Complex Analysis

1. Complex Analysis

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. If B(S)  S, i.e., if S contains all of its boundary points, then it is called a 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. 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

13. 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.

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

15. 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.

16. 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 .

17. 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|< .

18. 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

19. Properties of Limits 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.

20. 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.

21. 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.

22. 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.

23. Derivatives • Differentiation of complex-valued functions is completely analogous to the real case: • Definition. Derivative. Let f(z) be a complex-valued function defined in a neighborhood of z0. Then the derivative of f(z) at z0is given byProvided this limit exists. F(z) is said to be differentiable at z0.

24. Properties of Derivatives

25. Analytic. Holomorphic. • Definition. A complex-valued function f (z) is said to be analytic, or equivalently, holomorphic, on an open set  if it has a derivative at every point of . (The term “regular” is also used.) • It is important that a function may be differentiable at a single point only. Analyticity implies differentiability within a neighborhood of the point. This permits expansion of the function by a Taylor series about the point. • If f (z) is analytic on the whole complex plane, then it is said to be an entire function.

26. Rational Function. • Definition. If f and g are polynomials in z, then h (z) = f (z)/g(z), g(z)  0 is called a rational function. • Remarks. • All polynomial functions of z are entire. • A rational function of z is analytic at every point for which its denominator is nonzero. • If a function can be reduced to a polynomial function which does not involve , then it is analytic.

27. Example 1 Thus f1(z) is analytic at all points except z=1.

28. Example 2 Thus f2(z) is nowhere analytic.

29. Testing for Analyticity Determining the analyticity of a function by searching for in its expression that cannot be removed is at best awkward. Observe: It would be difficult and time consuming to try to reduce this expression to a form in which you could be sure that the could not be removed. The method cannot be used when anything but algebraic functions are used.

30. Cauchy-Riemann Equations (1) If the function f (z) = u(x,y) + iv(x,y) is differentiable at z0 = x0 + iy0, then the limit can be evaluated by allowing z to approach zero from any direction in the complex plane.

31. Cauchy-Riemann Equations (2) If it approaches along the x-axis, then z = x, and we obtain But the limits of the bracketed expression are just the first partial derivatives of u and v with respect to x, so that:

32. Cauchy-Riemann Equations (3) If it approaches along the y-axis, then z = y, and we obtain And, therefore

33. Cauchy-Riemann Equations (4) By definition, a limit exists only if it is unique. Therefore, these two expressions must be equivalent. Equating real and imaginary parts, we have that must hold at z0 = x0 + iy0 . These equations are called the Cauchy-Riemann Equations. Their importance is made clear in the following theorem.

34. Cauchy-Riemann Equations (5) • Theorem. Let f (z) = u(x,y) + iv(x,y) be defined in some open set  containing the point z0. If the first partial derivatives of u and v exist in , and are continuous at z0 , and satisfy the Cauchy-Riemann equations at z0, then f (z) is differentiable at z0. Consequently, if the first partial derivatives are continuous and satisfy the Cauchy-Riemann equations at all points of , then f (z) is analytic in .

35. Example 1 Hence, the Cauchy-Riemann equations are satisfied only on the line x = y, and therefore in no open disk. Thus, by the theorem, f (z) is nowhere analytic.

36. Example 2 Prove that f (z) is entire and find its derivative. The first partials are continuous and satisfy the Cauchy-Riemann equations at every point.

37. Harmonic Functions • Definition. Harmonic. A real-valued function (x,y) is said to be harmonic in a domain D if all of its second-order partial derivatives are continuous in D and if each point of D satisfies Theorem. If f (z) = u(x,y) + iv(x,y) is analytic in a domain D, then each of the functions u(x,y) and v(x,y) is harmonic in D.

38. Harmonic Conjugate • Given a function u(x,y) harmonic in, say, an open disk, then we can find another harmonic function v(x,y) so that u + iv is an analytic function of z in the disk. Such a function v is called a harmonic conjugate of u.

39. Example Construct an analytic function whose real part is: Solution: First verify that this function is harmonic.

40. Example, Continued Integrate (1) with respect to y:

41. Example, Continued Now take the derivative of v(x,y) with respect to x: According to equation (2), this equals 6xy – 1. Thus,

42. Example, Continued The desired analytic function f (z) = u + iv is:

43. Complex Exponential • We would like the complex exponential to be a natural extension of the real case, with f (z) = ezentire. We begin by examining ez = ex+iy = exeiy. • eiy = cos y + i sin y by Euler’s and DeMoivre’s relations. • Definition. Complex Exponential Function. If z = x + iy, then ez = ex(cos y + i sin y). • That is, |ez|= ex and arg ez = y.

44. More on Exponentials • Recall that a function f is one-to-one on a set S if the equation f (z1) = f (z2), where z1,z2  S, implies that z1 = z2. The complex exponential function is not one-to-one on the whole plane. • Theorem. A necessary and sufficient condition that ez = 1 is that z = 2ki, where k is an integer. Also, a necessary and sufficient condition that is that z1 = z2 +2ki, where k is an integer. Thus ez is a periodic function.