Chapter 2 Complex Numbers February 5 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 p46. A complex number has a real part and an imaginary part.

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: The original rectangular form x + iy. A point with the coordinates (x, y). A vector that starts from the origin and ends at the point (x, y). The polar form reiq that satisfies Example p48. Modulus (magnitude, absolute value) of a complex number: Angle (argument, phase) of a complex number: Example p50.

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.

Read: Chapter 2: 1-4 Homework: 2.4.1,3,5,15,18. Due: February 14

February 7, 10 Complex algebra 2.1 Complex algebra A. Simplifying to x + iy form Examples p51.1-4; Problems 5.3,7. B. Complex conjugate of a complex expression The complex conjugate of any expression is just to change all i’s into –i. Example p53.1

C. Finding the absolute value of z Example p53.2; Problems 5.26,28. D. Complex equation Example p54; Problems 5.36,43. E. Graphs Complex equations or inequalities have geometrical meanings. Example p55.1-4; Problems 5.52,53,59.

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 p56.

Read: Chapter 2: 5 Homework: 2.5.2,7,23,26,33,36,47,59,68. Due: February 21

February 14 Complex infinite series 2.6 Complex infinite series Convergence of a complex infinite series: Theorem: An absolutely convergent complex series converges.

Theorem: Ration test for a complex series an: Example p57.1-2; Problems 6.2,6,7.

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 p58.7.2a-c; Problems 7.5,8,15. Disk of convergence for the quotient of two power series: Example p59.

Read: Chapter 2: 6-7 Homework: 2.6.3,5,6,13; 2.7.8,11,15. Due: February 21

February 17 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.

Power of a complex number: Roots of a complex number: 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:

Notes about roots of a complex number: There are altogether n values for The first root is All roots are evenly distributed on the circle with a radius of . Each root has a incremental phase change of Example p.65.2-4; Problem 10.18.

Read: Chapter 2: 8-10 Homework: 2.9.7,13,23,38; 2.10.2,18,21,27. Due: February 28

February 19 Exponential and trigonometric functions 2.11 The exponential and trigonometric functions Notes on trigonometric functions: sinz and cosz are generally complex numbers. They can be more than 1 even if they are real. The trigonometric identities (such as ) and the derivative rules (such as ) still hold. Examples p.68.1-4; Problems 11.6,9.

2.12 Hyperbolic functions Example p.70; Problems 12.1,9,15.

Read: Chapter 2: 11-12 Homework: 2.11.6,8,10,11; 2.12.1,3,11,32,36. Due: February 28

February 21 Complex roots and powers 2.13 Logarithms Notes on logarithms: We use Lnr to represent the ordinary real logarithm of r. Because a complex number can have an infinite number of phases, it will have an infinite number of logarithms, differing by multiples of 2pi. The logarithm with the imaginary part 0q <2p is called the principle value. Examples p.72. 1-2; Problems 14.3,6,7.

2.14 Complex roots and powers Notes on roots and powers: There are many amplitudes as well as many phases for ab. For the amplitude in most cases only the principle value (0q a<2p and n=0) is needed. 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.

2.15 Inverse trigonometric and hyperbolic functions Example p.75. 1; Problems 15.3.

Read: Chapter 2: 13-15 Homework: 2.14.3,4,8,14,17; 2.15.1,3,15. Due: February 28