Complex Variables

1. 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. 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 z0which is completely contained in S. z0 S

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

5. Connected • An open set S is said to be connected if every pair of points z1and z2in 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

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. 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 .

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

9. 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.

10. 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.

11. Graph of Complex Function y v w = f(z) x u domain ofdefinition range z-plane w-plane

12. 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.

13. 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!

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

15. Example 1 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) = x2and 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 u x domain

16. Example 2 Describe the function f(z) = z3for 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.

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

18. 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)

19. Example 3 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

20. Sequence • Definition. A sequence of complex numbers, denoted , is a function f, such that f: N  C, i.e, it is a function whose domain is the set of natural numbers between 1 and k, and whose range is a subset of the complex numbers. If k = , then the sequence is called infinite and is denoted by , or more often, zn . (The notation f(n) is equivalent.) • Having defined sequences and a means for measuring the distance between points, we proceed to define the limit of a sequence.

21. Meaning of Zn Z0 Zn Z0 |zn-z0|=rn 0 Where rn= , xn x0,,, yn y0 I proved it in previous classes

22. Geometric Meaning of Zn Z0 zn tends to z0 in any linear or curvilinear way.

23. Limit of a Sequence • Definition. A sequence of complex numbers is said to have the limit z0 , or to converge to z0, if for any  > 0, there exists an integer N such that |zn – z0| <  for all n > N. We denote this by • Geometrically, this amounts to the fact that z0is the only point of znsuch that any neighborhood about it, no matter how small, contains an infinite number of points zn .

24. Theorems and Exercises Theorem. Show that zn=xn+iyn z0=x0+iy0 if and only if xn x0, yn y0 . Ex. Plot the first ten elements of the following sequences and find their limits if they exist: • 1/n +i 1/n • 1/n2 +i 1/n2 • n +i 1/n • (1-1/n )n +i (1+1/n)n

25. Limit of a Function • We say that the complex number w0is the limit of the function f(z) as z approaches z0 if f(z) stays close to w0whenever z is sufficiently near z0 . Formally, we state: • Definition. Limit of a Complex Sequence. Let f(z) be a function defined in some neighborhood of z0 except with the possible exception of the point z0is the number w0if for any real number  > 0 there exists a positive real number  > 0 such that |f(z) – w0|<  whenever 0<|z - z0|< .

26. Limits: Interpretation We can interpret this to mean that if we observe points z within a radius  of z0, we can find a corresponding disk about w0such that all the points in the disk about z0are mapped into it. That is, any neighborhood of w0 contains all the values assumed by f in some full neighborhood of z0, except possibly f(z0). v y w = f(z)   w0 z0 u x w-plane z-plane

27. Complex Functions : Limit and Continuity f: Ω1 Ω2 Ω1 and Ω2 are domain and codomain respectively. Let z0 be a limit point of Ω1 , w0 belongs to Ω2 . Let us take any B any nbd of w0 in Ω2 and take inverse of B, f-1{B}. f-1{B} contains a nbd of z0 in Ω1.. In the case of Continuity the only difference is w0 =f(z0)),

28. Properties of Limits • If as z z0, lim f(z)  A, then A is unique If as z z0, lim f(z)  A and lim g(z)  B, then • lim [ f(z)  g(z) ] = A  B • lim f(z)g(z) = AB, and • lim f(z)/g(z) = A/B. if B  0.

29. Continuity • Definition. Let f(z) be a function such that f: C C. We call f(z) continuous at z0 iff: • F is defined in a neighborhood of z0, • The limit exists, and • A function f is said to be continuous on a set S if it is continuous at each point of S. If a function is not continuous at a point, then it is said to be singular at the point.

30. Note on Continuity • One can show that f(z) approaches a limit precisely when its real and imaginary parts approach limits, and the continuity of f(z) is equivalent to the continuity of its real and imaginary parts.

31. Properties of Continuous Functions • If f(z) and g(z) are continuous at z0, then so are f(z)  g(z) and f(z)g(z). The quotient f(z)/g(z) is also continuous at z0provided that g(z0)  0. • Also, continuous functions map compact sets into compact sets.

32. Exercises • Find domain and range of the following functions and check their continuity: • f1(z)=z • f2(z)=|z| • f3(z)=z2 • f4(z)= • f5(z)=1/(z-2) • f6(z)=ez/log(z)/z1./2/cos(z)

33. Test for Continuity of Functions it is true in a general metric space but not in general topological space. f: <S1,d1> <S2,d2> is continuous at s in S1. For all sn s f(sn) f(s)