Chapter 2 Complex Numbers February 7 Complex numbers

Chapter 2 Complex Numbers February 7 Complex numbers 2. 1 Introduction 2. 2 Real and imaginary parts of a complex number 2. 3 The complex plane 2. 4 Terminology and notation Solution to a quadratic equation: Example p 46. A complex number has a real part and an imaginary part. 1

Representation of a complex number on the complex plane: A complex number x + iy can be specified or represented by the following equivalent methods on the complex plane: 1) The original rectangular form x + iy. 2) A point with the coordinates (x, y). 3) A vector that starts from the origin and ends at the point (x, y). 4) The polar form reiq that satisfies Example p 48. Modulus (magnitude, absolute value) of a complex number: Angle (argument, phase) of a complex number: Example p 50. 2

Complex conjugate The complex conjugate pair z=x + iy and z*=x − iy are symmetric with respect to the x-axis on the complex plane. Problems 4. 3, 9, 18. 3

Read: Chapter 2: 1 -4 Homework: 2. 4. 1, 3, 5, 18. Due: February 16 4

February 9, 14 Complex algebra 2. 1 Complex algebra A. Simplifying to x + iy form Examples p 51. 1 -4; Problems 5. 3, 7. B. Complex conjugate of a complex expression The complex conjugate of any expression is changing all i’s into –i. Care must be give to multi-value functions, i. e. , Example p 53. 1 5

C. Finding the absolute value of z Example p 53. 2; Problems 5. 26, 28, 33. D. Complex equation Example p 54; Problems 5. 36, 43. E. Graphs Complex equations or inequalities have geometrical meanings. Example p 55. 1 -4; Problems 5. 52, 53, 59. 6

F. Physical applications The position of a moving particle is represented by a vector. This vector also represents a complex number. Addition and subtraction of complex numbers is analogous to the addition and subtraction of vectors. Therefore the position, speed and acceleration of a particle can be well represented by complex numbers. However, because the multiplication of complex numbers is not in analogy with the multiplication of the vectors, physical principles involving multiplication of vectors can not be represented by complex algebra. Example: W=F·s. Example p 56. 7

Read: Chapter 2: 5 Homework: 2. 5. 2, 7, 23, 26, 33, 36, 47, 59, 68. Due: February 23 8

February 16 Complex infinite series 2. 6 Complex infinite series Convergence of a complex infinite series: Theorem: An absolutely convergent complex series converges. 9

Theorem: Ratio test for a complex series an: Example p 57. 1, 2; Problems 6. 2, 6, 7. 10

2. 7 Complex power series; disk of convergence Disk of convergence: An area on which the series is convergent. Radius of convergence: The radius of the disk of convergence. Example p 58. 7. 2 a-c; Problems 7. 5, 8, 15. Disk of convergence for the quotient of two power series: Example p 59. 11

Read: Chapter 2: 6 -7 Homework: 2. 6. 3, 5, 6, 13; 2. 7. 8, 11, 15. Due: February 23 12

February 19 Elementary functions 2. 8 Elementary functions of complex numbers Elementary functions: powers and roots; trigonometric and inverse trigonometric functions; logarithmic and exponential functions; and the combination of them. Functions of a complex variable can be defined using their corresponding infinite series. 2. 9 Euler’s formula Examples p. 62. 13

Multiplication and division of complex numbers using Euler’s formula: Example p. 63; Problems 9. 13, 22, 38. 2. 10 Powers and roots of complex numbers Power of a complex number: Example p. 64. 1. Roots of a complex number: Fundamental theorem of algebra: 14

Notes about roots of a complex number: 1) There altogether n values for 2) The first root is 3) All roots are evenly distributed on the circle with a radius of incremental phase change of . Each root has an Example p. 65. 2 -4; Problem 10. 18. 15

Read: Chapter 2: 8 -10 Homework: 2. 9. 7, 13, 23, 38; 2. 10. 2, 18, 21, 27. Due: March 2 16

February 21 Exponential and trigonometric functions 2. 11 The exponential and trigonometric functions Notes on trigonometric functions: 1) sinz and cosz are generally complex numbers. They can be more than 1 even if they are real. 2) The trigonometric identities (such as ) and the derivative rules (such as ) still hold. Examples p. 68. 1 -4; Problems 11. 6, 9. 17

2. 12 Hyperbolic functions Example p. 70; Problems 12. 1, 9, 15. 18

Read: Chapter 2: 11 -12 Homework: 2. 11. 6, 8, 10, 11; 2. 1, 3, 11, 32, 36. Due: March 2 19

February 23 Complex roots and powers 2. 13 Logarithms Notes on logarithms: 1) We use Lnr to represent the ordinary real logarithm of r. 2) Because a complex number can have an infinite number of phases, it will have an infinite number of logarithms, differing by multiples of 2 pi. 3) The logarithm with the imaginary part 0 q <2 p is called the principle value. Examples p. 72. 1 -2; Problems 14. 3, 6, 7. 20

2. 14 Complex roots and powers Notes on roots and powers: 1) There are many amplitudes as well as many phases for ab. 2) For the amplitude in most cases only the principle value (0 q a<2 p and n=0) is needed. 3) by=0, bx=m or 1/m gives us the real powers and real roots of a complex number. Examples p. 73. 1 -3; Problems 14. 8, 12, 14. 21

2. 15 Inverse trigonometric and hyperbolic functions Example p. 75. 1; Problems 15. 3. 22

Read: Chapter 2: 13 -15 Homework: 2. 14. 3, 4, 8, 14, 17; 2. 15. 1, 3, 15. Due: March 2 23