp is preserved e z 1+z 2 = e z 1 e z 2 for all z 1, z 2 C. The complex exponential function for purely imaginary arguments: e ϕi = cos(ϕ) + sin(ϕ)i, for any ϕ R, and these complex numbers constitute the unit circle in C (as ϕ varies over R or only over an interval of length 2π). Section 3: Complex numbers / 3.2 Complex exponential function 4

5 Euler representation of a complex number z = α + βi : z = r e ϕi, where r = z and ϕ [0, 2π[ ; (corresponds to polar coordinates of points in the euclidean plane). Note: By the Euler representation geometric interpretation of complex multiplication, via z 1 z 2 = r 1 r 2 e (ϕ 1+ϕ 2 )i, when z j = r j e ϕ ji, j = 1, 2. Complex representations of sine and cosine cos ϕ = Re ( e ϕi) = 1 2 ( e ϕi + e ϕi), sin ϕ = Im ( e ϕi) = 1 2i ( e ϕi e ϕi). Note: e ϕi = cos(ϕ) sin(ϕ)i = cos( ϕ) + sin( ϕ)i = e ϕi. Example: cos 2 ϕ = ( 1 2 ( e ϕi +e ϕi)) 2 ( = 1 4 = ( e 2ϕi + e 2ϕi) = 1 2 cos(2ϕ) ) e 2ϕi +e 2ϕi +2 }{{} e 0 = 1 Section 3: Complex numbers / 3.2 Complex exponential function 5

6 Example: One-dim. harmonic oscillation, complex representation s(t) = a cos(ωt + α), t a real variable (time), a > 0 amplitude, ω > 0 ang. frequency, α [0, 2π[ phase shift. Complex representation: S(t) = A e ωti, A = ae αi the complex amplitude. We have: s(t) = Re ( S(t) ), since: a cos(ωt + α) = Re ( ae (ωt+α)i) = Re ( } ae {{ αi } e ωti) = A Section 3: Complex numbers / 3.2 Complex exponential function 6

7 Example: Superposition of two harmonic oscillations with same frequency but different amplitudes and phases: s 1 (t) = a cos(ωt + α), s 2 (t) = b cos(ωt + β). Superposition: s(t) = s 1 (t) + s 2 (t). Use complex representations: S 1 (t) = Ae ωti, A = ae αi ; S 2 (t) = Be ωti, B = be βi. So: s(t) = Re ( S(t) ), where S(t) = S 1 (t) + S 2 (t) = Ae ωti + Be ωti = (A + B)e ωti. Find Euler representation: C = A + B = c e γi. Then: s(t) = Re ( S(t) ) = c cos(ωt + γ). Section 3: Complex numbers / 3.2 Complex exponential function 7

8 Complex polynomials Well-known: A real polynomial of degree n P (x) = c n x n + c n 1 x n c 1 x + c 0, x R, (with real coefficients c 0, c 1,..., c n, where c n 0) may not have any zero, i.e., there may not exist an x 0 R with P (x 0 ) = 0. Example: A second degree polynomial P (x) = x 2 + px + q, where p, q R are such that = ( p 2 2) q < 0, does not have any (real) zeros. However, for a complex polynomial of degree n, P (z) = c n z n + c n 1 z n c 1 z + c 0, z C, (with complex coefficients c 0, c 1,..., c n, where c n 0) the situation is completely different. Section 3: Complex numbers / 3.3 Complex polynomials 8

9 Fundamental theorem on complex polynomials If P (z) is a complex polynomial of degree n 1, then there exists a zero of P (z), i.e., a z 0 C such that P (z 0 ) = 0 ; and, moreover: There exist z 1, z 2,..., z n C such that P (z) factorizes to P (z) = c n (z z 1 ) (z z 2 ) (z z n ), for all z C, (where c n 0 is the highest coefficient of P (z) ). Note: The complex numbers z 1, z 2,..., z n (the zeros of P (z)) need not be all distinct. Example: The above 2nd degree polynomial, but now with a complex variable: P (z) = z 2 + pz + q, z C, where p, q R such that = ( p 2 2) q < 0. Not difficult to check: P (z) = (z z 1 ) (z z 2 ), where z 1,2 = p 2 ± i. Section 3: Complex numbers / 3.3 Complex polynomials 9

10 Complex eigenvalues and eigenvectors For a (real) n n matrix A consider its characteristic polynomial P A (x) and replace the real variable x by a complex variable z. So: Consider the complex characteristic polynomial P A (z). By the fundamental theorem on complex polynomials: P A (z) = ( 1) n (z λ 1 ) (z λ 2 ) (z λ n ), with some complex numbers λ 1, λ 2,..., λ n (not necessarily all distinct). These λ 1, λ 2,..., λ n are called the (complex) eigenvalues of A. Note: Any real eigenvalue of A (as discussed in Section 2) is among the complex eigenvalues λ 1, λ 2,..., λ n. If A is diagonalizable then all the complex eigenvalues λ 1, λ 2,..., λ n are real. In particular, all the eigenvalues λ 1, λ 2,..., λ n of a symmetric matrix A are real. Section 3: Complex numbers / 3.4 Complex eigenvalues and eigenvectors 10

11 Complex eigenvector If λ C is an eigenvalue of the (real) n n matrix A (i.e., if λ is one of the above λ 1, λ 2,..., λ n ), then there exists a complex vector z C n 1, z 0, such that Az = λz. Remark: For a complex column vector z = z 1 z 2. z n C n 1 multiplication with a complex number λ is as usual defined by coordinatewise multiplication; the matrix-vector product Az is formally defined as for real vectors: So Az is the complex n 1 vector with coordinates n a ij z j, (i = 1, 2,..., n). j=1 Section 3: Complex numbers / 3.4 Complex eigenvalues and eigenvectors 11

### 3 + 4i 2 + 3i. 3 4i Fig 1b

The introduction of complex numbers in the 16th century was a natural step in a sequence of extensions of the positive integers, starting with the introduction of negative numbers (to solve equations of

### Lecture 5. Complex Numbers and Euler s Formula

Lecture 5. Complex Numbers and Euler s Formula University of British Columbia, Vancouver Yue-Xian Li March 017 1 Main purpose: To introduce some basic knowledge of complex numbers to students so that they

### Complex 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

### Integrating Algebra and Geometry with Complex Numbers

Integrating Algebra and Geometry with Complex Numbers Complex numbers in schools are often considered only from an algebraic perspective. Yet, they have a rich geometric meaning that can support developing

### MATH 135: COMPLEX NUMBERS

MATH 135: COMPLEX NUMBERS (WINTER, 010) The complex numbers C are important in just about every branch of mathematics. These notes 1 present some basic facts about them. 1. The Complex Plane A complex

### 1 (2n)! (-1)n (θ) 2n

Complex Numbers and Algebra The real numbers are complete for the operations addition, subtraction, multiplication, and division, or more suggestively, for the operations of addition and multiplication

### Sinusoids. Amplitude and Magnitude. Phase and Period. CMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation

Sinusoids CMPT 889: Lecture Sinusoids, Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University September 6, 005 Sinusoids are

### Complex Numbers. 1 Introduction. 2 Imaginary Number. December 11, Multiplication of Imaginary Number

Complex Numbers December, 206 Introduction 2 Imaginary Number In your study of mathematics, you may have noticed that some quadratic equations do not have any real number solutions. For example, try as

### CMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation

CMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University September 26, 2005 1 Sinusoids Sinusoids

### 2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

### Chapter 10: Sinusoidal Steady-State Analysis

Chapter 10: Sinusoidal Steady-State Analysis 1 Objectives : sinusoidal functions Impedance use phasors to determine the forced response of a circuit subjected to sinusoidal excitation Apply techniques

### Math Matrix Algebra

Math 44 - Matrix Algebra Review notes - 4 (Alberto Bressan, Spring 27) Review of complex numbers In this chapter we shall need to work with complex numbers z C These can be written in the form z = a+ib,

### CMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation

CMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 23, 2006 1 Exponentials The exponential is

### Lecture 2. Spring Quarter Statistical Optics. Lecture 2. Characteristic Functions. Transformation of RVs. Sums of RVs

s of Spring Quarter 2018 ECE244a - Spring 2018 1 Function s of The characteristic function is the Fourier transform of the pdf (note Goodman and Papen have different notation) C x(ω) = e iωx = = f x(x)e

### 10.1 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

### Section 8.2 : Homogeneous Linear Systems

Section 8.2 : Homogeneous Linear Systems Review: Eigenvalues and Eigenvectors Let A be an n n matrix with constant real components a ij. An eigenvector of A is a nonzero n 1 column vector v such that Av

### 1 Review of complex numbers

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

### APPLICATIONS The eigenvalues are λ = 5, 5. An orthonormal basis of eigenvectors consists of

CHAPTER III APPLICATIONS The eigenvalues are λ =, An orthonormal basis of eigenvectors consists of, The eigenvalues are λ =, A basis of eigenvectors consists of, 4 which are not perpendicular However,

### Chapter 5 Eigenvalues and Eigenvectors

Chapter 5 Eigenvalues and Eigenvectors Outline 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalization 5.3 Complex Vector Spaces 2 5.1 Eigenvalues and Eigenvectors Eigenvalue and Eigenvector If A is a n n

### Section 5.5. Complex Eigenvalues

Section 5.5 Complex Eigenvalues Motivation: Describe rotations Among transformations, rotations are very simple to describe geometrically. Where are the eigenvectors? A no nonzero vector x is collinear

### SPRING 2006 PRELIMINARY EXAMINATION SOLUTIONS

SPRING 006 PRELIMINARY EXAMINATION SOLUTIONS 1A. Let G be the subgroup of the free abelian group Z 4 consisting of all integer vectors (x, y, z, w) such that x + 3y + 5z + 7w = 0. (a) Determine a linearly

### SPECIALIST MATHEMATICS

SPECIALIST MATHEMATICS (Year 11 and 12) UNIT A A1: Combinatorics Permutations: problems involving permutations use the multiplication principle and factorial notation permutations and restrictions with

### 1 Complex Numbers. 1.1 Sums and Products

1 Complex Numbers 1.1 Sums Products Definition: The complex plane, denoted C is the set of all ordered pairs (x, y) with x, y R, where Re z = x is called the real part Imz = y is called the imaginary part.

### Mathematics Specialist Units 3 & 4 Program 2018

Mathematics Specialist Units 3 & 4 Program 018 Week Content Assessments Complex numbers Cartesian Forms Term 1 3.1.1 review real and imaginary parts Re(z) and Im(z) of a complex number z Week 1 3.1. review

### Complex Numbers. The set of complex numbers can be defined as the set of pairs of real numbers, {(x, y)}, with two operations: (i) addition,

Complex Numbers Complex Algebra The set of complex numbers can be defined as the set of pairs of real numbers, {(x, y)}, with two operations: (i) addition, and (ii) complex multiplication, (x 1, y 1 )

### MATH 423 Linear Algebra II Lecture 3: Subspaces of vector spaces. Review of complex numbers. Vector space over a field.

MATH 423 Linear Algebra II Lecture 3: Subspaces of vector spaces. Review of complex numbers. Vector space over a field. Vector space A vector space is a set V equipped with two operations, addition V V

### Complex Numbers. The set of complex numbers can be defined as the set of pairs of real numbers, {(x, y)}, with two operations: (i) addition,

Complex Numbers Complex Algebra The set of complex numbers can be defined as the set of pairs of real numbers, {(x, y)}, with two operations: (i) addition, and (ii) complex multiplication, (x 1, y 1 )

### 1 Holomorphic functions

Robert Oeckl CA NOTES 1 15/09/2009 1 1 Holomorphic functions 11 The complex derivative The basic objects of complex analysis are the holomorphic functions These are functions that posses a complex derivative

### 01 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

### 1 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.

### CM2202: 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

### Chapter 2: Complex numbers

Chapter 2: Complex numbers Complex numbers are commonplace in physics and engineering. In particular, complex numbers enable us to simplify equations and/or more easily find solutions to equations. We

### Section 9.3 Phase Plane Portraits (for Planar Systems)

Section 9.3 Phase Plane Portraits (for Planar Systems) Key Terms: Equilibrium point of planer system yꞌ = Ay o Equilibrium solution Exponential solutions o Half-line solutions Unstable solution Stable

### 2 Vector Products. 2.0 Complex Numbers. (a~e 1 + b~e 2 )(c~e 1 + d~e 2 )=(ac bd)~e 1 +(ad + bc)~e 2

Vector Products Copyright 07, Gregory G. Smith 8 September 07 Unlike for a pair of scalars, we did not define multiplication between two vectors. For almost all positive n N, the coordinate space R n cannot

### Announcements Wednesday, November 7

Announcements Wednesday, November 7 The third midterm is on Friday, November 16 That is one week from this Friday The exam covers 45, 51, 52 53, 61, 62, 64, 65 (through today s material) WeBWorK 61, 62

### c Igor Zelenko, Fall

c Igor Zelenko, Fall 2017 1 18: Repeated Eigenvalues: algebraic and geometric multiplicities of eigenvalues, generalized eigenvectors, and solution for systems of differential equation with repeated eigenvalues

### In these chapter 2A notes write vectors in boldface to reduce the ambiguity of the notation.

1 2 Linear Systems In these chapter 2A notes write vectors in boldface to reduce the ambiguity of the notation 21 Matrix ODEs Let and is a scalar A linear function satisfies Linear superposition ) Linear

### EIGENVALUES AND EIGENVECTORS 3

EIGENVALUES AND EIGENVECTORS 3 1. Motivation 1.1. Diagonal matrices. Perhaps the simplest type of linear transformations are those whose matrix is diagonal (in some basis). Consider for example the matrices

### 2nd-Order Linear Equations

4 2nd-Order Linear Equations 4.1 Linear Independence of Functions In linear algebra the notion of linear independence arises frequently in the context of vector spaces. If V is a vector space over the

### 10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections )

c Dr. Igor Zelenko, Fall 2017 1 10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections 7.2-7.4) 1. When each of the functions F 1, F 2,..., F n in right-hand side

### Complex numbers. Reference note to lecture 9 ECON 5101 Time Series Econometrics

Complex numbers. Reference note to lecture 9 ECON 5101 Time Series Econometrics Ragnar Nymoen March 31 2014 1 Introduction This note gives the definitions and theorems related to complex numbers that are

### Mathematics of Imaging: Lecture 3

Mathematics of Imaging: Lecture 3 Linear Operators in Infinite Dimensions Consider the linear operator on the space of continuous functions defined on IR. J (f)(x) = x 0 f(s) ds Every function in the range

### Math 814 HW 3. October 16, p. 54: 9, 14, 18, 24, 25, 26

Math 814 HW 3 October 16, 2007 p. 54: 9, 14, 18, 24, 25, 26 p.54, Exercise 9. If T z = az+b, find necessary and sufficient conditions for T to cz+d preserve the unit circle. T preserves the unit circle

### Chapter 10: Sinusoids and Phasors

Chapter 10: Sinusoids and Phasors 1. Motivation 2. Sinusoid Features 3. Phasors 4. Phasor Relationships for Circuit Elements 5. Impedance and Admittance 6. Kirchhoff s Laws in the Frequency Domain 7. Impedance

### Homework 3 Solutions Math 309, Fall 2015

Homework 3 Solutions Math 39, Fall 25 782 One easily checks that the only eigenvalue of the coefficient matrix is λ To find the associated eigenvector, we have 4 2 v v 8 4 (up to scalar multiplication)

### Differential Equations 2280 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 2015 at 12:50pm

Differential Equations 228 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 25 at 2:5pm Instructions: This in-class exam is 5 minutes. No calculators, notes, tables or books. No answer check is

### COMPLEX ANALYSIS TOPIC V: HISTORICAL REMARKS

COMPLEX ANALYSIS TOPIC V: HISTORICAL REMARKS PAUL L. BAILEY Historical Background Reference: http://math.fullerton.edu/mathews/n2003/complexnumberorigin.html Rafael Bombelli (Italian 1526-1572) Recall

### In Z: x + 3 = 2 3x = 2 x = 1 No solution In Q: 3x = 2 x 2 = 2. x = 2 No solution. In R: x 2 = 2 x = 0 x = ± 2 No solution Z Q.

THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH 1141 HIGHER MATHEMATICS 1A ALGEBRA. Section 1: - Complex Numbers. 1. The Number Systems. Let us begin by trying to solve various

### Background ODEs (2A) Young Won Lim 3/7/15

Background ODEs (2A) Copyright (c) 2014-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any

### Circuit Analysis-III. Circuit Analysis-II Lecture # 3 Friday 06 th April, 18

Circuit Analysis-III Sinusoids Example #1 ü Find the amplitude, phase, period and frequency of the sinusoid: v (t ) =12cos(50t +10 ) Signal Conversion ü From sine to cosine and vice versa. ü sin (A ± B)

### Complex numbers. Reference note to lecture 1 ECON 5101/9101, Time Series Econometrics

Complex numbers. Reference note to lecture 1 ECON 5101/9101, Time Series Econometrics Ragnar Nymoen Feb 3 2011 1 Introduction This note gives the definitions and theorems related to complex numbers that

### Mathematical Preliminaries and Review

Mathematical Preliminaries and Review Ruye Wang for E59 1 Complex Variables and Sinusoids AcomplexnumbercanberepresentedineitherCartesianorpolarcoordinatesystem: z = x + jy= z z = re jθ = r(cos θ + j sin

### Chapter 1. Complex Numbers. Dr. Pulak Sahoo

Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University Of Kalyani West Bengal, India E-mail : sahoopulak1@gmail.com 1 Module-1: Basic Ideas 1 Introduction

### z = x + iy ; x, y R rectangular or Cartesian form z = re iθ ; r, θ R polar form. (1)

11 Complex numbers Read: Boas Ch. Represent an arb. complex number z C in one of two ways: z = x + iy ; x, y R rectangular or Cartesian form z = re iθ ; r, θ R polar form. (1) Here i is 1, engineers call

### Spectral Theorem for Self-adjoint Linear Operators

Notes for the undergraduate lecture by David Adams. (These are the notes I would write if I was teaching a course on this topic. I have included more material than I will cover in the 45 minute lecture;

### and let s calculate the image of some vectors under the transformation T.

Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =

### 3.024 Electrical, Optical, and Magnetic Properties of Materials Spring 2012 Recitation 1. Office Hours: MWF 9am-10am or by appointment

Adam Floyd Hannon Office Hours: MWF 9am-10am or by e-mail appointment Topic Outline 1. a. Fourier Transform & b. Fourier Series 2. Linear Algebra Review 3. Eigenvalue/Eigenvector Problems 1. a. Fourier

### AH Complex Numbers.notebook October 12, 2016

Complex Numbers Complex Numbers Complex Numbers were first introduced in the 16th century by an Italian mathematician called Cardano. He referred to them as ficticious numbers. Given an equation that does

### COMPLEX 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,

### Complex Numbers and Phasor Technique

A P P E N D I X A Complex Numbers and Phasor Technique In this appendix, we discuss a mathematical technique known as the phasor technique, pertinent to operations involving sinusoidally time-varying quantities.

### Midterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015

Midterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015 The test lasts 1 hour and 15 minutes. No documents are allowed. The use of a calculator, cell phone or other equivalent electronic

### Complex Numbers. Introduction

10 Assessment statements 1.5 Complex numbers: the number i 5 1 ; the term s real part, imaginary part, conjugate, modulus and argument. Cartesian form z 5 a 1 ib. Sums, products and quotients of complex

### systems of linear di erential If the homogeneous linear di erential system is diagonalizable,

G. NAGY ODE October, 8.. Homogeneous Linear Differential Systems Section Objective(s): Linear Di erential Systems. Diagonalizable Systems. Real Distinct Eigenvalues. Complex Eigenvalues. Repeated Eigenvalues.

### Quick 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:

### Control Systems. Dynamic response in the time domain. L. Lanari

Control Systems Dynamic response in the time domain L. Lanari outline A diagonalizable - real eigenvalues (aperiodic natural modes) - complex conjugate eigenvalues (pseudoperiodic natural modes) - phase

### Announcements Wednesday, November 7

Announcements Wednesday, November 7 The third midterm is on Friday, November 6 That is one week from this Friday The exam covers 45, 5, 52 53, 6, 62, 64, 65 (through today s material) WeBWorK 6, 62 are

### 8. Complex Numbers. sums and products. basic algebraic properties. complex conjugates. exponential form. principal arguments. roots of complex numbers

EE202 - EE MATH II 8. Complex Numbers Jitkomut Songsiri sums and products basic algebraic properties complex conjugates exponential form principal arguments roots of complex numbers regions in the complex

### 1 Fields and vector spaces

1 Fields and vector spaces In this section we revise some algebraic preliminaries and establish notation. 1.1 Division rings and fields A division ring, or skew field, is a structure F with two binary

Symmetric and self-adjoint matrices A matrix A in M n (F) is called symmetric if A T = A, ie A ij = A ji for each i, j; and self-adjoint if A = A, ie A ij = A ji or each i, j Note for A in M n (R) that

### Complex Analysis Homework 1: Solutions

Complex Analysis Fall 007 Homework 1: Solutions 1.1.. a) + i)4 + i) 8 ) + 1 + )i 5 + 14i b) 8 + 6i) 64 6) + 48 + 48)i 8 + 96i c) 1 + ) 1 + i 1 + 1 i) 1 + i)1 i) 1 + i ) 5 ) i 5 4 9 ) + 4 4 15 i ) 15 4

### The modulus, or absolute value, of a complex number z a bi is its distance from the origin. From Figure 3 we see that if z a bi, then.

COMPLEX NUMBERS _+i _-i FIGURE Complex numbers as points in the Arg plane i _i +i -i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with

### or i 2 = -1 i 4 = 1 Example : ( i ),, 7 i and 0 are complex numbers. and Imaginary part of z = b or img z = b

1 A- LEVEL MATHEMATICS P 3 Complex Numbers (NOTES) 1. Given a quadratic equation : x 2 + 1 = 0 or ( x 2 = -1 ) has no solution in the set of real numbers, as there does not exist any real number whose

### function independent dependent domain range graph of the function The Vertical Line Test

Functions A quantity y is a function of another quantity x if there is some rule (an algebraic equation, a graph, a table, or as an English description) by which a unique value is assigned to y by a corresponding

### A Learning Progression for Complex Numbers

A Learning Progression for Complex Numbers In mathematics curriculum development around the world, the opportunity for students to study complex numbers in secondary schools is decreasing. Given that the

### Matrix Representation

Matrix Representation Matrix Rep. Same basics as introduced already. Convenient method of working with vectors. Superposition Complete set of vectors can be used to express any other vector. Complete set

### Topic 4 Notes Jeremy Orloff

Topic 4 Notes Jeremy Orloff 4 Complex numbers and exponentials 4.1 Goals 1. Do arithmetic with complex numbers.. Define and compute: magnitude, argument and complex conjugate of a complex number. 3. Euler

### CURRICULUM GUIDE. Honors Algebra II / Trigonometry

CURRICULUM GUIDE Honors Algebra II / Trigonometry The Honors course is fast-paced, incorporating the topics of Algebra II/ Trigonometry plus some topics of the pre-calculus course. More emphasis is placed

### Notice that these numbers march out along a spiral. This continues for all powers of 1+i, even negative ones.

18.03 Class 6, Feb 16, 2010 Complex exponential [1] Complex roots [2] Complex exponential, Euler's formula [3] Euler's formula continued [1] More practice with complex numbers: Magnitudes Multiply: zw

### Chapter 3: Complex Numbers

Chapter 3: Complex Numbers Daniel Chan UNSW Semester 1 2018 Daniel Chan (UNSW) Chapter 3: Complex Numbers Semester 1 2018 1 / 48 Philosophical discussion about numbers Q In what sense is 1 a number? DISCUSS

### MATH 1553 PRACTICE MIDTERM 3 (VERSION B)

MATH 1553 PRACTICE MIDTERM 3 (VERSION B) Name Section 1 2 3 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 10 points. The maximum score on this exam is 50 points.

### Autonomous system = system without inputs

Autonomous system = system without inputs State space representation B(A,C) = {y there is x, such that σx = Ax, y = Cx } x is the state, n := dim(x) is the state dimension, y is the output Polynomial representation

### Review of Linear System Theory

Review of Linear System Theory The following is a (very) brief review of linear system theory and Fourier analysis. I work primarily with discrete signals. I assume the reader is familiar with linear algebra

### A = 3 1. We conclude that the algebraic multiplicity of the eigenvalues are both one, that is,

65 Diagonalizable Matrices It is useful to introduce few more concepts, that are common in the literature Definition 65 The characteristic polynomial of an n n matrix A is the function p(λ) det(a λi) Example

### Remark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial

### Discrete mathematics I - Complex numbers

Discrete mathematics I - Emil Vatai (based on hungarian slides by László Mérai) 1 January 31, 018 1 Financed from the financial support ELTE won from the Higher Education Restructuring

### Overview of Complex Numbers

Overview of Complex Numbers Definition 1 The complex number z is defined as: z = a+bi, where a, b are real numbers and i = 1. General notes about z = a + bi Engineers typically use j instead of i. Examples

### 2.3 Oscillation. The harmonic oscillator equation is the differential equation. d 2 y dt 2 r y (r > 0). Its solutions have the form

2. Oscillation So far, we have used differential equations to describe functions that grow or decay over time. The next most common behavior for a function is to oscillate, meaning that it increases and

### Eigenvalues and Eigenvectors

Sec. 6.1 Eigenvalues and Eigenvectors Linear transformations L : V V that go from a vector space to itself are often called linear operators. Many linear operators can be understood geometrically by identifying

### C. Complex Numbers. 1. Complex arithmetic.

C. Complex Numbers. Complex arithmetic. Most people think that complex numbers arose from attempts to solve quadratic equations, but actually it was in connection with cubic equations they first appeared.

### Complex Numbers. Math 3410 (University of Lethbridge) Spring / 8

Complex Numbers Consider two real numbers x, y. What is 2 + x? What is x + y? What is (2 + x)(3 + y)? What is (2x + 3y)(3x + 5y)? What is the inverse of 3 + x? What one fact do I know for sure about x

### Eigenvalues in Applications

Eigenvalues in Applications Abstract We look at the role of eigenvalues and eigenvectors in various applications. Specifically, we consider differential equations, Markov chains, population growth, and

### A Primer on Complex Numbers

ams 10/10A supplementary notes ucsc A Primer on Complex Numbers c 2013, Yonatan Katznelson 1. Imaginary and complex numbers. The real numbers are can be thought of as numbers that represent quantities

### Complex Numbers Review

Complex Numbers view ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The real numbers (denoted R) are incomplete

### Class test: week 10, 75 minutes. (30%) Final exam: April/May exam period, 3 hours (70%).

17-4-2013 12:55 c M. K. Warby MA3914 Complex variable methods and applications 0 1 MA3914 Complex variable methods and applications Lecture Notes by M.K. Warby in 2012/3 Department of Mathematical Sciences

### Topic 2 Quiz 2. choice C implies B and B implies C. correct-choice C implies B, but B does not imply C

Topic 1 Quiz 1 text A reduced row-echelon form of a 3 by 4 matrix can have how many leading one s? choice must have 3 choice may have 1, 2, or 3 correct-choice may have 0, 1, 2, or 3 choice may have 0,

### Properties of Linear Transformations from R n to R m

Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation

### AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES

AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral radius of a matrix A may be obtained using the formula ρ(a) lim