1. Mohammed Nasser Acknowledgement: Steve Cunningham
Complex Variables
Mohammed Nasser
Acknowledgement:
Steve Cunningham

2. Open Disks or Neighborhoods
Definition. The set of all points z which satisfy the inequality |z – z0|<, where  is a positive real number is called an open disk or neighborhood of z0 .
Remark. The unit disk, i.e., the neighborhood |z|< 1, is of particular significance.
Fig 1 1

3. Interior Point
Definition. A point is called an interior point of S if and only if there exists at least one neighborhood of z0 which is completely contained in S. z0 S
Fig 2

4. Open Set. Closed Set.
Definition. If every point of a set S is an interior point of S, we say that S is an open set.
Definition. S is closed iff Sc is open. Theorem: S`  S, i.e., S contains all of its limit points S is closed set.
Sets may be neither open nor closed.
Neither
Open
Closed
Fig 3

5. Connected
An open set S is said to be connected if every pair of points z1 and z2 in S can be joined by a polygonal line that lies entirely in S. Roughly speaking, this means that S consists of a “single piece”, although it may contain holes. S z1 z2
Fig 4

6. Domain, Region, Closure, Bounded, Compact
An open, connected set is called a domain. A region is a domain together with some, none, or all of its boundary points. The closure of a set S denoted , is the set of S together with all of its boundary. Thus
A set of points S is bounded if there exists a positive real number r such that |z|<r for every z  S.
A region which is both closed and bounded is said to be compact.

7. Open Ball
It is open.
Prove it.
Is it connected?
Yes
Fig 5

8. Problems
The graph of |z – ( i)| < 0.05 is shown in Fig 6 It is an open set.
Is it connected?
Yes
Fig 6

9. Problems
The graph of Re(z)  1 is shown in Fig 7.It is not an open set.
It is a closed set.
Prove it.
Is it connected?
Yes
Fig 7

10. Problems Fig 17.10 illustrates some additional open sets. Fig 8
Both are open.
Prove it.
Are they connected?
Yes
Fig 8

11. Problems
Both are open.
Prove it.
Are they connected?
Yes
Fig 9

12. Problems
It is open.
Prove it.
Is it connected?
Yes
Fig 10

13. Problems
The graph of |Re(z)|  1 is shown in Fig 11.It is not an open set.
X= -1
Is it connected? No
Fig 11

14. Polar Form
Polar Form Referring to Fig , we have z = r(cos  + i sin ) where r = |z| is the modulus of z and  is the argument of z,  = arg(z). If  is in the interval − <   , it is called the principal argument, denoted by Arg(z).

15. Fig 12

16. Example
Solution See Fig 13 that the point lies in the fourth quarter.

17. Fig 13

18. Review: Real Functions of Real Variables
Definition. Let   . A function f is a rule which assigns to each element a   one and only one element b  ,   . We write f:  , or in the specific case b = f(a), and call b “the image of a under f.” We call  “the domain of definition of f ” or simply “the domain of f ”. We call  “the range of f.” We call the set of all the images of , denoted f (), the image of the function f . We alternately call f a mapping from  to .

19. Real Function
In effect, a function of a real variable maps from one real line to another. f  
Fig 14

20. Complex Function
Definition. Complex function of a complex variable. Let   C. A function f defined on  is a rule which assigns to each z   a complex number w. The number w is called a value of f at z and is denoted by f(z), i.e., w = f(z). The set  is called the domain of definition of f. Although the domain of definition is often a domain, it need not be.

21. Remark
Properties of a real-valued function of a real variable are often exhibited by the graph of the function. But when w = f(z), where z and w are complex, no such convenient graphical representation is available because each of the numbers z and w is located in a plane rather than a line.
We can display some information about the function by indicating pairs of corresponding points z = (x,y) and w = (u,v). To do this, it is usually easiest to draw the z and w planes separately.

22. Graph of Complex Functiony v
w = f(z) x u
domain of definition
range
z-plane
w-plane
Fig 15

23. Example 1 Find the image of the line Re(z) = 1 under f(z) = z2.
Solution Now Re(z) = x = 1, u = 1 – y2, v = 2y.

24. Fig 16

25. Complex Exponential Function
Evaluate e i.
Solution:
You prove them.

26. Periodicity

27. Polar From of a Complex Number Revisited
Arithmetic Operations in Polar Form
The representation of z by its real and imaginary parts is useful for addition and subtraction.
For multiplication and division, representation by the polar form has apparent geometric meaning.

28. Suppose we have 2 complex numbers, z1 and z2 given by :
Easier with normal form than polar form
Easier with polar form than normal form
magnitudes multiply!
phases add!

29. For a complex number z2 ≠ 0,
phases subtract!
magnitudes divide!

30. Some Exercises (Example2)
Express and in terms of powers of and
Ex. 2:

31. find the nth roots of unity
k is an integer
Ex. :
Find the solutions to the equation
Imz
Rez
Ex. :
The three roots of are
Proof:

32. Ex: Solve the polynomial equation
(1)
(2)
(3)

33. Example 3
Describe the range of the function f(z) = x2 + 2i, defined on (the domain is) the unit disk |z| 1.
Solution: We have u(x,y) = x2 and v(x,y) = 2. Thus as z varies over the closed unit disk, u varies between 0 and 1, and v is constant (=2).
Therefore w = f(z) = u(x,y) + iv(x,y) = x2 +2i is a line segment from w = 2i to w = 1 + 2i. v y
f(z)
range x u
domain

34. Example 4
Describe the function f(z) = z3 for z in the semidisk given by |z| 2, Im z  0.
Solution: We know that the points in the sector of the semidisk from Arg z = 0 to Arg z = 2/3, when cubed cover the entire disk |w| 8 because
The cubes of the remaining points of z also fall into this disk, overlapping it in the upper half-plane as depicted on the next screen.

35. w = z3 y v 8 2 x u -2 2 -8 8 -8
Fig 17

36. If Z is in x+iy form
z3=z2..z=(x2-y2+i2xy)(x+iy)=(x3-xy2+i2x2y+ix2y-iy3 -2xy2)
=(x3-3xy2) +i(3x2y-y3)
If u(x,y)= (x3-3xy2) and v(x,y)=(3x2y-y3), we can write
z3=f(z)=u(x,y) +iv(x,y)

37. Example 5 f(z)=z2, g(z)=|z| and h(z)= D={(x,x)|x is a real number}
D={|z|<4| z is a complex number}
Draw the mappings

38. Logarithm Function
Given a complex number z = x + iy, z  0, we define w = ln z if z = ew Let w = u + iv, then We have and also

39. Logarithm of a Complex Number
DEFINITION
For z  0, and  = arg z,
Example 6
Find the values of (a) ln (−2) (b) ln i, (c) ln (−1 – i ).

40. Solution

41. Example 7
Find all values of z such that
Solution

42. Principal Value
Since Arg z is unique, there is only one value of Ln z for which z  0.

43. Example 8
The principal values of example 6 are as follows.

44. Important Point
Each function in the collection of ln z is called a branch. The function Ln z is called the principal branch or the principal logarithm function.
Some familiar properties of logarithmic function hold in complex case:

45. Example 9
Suppose z1 = 1 and z2 = -1. If we take ln z1 = 2i, ln z2 = i, we get