Chapter 1. Complex Numbers Weiqi Luo ( 骆伟祺 ) School of Software Sun Yat-Sen University ： Office ： # A313

School of Software  Textbook: James Ward Brown, Ruel V. Churchill, Complex Variables and Applications (the 8th ed.), China Machine Press, 2008  Reference:  王忠仁 张静 《工程数学 - 复变函数与积分变换》高等教育出 版社，

School of Software Numbers System 3 Refer to: Natural Numbers Zero & Negative Numbers IntegersFraction Rational numbersIrrational numbers Real numbers Imaginary numbers Complex numbers … More advanced number systems

School of Software  Sums and Products; Basic Algebraic Properties  Further Properties; Vectors and Moduli  Complex Conjugates; Exponential Form  Products and Powers in Exponential Form  Arguments of Products and Quotients  Roots of Complex Numbers  Regions in the Complex Plane 4 Chapter 1: Complex Numbers

School of Software  Definition Complex numbers can be defined as ordered pairs (x, y) of real numbers that are to be interpreted as points in the complex plane 5 1. Sums and Products Complex plane (x, y) (x, 0) (0, y) Real axis imaginary axis Note: The set of complex numbers Includes the real numbers as a subset x y O

School of Software  Notation It is customary to denote a complex number (x,y) by z, 6 1. Sums and Products x y z=(x, y) (x, 0) (0, y) x = Rez (Real part); y = Imz (Imaginary part) z 1 =z 2 iff 1.Rez 1 = Rez 2 2.Imz 1 = Imz 2 O Q: z 1 <z 2 ?

School of Software  Two Basic Operations  Sum (x 1, y 1 ) + (x 2, y 2 ) = (x 1 +x 2, y 1 +y 2 )  Product (x 1, y 1 ) (x 2, y 2 ) = (x 1 x 2 - y 1 y 2, y 1 x 2 +x 1 y 2 ) 7 1. Sums and Products 1.when y 1 =0, y 2 =0, the above operations reduce to the usual operations of addition and multiplication for real numbers. 2. Any complex number z= (x,y) can be written z = (x,0) + (0,y) 3. Let i be the pure imaginary number (0,1), then z = x (1, 0) + y (0,1) = x + i y, x & y are real numbers i 2 =(0,1) (0,1) =(-1, 0)  i 2 =-1

School of Software  Two Basic Operations (i 2  -1)  Sum (x 1, y 1 ) + (x 2, y 2 ) = (x 1 +x 2, y 1 +y 2 )  (x 1 + iy 1 ) + (x 2 + iy 2 ) = (x 1 +x 2 )+i(y 1 +y 2 )  Product (x 1, y 1 ) (x 2, y 2 ) = (x 1 x 2 - y 1 y 2, y 1 x 2 +x 1 y 2 )  (x 1 + iy 1 ) (x 2 + iy 2 ) = (x 1 x 2 + x 1 iy 2 ) + (iy 1 x 2 + i 2 y 1 y 2 ) = (x 1 x 2 + x 1 iy 2 ) + (iy 1 x 2 - y 1 y 2 ) = (x 1 x 2 - y 1 y 2 ) +i(y 1 x 2 +x 1 y 2 ) 8 1. Sums and Products

School of Software  Various properties of addition and multiplication of complex numbers are the same as for real numbers  Commutative Laws z 1 + z 2 = z 2 +z 1, z 1 z 2 =z 2 z 1  Associative Laws (z 1 + z 2 )+ z 3 = z 1 + (z 2 +z 3 ) (z 1 z 2 ) z 3 =z 1 (z 2 z 3 ) 9 2. Basic Algebraic Properties e.g. Prove that z 1 z 2 =z 2 z 1 (x 1, y 1 ) (x 2, y 2 ) = (x 1 x 2 - y 1 y 2, y 1 x 2 +x 1 y 2 ) = (x 2 x 1 - y 2 y 1, y 2 x 1 +x 2 y 1 ) = (x 2, y 2 ) (x 1, y 1 )

School of Software  For any complex number z(x,y)  z + 0 = z; z ∙ 0 = 0; z ∙ 1 = z  Additive Inverse -z = 0 – z = (-x, -y)  (-x, -y) + (x, y) =(0,0)=0  Multiplicative Inverse when z ≠ 0, there is a number z -1 (u,v) such that z z -1 =1, then (x,y) (u,v) =(1,0)  xu-yv=1, yu+xv= Basic Algebraic Properties

School of Software pp. 5 Ex. 1, Ex.4, Ex. 8, Ex. 9 Homework 11

School of Software  If z 1 z 2 =0, then so is at least one of the factors z 1 and z 2 3. Further Properties 12 Proof: Suppose that z 1 ≠ 0, then z 1 -1 exists z 1 -1 (z 1 z 2 )=( z 1 -1 z 1 ) z 2 =1 z 2 = z 2 Associative Laws z 1 -1 (z 1 z 2 )=z =0 Therefore we have z 2 =0

School of Software  Other two operations: Subtraction and Division  Subtraction: z 1 -z 2 =z 1 +(-z 2 ) (x 1, y 1 ) - (x 2, y 2 ) = (x 1, y 1 )+(-x 2, -y 2 ) = (x 1 -x 2, y 1 -y 2 )  Division: 3. Further Properties 13

School of Software  An easy way to remember to computer z 1 /z 2 3. Further Properties 14 Note that commonly used For instance

School of Software 3. Further Properties 15 Binomial Formula Where

School of Software pp.8 Ex. 1. Ex. 2, Ex. 3, Ex Further Properties 16

School of Software  Any complex number is associated a vector from the origin to the point (x, y) 4. Vectors and Moduli 17 x x y O z1z1 z2z2 z 1 +z 2 Sum of two vectors y z 1 =(x 1, y 1 ) O z 2 =(x 2, y 2 ) The moduli or absolute value of z is a nonnegative real number Product: refer to pp.21

School of Software  Example 1 The distance between two point z 1 (x 1, y 1 ) and z 2 (x 2, y 2 ) is |z 1 -z 2 |. 4. Vectors and Moduli 18 x y O z2z2 z1z1 |z 1 - z 2 | -z 2 z 1 - z 2 Note: |z 1 - z 2 | is the length of the vector representing the number z 1 -z 2 = z 1 + (-z 2 ) Therefore

School of Software  Example 2 The equation |z-1+3i|=2 represents the circle whose center is z 0 = (1, -3) and whose radius is R=2 4. Vectors and Moduli 19 x y O z 0 (1, -3) Note: | z-1+3i | = | z-(1-3i) | = 2

School of Software  Some important inequations  Since we have  Triangle inequality 4. Vectors and Moduli 20 x y z 1 =(x, y) O x y O z1z1 z2z2 z 1 +z 2

School of Software  4. Vectors and Moduli 21 Proof: when |z 1 | ≥ |z 2 |, we write Similarly when |z 2 | ≥ |z 1 |, we write Triangle inequality

School of Software       4. Vectors and Moduli 22

School of Software  Example 3 If a point z lies on the unit circle |z|=1 about the origin, then we have 4. Vectors and Moduli 23 2 x y O z 1

School of Software pp. 12 Ex. 2, Ex. 4, Ex Homework 24

School of Software  Complex Conjugate (conjugate) The complex conjugate or simply the conjugate, of a complex number z=x+iy is defined as the complex number x-iy and is denoted by z 5. Complex Conjugates 25 x y O z(x,y) z (x,-y) Properties:

School of Software  If z 1 =x 1 +iy 1 and z 2 =x 2 +iy 2, then  Similarly, we have 5. Complex Conjugates 26

School of Software  If, then  5. Complex Conjugates 27

School of Software  Example 1 5. Complex Conjugates 28

School of Software  Example 2 5. Complex Conjugates 29 Refer to pp. 14

School of Software pp. 14 – 16 Ex. 1, Ex. 2, Ex. 7, Ex Homework 30

School of Software  Polar Form Let r and θ be polar coordinates of the point (x,y) that corresponds to a nonzero complex number z=x+iy, since x=rcosθ and y=rsinθ, the number z can be written in polar form as z=r(cosθ + isinθ), where r>0 6. Exponential Form 31 y x O z(x,y) θ r argz: the argument of z Argz: the principal value of argz x y O z(x,y) 1 r θ θ Θ

School of Software  Example 1 The complex number -1-i, which lies in the third quadrant has principal argument -3π/4. That is It must be emphasized that the principal argument must be in the region of (-π, +π ]. Therefore, However, 6. Exponential Form 32 argz = α + 2nπ Here: α can be any one of arguments of z

School of Software  The symbol e iθ, or exp(iθ) 6. Exponential Form 33 Why? Refer to Sec. 29 cosθsinθ Let x=iθ, then we have

School of Software  Example 2 The number -1-i in Example 1 has exponential form 6. Exponential Form 34

School of Software  z=Re iθ where 0≤ θ ≤2 π 6. Exponential Form 35 x y O Re iθ R θ x y θ O z0z0 z z=z 0 +Re iθ Re iθ |z-z 0 |=R

School of Software  Product in exponential form 7. Products and Powers in Exponential Form 36

School of Software  Example 1 In order to put in rectangular form, one need only write 7. Products and Powers in Exponential Form 37

School of Software  Example 2 7. Products and Powers in Exponential Form 38 de Moivre’s formula pp. 23, Exercise 10, 11

School of Software 8. Arguments of products and quotients 39 argz 1 z 2 = θ 1 +θ 2 +2(n 1 +n 2 )π = (θ 1 +2n 1 π)+ (θ 2 +2n 2 π) = argz 1 +argz 2 arg(z 1 z 2 )= θ 1 +θ 2 +2nπ, n=0, ±1, ±2 … θ 1 is one of arguments of z 1 and θ 2 is one of arguments of z 2 then θ 1 +θ 2 is one of arguments of z 1 z 2 Here: n 1 and n 2 are two integers with n 1 +n 2 =n Q: Argz 1 z 2 = Argz 1 +Argz 2 ?

School of Software  Example 1 When z 1 =-1 and z 2 =i, then Arg(z 1 z 2 )=Arg(-i) = -π/2 but Arg(z 1 )+Arg(z 2 )=π+π/2=3π/2 8. Arguments of products and quotients 40 ≠ Note: Argz 1 z 2 =Argz 1 +Argz 2 is not always true.

School of Software  Arguments of Quotients 8. Arguments of products and quotients 41

School of Software  Example 2 In order to find the principal argument Arg z when observe that since 8. Arguments of products and quotients 42 Argz

School of Software pp Ex. 1, Ex. 6, Ex. 8, Ex Homework 43

School of Software  Two equal complex numbers 9. Roots of Complex Numbers 44 If and only if for some integer k At the same point

School of Software  Roots of Complex Number Given a complex number, we try to find all the number z, s.t. Let then thus we get 9. Roots of Complex Numbers 45 The unique positive nth root of r 0

School of Software The nth roots of z 0 are 9. Roots of Complex Numbers 46 Note: 1.All roots lie on the circle |z|; 2.There are exactly n distinct roots! |z|

School of Software 9. Roots of Complex Numbers 47 Let then Therefore where Note: the number c 0 can be replaced by any particular nth root of z 0

School of Software  Example 1 Let us find all values of (-8i) 1/3, or the three roots of the number -8i. One need only write To see that the desired roots are 10. Examples 48 2i

School of Software  Example 2 To determine the nth roots of unity, we start with And find that 10. Examples 49 n=3n=4 n=6

School of Software  Example 3 the two values c k (k=0,1) of, which are the square roots of, are found by writing 10. Examples 50

School of Software pp Ex. 2, Ex. 4, Ex. 5, Ex. 7, Ex Homework 51

School of Software  ε- neighborhood The ε- neighborhood of a given point z 0 in the complex plane as shown below 11. Regions in the Complex Plane 52 x y O z0z0 ε z Deleted neighborhood x y O z0z0 ε z Neighborhood

School of Software  Interior Point A point z 0 is said to be an interior point of a set S whenever there is some neighborhood of z 0 that contains only points of S  Exterior Point A point z 0 is said to be an exterior point of a set S when there exists a neighborhood of it containing no points of S;  Boundary Point (neither interior nor exterior) A boundary point is a point all of whose neighborhoods contain at least one point in S and at least one point not in S. The totality of all boundary points is called the boundary of S. 11. Regions in the Complex Plane 53

School of Software  Consider the set S={z| |z|≤1} 11. Regions in the Complex Plane 54 z0z0 x y O All points z, where |z|<1 are Interior points of S; z0z0 All points z, where |z|>1 are Exterior points of S; All points z, where |z|=1 are Boundary points of S; z0z0 S={z| |z|≤1-{1,0}} ?

School of Software  Open Set A set is open if it and only if each of its points is an interior point.  Closed Set A set is closed if it contains all of its boundary points.  Closure of a set The closure of a set S is the closed set consisting of all points in S together with the boundary of S. 11. Regions in the Complex Plane 55

School of Software  Examples  S={z| |z|<1} ? Open Set  S={z| |z|≤1} ? Closed Set  S={z| |z|≤1} – {(0,0)} ? Neither open nor closed  S= all points in complex plane ? Both open and closed Key: identify those boundary points of a given set 11. Regions in the Complex Plane 56

School of Software  Connected An open set S is connected if each pair of points z 1 and z 2 in it can be joined by a polygonal line, consisting of a finite number of line segments joined end to end, that lies entirely in S. 11. Regions in the Complex Plane 57 The open set 1<|z|<2 is connected. x O The set S={z| |z|<1 U |z-(2+i)|<1} is open However, it is not connected. y

School of Software  Domain A set S is called as a domain iff 1.S is open; 2.S is connected. e.g. any neighborhood is a domain.  Region A domain together with some, none, or all of it boundary points is referred to as a region. 11. Regions in the Complex Plane 58

School of Software  Bounded A set S is bounded if every point of S lies inside some circle |z|=R; Otherwise, it is unbounded. 11. Regions in the Complex Plane 59 x y O S R e.g. S={z| |z|≤1} is bounded S={z| Rez≥0} is unbounded

School of Software  Accumulation point A point z 0 is said to be an accumulation point of a set S if each deleted neighborhood of z 0 contains at least one point of S.  If a set S is closed, then it contains each of its accumulation points. Why?  A set is closed iff it contains all of its accumulation points 11. Regions in the Complex Plane 60 The relationships among the Interior, Exterior, Boundary and Accumulation Points! e.g. the origin is the only accumulation point of the set Z n =i/n, n=1,2,…  An Interior point must be an accumulation point.  An Exterior point must not be an accumulation point.  A Boundary point must be an accumulation point?

School of Software pp. 33 Ex. 1, Ex. 2, Ex. 5, Ex. 6, Ex Homework 61