Arithmetic. Today I will talk about the definition of the complex number system. Complex arithmetic is elementary and easy, perhaps trivial and familiar to you. It's a matter of basic language and technique.

Assignment: If you like, you may ignore Chapter 1 until and unless I make a specific reference to material that we need. Depending on your background, much of what's there may be familiar. However, you may wish at least to look over Chapter 1 during this first week; and if you wish to study parts of it in a systematic way, here are some suggested problems to do on your own (not to turn in): Section 1.1: 1-4, 9-12. Section 1.2: 1-7, 9, 11-15. Section 1.3: 2-13. Section 1.4: 1-4, 6, 7, 10, 11, 19, 20. Section 1.5: 1, 3, 4, 6, 7. Section 1.6: 1-11, 13, 14, 17-19, 21, 25. Note: "On your own" means, not to turn in. Feel free to ask me questions about any of this material.

The following problems in Chapter 1 may be written up and turned in for credit, but no later than Friday, June 15. In 1.1, no more than one of these: 8, 9, 10. In 1.3, no more than one of these: 7, 11, 14, 15. In 1.5, you may do 7. In 1.6, no more than two of these: 10, 19, 20, 23, 26, 27.

General instructions about the problems that you turn in: Please turn in no more than one or two problems per day, starting June 5. Please turn in each problem (or, sometimes, a specified combination of problems) on a separate paper. For the scoring criteria, see this PDF file.
 
 

I'll continue to talk about complex arithmetic, and about complex notation for objects (like circles and lines). I'll also talk about ways to deal with certain elementary mappings on the plane.

Assignment: Read Section 2.1. Complex arithmetic should become second nature to you. The polar-coordinates definition of multiplication conveys the geometric meaning that enlivens complex-arithmetic computations like multiplication, division, conjugation, taking powers, and finding roots (for example, the 8th roots of 2). Notice the cute confirmation that the angles of a triangle sum to pi radians. What are the linear maps from the complex plane to the complex plane? How are they different from the linear maps from the real plane to the real plane? The complex equations of lines and planes make it easier to understand how they are mapped by the reciprocal map and by reflection in the unit circle, and we'll need these results in Section 5.2.

It will be a good investment of your time to work on getting a good sense of basic complex arithmetic and the geometry of the plane as discussed in this Section. Some problems to do on your own: 2.1: 1-5, 7, 8, 9-15, 18-22, 33, and 38.

For credit, among the problems in Section 2.1, you may do either (1) 30 AND 31 (counting as one problem); or (2) 34, 35, AND 36 (which must all three be done, and which will count as two problems); but not both. Deadline: Friday, June 15.

An extra note: I hope you find that the conventions regarding the point at infinity, stated on page 93, work well. They work for deeper reasons than are now apparent. Some books present the Riemann sphere quite early, but I find that it's not really motivated until singularities are studied. That's why it occurs as Section 3.6 in my book. Some of you may find it interesting and worthwhile at least to read over Section 3.6 quickly and look at the pictures.
 
 

Complex analysis is about mappings from the plane to the plane. A certain amount of technique is required to visualize such a map and understand how it behaves. I'll develop and illustrate some techniques for dealing with a few specific mappings, chosen because they are important and useful: powers, exponential, logarithm, and sine.

Assignment: Read Section 2.2. Problems to do on your own: 1, 2, 3, 5, 7, 11, 14, 16, 25. It may be important to do Problem 2. It deals with an elementary point about the definition of powers.

You may turn in any one of these from Section 2.2: 19; {21, 22, AND 23} (counting as one); or 27. Deadline: Friday, June 15.

An extra note: By doing a search on the internet for "Complex Analysis," you may find a good many websites, providing the syllabi of courses at various universities, as well as colorful and dynamic graphics. When you have studied mappings, you may find it amusing and instructive to look at some of these. Here is a link to one site that is nice to look at, and that has links to some other sites.
 
 

Today I'll talk a bit more about the sine. By way of preview, I'll tell you briefly about how the sine helps to solve certain boundary value problems. And I'll begin to talk about differentiability.

The condition of approximability by a linear function unifies all the definitions of differentiability. A linear function from the complex plane to the complex plane is given by multiplication by a complex number. It is thus a restricted, special case of a linear function from the real plane to the real plane. This makes the Cauchy-Riemann Equations understandable.

Assignment: Read Section 2.3. On your own, do problems 2, 3, 5, 7, 8, 9, 10.
 
 

As given in Section 2.3, the definition of a holomorphic function f(x+iy)=u(x,y)+iv(x,y) on an open set entails, at each point, the existence of the four first-order partial derivatives (of u and v with respect to x and y) and their obedience to the Cauchy-Riemann equations, plus the condition of the good linear approximation. It may appear to you, interestingly enough, that holomorphy does not imply the continuity of the four first-order partials, or even the existence of the second-order partials, much less the obedience of u and v to Laplace's equation. More interestingly, as we shall see before long, it does imply those things and much more.

I'm going to provide some handouts showing several of the Figures in the book and, in case it's of interest, the Mathematica code that generated them.

I'm going to give you a brief survey of what's coming, including Goursat's Lemma, which we will cover in this course, and the Looman-Menchoff Theorem, which we will not.

Sequences. Upper bounds, least upper bounds, limit superior of a sequence, geometric series.

Assignment: Read Section 2.4. On your own, do at least eight of the problems.

For credit, you may write up any two of the problems 1-15 in Section 2.4. The two will count as one. OR, you may write up {16 AND 17}, which will count double. I'll give you another option which may interest some of you, pertaining to the Looman-Menchoff Theorem; see me for details. Deadline: Friday, June 22.
 
 

Boundary. Closed sets, open sets. Compact sets; compactness in terms of sequences. Integration. Curves.

Assignment: Read Section 2.5, subsections ABCDE. On your own, do problems 1-9.
 
 

Taking derivatives. A complex integral as a combination of real-valued integrals. Interpretation of a complex integral of a conjugate-holomorphic function as dealing with an irrotational, incompressible fluid flow vector field.

Assignment: Read Section 2.5, subsections FGHIJ. On your own, do problems 10, 11, 12, 13, 17.

For credit, you may write up any two of the problems in Section 2.5. The two will count as one. Deadline: Friday, June 22. Note: Wednesday 6/20 is the deadline fur turning in problems from Chapter 1.
 
 

Equivalence classes of curves. Paths. The equivalence of connectedness and pathwise connectedness for open sets. If a continuous function has an antiderivative on an open set, then its integrals are independent of path on that set.
 
 

Morera's Theorem: If a continuous function on a connected open set has integrals independent of path, then it has an antiderivative on that set. Goursat's Lemma: A function that is holomorphic on an open set has a zero integral over the boundary of every triangle contained in the set.

Assignment: I plan to distribute copies of the code that generated Figure 2.5-1. Problems which may be turned in for credit: Use Mathematica or other software to produce a pair of Polya diagrams, like those in Figure 2.5-1, but for one or more of the functions u+iv given on page 28.
 
 

A summary of some topics from advanced calculus: Uniform continuity. A continuous function on a compact domain is uniformly continuous. Subuniform continuity, subuniform convergence. The uniform limit of continuous functions is continuous.

"Goursat's Lemma Slightly Improved." An important example of a holomorphic function (on a punctured plane) whose integrals are not independent of path. Winding numbers: techniques, examples.

Assignment: I'll distribute the code that generated Figure 2.2-3. That may help you if you do problem 27 in Section 2.3 and use Mathematica. (The picture that results from problem 27 appears in many undergraduate textbooks, and has meaning for several physical applications: fluid flow, magnetic fields, electric fields, and temperature distribution.) If you haven't done problems 34-36 in Section 2.1, you might should consider doing them. If you need a little help, feel free to come see me. Note: In problem 34, one must exclude the case when z is on the line through B perpendicular to the given line.
 
 

Infinite series. The geometric series. Power series in general. The Cauchy-Hadamard Theorem.

Starting with a power series that converges to a sum f(z) for z in the disk of convergence. We define the derived series in a formal way. We prove that its radius of convergence is the same as that of the original series, and that it converges to the derivative f'(z). Then we can show that the power series representation for f is unique; it must be the Taylor series for f.

Assignment: Read Section 2.6 carefully; it's designed to cover just what's needed and no more. On your own, do problems 2, 5, 8, 9, 17, 18.

To turn in for credit, you may do any five of these: 1, 4, 6, 7, 9, 10, 11, 14, 15, 19*, 20*, 21, 22, 25*, 26*, 27*. The ones marked with an asterisk will count double. Please turn in one or two problems per day, no more. DEADLINE FOR ALL CHAPTER 2 PROBLEMS: WEDNESDAY, JUNE 27.
 
 

Cauchy's Theorem for a Convex Domain. The Power Series Representation.

Assignment: Read Section 3.1. On your own, do problems 1, 2, 5, 6.

For credit, you may do any three of these: 4, 7*, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24. The one marked with an asterisk will count double. Please turn in one or two problems per day, no more. DEADLINE FOR PROBLEMS IN SECTION 3.1: THURSDAY, JUNE 28.
 
 

Local independence of path implies holomorphy. Holomorphy except at one point plus continuity at the point implies holomorphy everywhere. Another uniqueness theorem for the power series representation. Liouville's Theorem.

Assignment: Today I'll distribute a couple of old Math 7350 final exams. The problems marked with an asterisk may be turned in for credit at any time, with the last day of class as the deadline. The idea will is to read and understand the problems, and then watch for when we cover the results and methods that will allow you to solve them.
 
 

The Fundamental Theorem of Algebra. The range of a non-constant entire function is dense in the plane. Subuniform convergence preserves holomorphy. Preparation for Cauchy's Theorem.

Assignment: Read Section 3.2.
 
 

Cauchy's Theorem.

Assignment: In Section 3.2, for double credit, you may do any one of the Exercises 5, 6, and 7 on page 211. (The hint for Exercise 7 is mis-stated; there's now a correction in the Errata file.) Deadline: Monday, July 2.
 
 

Simply connected sets, logarithms, and roots. Subuniform boundedness, subuniform convergence.
 
 

Laurent series representations. The classification of isolated singularities.

Assignment: In Section 3.3, on your own, do Exercises 3, 5, 7, 9, 17, 31, 32.

In Section 3.3, for credit, you may do any one of Exercises 1, 2, 4, 6, 8, 10; and/or any one of Exercises 11, 12, 14, 16, 18, 20; and/or any one of Exercises 21, 22, 24, 25, 26, 27, 28, 29, 30; and/or any one of Exercises 30-40. Deadline: Thursday, July 5.
 
 

If the function is bounded near an isolated singularity, then the singularity is removable. The Casorati-Weierstrass Theorem. Residues. Examples, classifying isolated singularities and evaluating residues.
 
 

Singularities at infinity. Now that we understand isolated singularities, we can see the Residue Theorem as a staightforward consequence of Cauchy's Theorem. I'll discuss the Residue Calculus methods of Section 4.1, in which the integral of a rational function in the sine and cosine is recognized as the pullback of a complex integral around the circle, and the Residue Theorem is applied to find the value of the integral.

Assignment: Read Section 3.4, subsections A and B. Read Section 4.1. On your own, do Exercises 1 and 3 in Section 4.1.

To turn in for credit, you may do any one of these Exercises in Section 4.1: 4, 5*, 7*, 8*, 9*, 10*, 11*. The ones with an asterisk will count double. Deadline: July 17.
 
 

Classifying an isolated singularity and evaluating a residue at infinity. A meromorphic function on the extended plane is rational. The valence function. The valence function for every rational function is constant. Applying the Residue Theorem to the singularities and residues of a function's logarithmic derivative will lead to the Argument Principle (a counting integral that will compute values of a valence function).

Read Section 3.4, subsections C and D. On your own, do Exercises 1, 8, 10, 11, 12, and 15. To turn in for credit, you may do any five of the other Exercises in Section 3.4. Deadline: July 17.
 
 

The Fresnel integrals. The Argument Principle.

Assignment: Read Sections 4.2, 4.3. On your own, do Exercise 1 in Section 4.2 and Exercise 9 in Section 4.3.

To turn in for credit, you may do any two of the Exercises 2-6 in Section 4.2; and in Section 4.3, any one of 1-8. Deadline: July 18.
 
 

Examples. Rouche's Theorem. Examples.
 
 

Mapping properties. How a holomorphic function acts locally.

Assignment: Read Section 3.5. On your own, do Exercises 1, 3, 4.

To turn in for credit, you may do any one of the Exercises 2, 5*, 6. Exercise 5 counts double. Deadline: July 19.
 
 

More on local action. Invertibility, and the derivative of the inverse. Schwartz's Lemma. Hurwitz's Theorem.
 
 

I'll discuss the conformal mapping method in general and talk about some of the Examples in Section 5.1.

In particular, I'll discuss boundary value problems in which the domain is the upper half-plane, so that the boundary is the real axis; and in which the boundary-value function takes on a finite number of values, each on an interval of the real axis. The method, the linear combination of arguments, is "easy" in the sense described above, at least after it's once pointed out to you. I'll talk about the Example illustrated in Figure 5.1-2 and show you the Mathematica commands used to draw the graphic on the right. Rather than just produce expressions for a solution, take the time to understand what the solution means physically in, for example, a heat flow problem. In this problem, for example: What's the limit of the temperature along each vertical half-line? What's the limit of the temperature along each horizontal half-line to the left? --to the right? Observe that the 50-degree isotherm is the only one that meets the x-axis at a point other than one of the two insulated points. Observe the behavior of the isotherms for temperatures greater than/less than 50 degrees; draw, for example, the 49-degree isotherm.

Assignment: Read Section 5.1.

To turn in for credit, you may do any two of the Exercises 3, 5, 7, 9, 11. Deadline: July 19.
 
 

Now it's clear that a larger collection of conformal mappings would be useful. It is quite worthwhile to become an expert on Mobius maps. They are the only mappings which are conformal and one-to-one from the extended plane to itself. To become familiar with them, learn the logic of their basic properties. For example: Mobius maps preserve Circles. A Mobius map, when its values are known at three points, is completely determined.

Assignment: Read Section 5.2. It is highly recommended that you do several of the Exercises in this Section.

To turn in for credit, you may do any four of the Exercises 10-24 in Section 5.2. If you like, read Section 3.6 on the Riemann Sphere, and do any one of the Exercises in that Section for double credit. Deadline: July 20.
 
 

How to analyze a Mobius map using its fixed points. The Mobius maps of Example 5.2.18 are conformal equivalences of the unit disk with itself. Figure 5.2-4 shows a typical case. They have applications, for example, in electric-field problems, as in Examples 5.3.2 and 5.3.3.

For triple credit, present a complete treatment of Exercises 34-37 in Section 2.1, and show how those results can be used to prove 5.2.21. Deadline: July 23.
 
 

One more Mobius map example. The symmetry principle for Mobius maps. The Joukowski map and an application to the flow of a perfect fluid (Example 5.4.2). How to get Mathematica to produce the picture on the right in Figure 1.1-9.

Assignment: Read Sections 5.3 and 5.4.

To turn in for double credit: You may do up to two of the Exercises in 5.3, and up to two in 5.4. Deadline: July 23.
 
 

A summary: Harmonic functions; the Poisson integral; the mean value property as a characterization of harmonic functions; and the reflection principle.

Assignment: Read Section 5.5.

To turn in for double credit, you may do up to three of the Exercises in Section 5.5. Deadline: July 23.
 
 

Proof of the Reflection Principle in a simple case. The Riemann Mapping Theorem.

Assignment: Understand the statement and proof of the Reflection Principle as given in class. Understand the statement of the Riemann Mapping Theorem. No problems will be accepted after class on Monday the 23rd. Anyone who really needs more problems to do should come see me.
 
 

Proof of the Riemann Mapping Theorem. Note: The Final Exam will be closed-book, closed-notes, and will be held 7:30 to 9:30 a.m. on Wednesday the 25th.
 
 

Summaries of selected further results: The uniqueness question for the function produced by the Riemann Mapping Theorem. The Osgood-Taylor-Caratheodory Theorem. How the O-T-C Theorem and the Reflection Theorem allow one to derive Schwarz-Christoffel Formulas.