Theory of functions of a complex variable (M361): Fall 2012

T I M P E R U T Z
	Theory of functions of a complex variable (M361) University of Texas at Austin, Fall semester, 2012 Course number: M 361. Unique identifier: 56200 Tuesday, Thursday, 2 - 3:30 pm, RLM 6.118 Instructor: Timothy Perutz (Assistant Professor) Office hours: Tuesday, Wednesday 4:30-5:30 p.m., RLM 10.136. Email: perutz AT math DOT utexas DOT edu Textbook: *Jerrold E. Marsden and Michael J. Hoffman, Basic Complex Analysis,* 3rd edition.** Freeman, New York, 1999. Available from the University Co-Op bookstore. Here's a Google Books preview. I'm happy to answer questions about this course by email or in person (RLM 10.136).
	Aims The main aims of the course are (i) to learn to work algebraically with complex number system; (ii) to learn the main results of complex analysis, both in their theoretic development and in examples; (iii) to learn to apply contour integration to problems concerning real numbers. The theoretical development, when told without shortcuts, is an unfolding narrative. I hope to tell much of this story, though I shall place less emphasis on proofs requiring skills developed in real analysis courses.
	Syllabus We shall cover material from chapters 1-4 from Hoffman-Marsden. A syllabus and (approximate!) schedule is available in PDF format here.
	Assessment Homework (15%). There will be homework assignments each week except when there is a test. The two lowest homework grades are dropped. Homework is due at the beginning of class on Thursdays, every week except when there is a test. Late work will not be accepted. You may discuss homework problems with others, but you should write out your solutions alone. Final exam (45%). Friday, December 14, 9:00 - 12:00 noon. There won't be an alternative date, so check this one now! Write in your calendar. Two midterm exams. (20% each) In class, Tuesday Oct 2, Thursday Nov 8. Put the dates in your calendar.
	Further class policies Grading. I will convert your weighted average to a (plus/minus) letter grade at the end of the semester. The conversion from numerical to letter grades will not be based on a pre-determined scale. However, to get an A, you will have to demonstrate fluency in most of the topics of the course. Missed tests. In case of a test missed because of illness or a serious family emergency, confirmed by a letter from an appropriate authority, I will shift the weight from the missed test to the final exam. Disabilities. Students with disabilities may request appropriate academic accommodations from the Division of Diversity and Community Engagement, Services for Students with Disabilities, 512-471-6259. Please contact them as early as possible in the semester. Religious holidays. If your observance of religious holidays clashes with the schedule for classes, tests and homework deadlines, I will make reasonable allowances provided that you notify me early. Please tell me by the 12th class day that this is an issue, and additionally notify me at least 14 days before each specific class, homework deadline or test that you need to miss. Academic dishonesty will not be tolerated, and will result in a failing grade for this course. If you are feeling overwhelmed by the demands on your time, or by the difficulty of the class, I will make time to talk it through with you. The wrong thing to do is to take a shortcut by cheating (for instance, plagiarizing work). In class I expect your attention. I expect you to write down what I write on the board. It's a good idea to make a note of things I say but don't write on the board, too. Texting, instant messaging, and catching up on sleep, are not acceptable in class; I may ask you to leave if you do these things.
	Additional resources The central theorem in complex function is Augustin-Louis Cauchy's theorem about integrals of holomorphic functions around closed contours. This result was first stated, with a brief argument that falls short of what by modern-day mathematicians deem a proof, in his Mémoire sur les intégrales définies prises entre des limites imaginaires of 1825. (Look at p. 5, where he observes that the integral of a holomorphic function along a path is independent of the path.) Terry Tao has several Java applets illustrating aspects of complex complex function theory. You will need to enable Java in your browser. Here they are: The complex plane. Complex maps. Moebius transformations. Multi-valued complex maps. The complex derivative. Complex integration. Taylor and Laurent series.
	Here's a picture of one of the most remarkable things in complex function theory, the Mandelbrot set. David Eck has some very nice Mandelbrot-zooming applets here (I especially recommend his MandelbrotOrbits applets).
	Notes I don't expect to type notes for the whole course, but when I do type up the lectures I'll make them available here. Please let me know if you find (or suspect) an error. Notes for lectures 1-3. Notes for lectures 4-5.
	Corrections log I'll make a note here of corrections to the web-page and class notes. Aug 30. In today's class, I made an error when I said that for 3-vectors x and y, the quaternion product of (0, x) and (0, y) is (0, x × y). I should have said (-x . y, x × y), where x . y is the dot product. Aug 31. Corrected numbering of homeworks on webpage. Aug 31. Corrected test-date on webpage from Thursday Oct 2 to Tuesday Oct 2. Sept 8. In HW2, additional question B, I've now made s positive. Sept. 16. In HW3, I've corrected errors in additional questions 1(b) and (c). Sept. 21. In the PDF file for HW3, in question 2, the definition of the symbol ε was not specified. As the class was notified by email, its intended definition was ε = \|z_N(c)\| - 2. Oct. 30. In HW8, I've amended question A (the antiderivative now has to be found locally, not globally). Also, I've deleted a not-so-useful hint about 2.4.12.
	Homework assignments Homeworks are due on Thursdays, at the beginning of class, every week except when there's a test that week. In all cases, you should show your working, and EXPLAIN your answer as clearly and precisely as you can. Typically you will need both words and symbols. I have asked the grader to give full credit only for correct answers, correctly explained. There will be deductions for correct answers with incomplete or incorrect explanations. What you write should be literally correct! (Reread it.) It's not enough if a generous reader can figure out what you meant to say if you didn't actually say it. Look out for "type errors" - for instance, an assertion that A equals B when A and B are different types of things. Also, if your answer is correct but laborious, when a standard method gives a much more efficient solution, you may lose some credit (but not much). HW1. Due at the beginning of class, Thursday September 6. From Marsden-Hoffman: section 1.1, questions 2, 6, 10, 12, 16. Section 1.2, questions 4, 14. Solutions now available on Blackboard. HW2. Due September 13. Section 1.2, questions 2, 6, 8 (give an explicit numerical answer), 10, 20. Additional question A: look at q.25 and then establish a similar formula for sinθ + sin 2θ + ... + sin nθ. Additional question B: Consider the set S of complex numbers whose distance from 1 is s times their distance from -1, where s is a positive real number. Show that, except for one particular value of s, the set S is a circle. Determine its center and radius. What is the exception, and what happens in that case? Section 1.3: q. 10, 14. HW3. Due September 20. Section 1.3: q. 2, 4, 6, 8, 22, 30. Additional questions on sequences and the Mandelbrot set: PDF file [note correction to 1(b) and (c)]. HW4. Due September 27. Section 1.4. Question 2(b), 8. Section 1.5, q. 2, 10, 14, 18(a,b,c), 22, 28, 32. In class, I sketched a proof of the chain rule for holomorphic functions, but I omitted the verification that the error term E(h) was o(h). Prove that this is so (i.e., prove that E(h)/h goes to zero as h goes to zero). HW5. Due October 11. Section 1.6, q. 2, 4, 10. Section 2.1, q. 2, 4, 10, 12. HW6. Due October 18. Work through your class notes on the proof of Cauchy's theorem carefully, making sure you understand everything. Also read sections 2.2. and 2.3 carefully. Section 2.3, q.1. Carefully state Green's theorem for the case of a rectangular domain. Prove it using the (real) fundamental theorem of calculus [Hint: double integrals can be done by integrating first with respect to x, then with respect to y; or vice versa. If you get stuck, try a calculus text.] Explain how to deduce Green's theorem for domains that can be tiled by rectangles. HW7. Due October 25. Section 2.3 q.7. [Hint: you can do parts (a), (b) and (c) using Cauchy's theorem, the deformation theorem, and the fundamental theorem of calculus; for (d), I suggest using Cauchy's integral formula, though perhaps the authors have something else in mind.] Section 2.4. q.2, 4, 8, 13, 18. Prove that for every polynomial p(z) of degree d>0, there are positive real constants r and c such that for all z of modulus >r one has \|p(z)\| > c \|z\|^d. (This is useful for the proof of the Fundamental Theorem of Algebra.) HW8. Due November 1. Section 2.4, q. 12, 20. Section 2.5, q. 2 (you'll need the maximum modulus theorem, to be covered in Tuesday's class). A. Suppose that f is a continuous complex-valued function on an open set U in the complex plane, with the following property: the integral of f along any triangular loop is zero. Prove that on every open disc D contained in U, f has an antiderivative; i.e., there is a holomorphic function F on D such that F'=f. (Cf. Thm. 2.1.9 and your class notes on Cauchy's theorem for star-domains.) Hence prove that f is holomorphic on U. (This is a form of Morera's theorem.) B. Use Morera's theorem to help you prove that if f is continuous on U and holomorphic except perhaps at a point z of U then it is in fact holomorphic on U. C. (a) Use path-integral methods to integrate the function 1/(1+x² + x⁴) from zero to infinity. (b) Do the same for the function 1/(1+x²)². HW9. Due ~~November 15~~ Tuesday November 20. Section 3.1, questions 4, 5, 14. Section 3.2, questions 6, 8, 13, 16, 18. HW10. Due November 29. A short homework. Warning: next week's, mainly about contour integration, will be long. Read the material in section 3.2 about the isolation of zeros. Question: Suppose that a holomorphic function f has vanishing derivatives of all orders at some point z₀ in its domain U. (i) Carefully prove that f is identically zero in some open disc D(z₀;r). (ii) Suppose that a_n is a sequence of points in U such that f(a_n) = 0 for all n. If a_n → a, show that f(a)=0. (iii) Suppose that z₁ is connected to z₀ by a path g : [0,1] → U. Use (i) and (ii) to show that f(z₁)=0. [Hint: if not, consider the smallest t such that f(g(t)) is non-zero.] HW11. Due December 6. Section 3.3, questions 2, 6, 11. Section 4.1, questions 2, 8. Section 4.3, questions 1, 2, 5, 14, 25. Section 4.4, questions 1, 4.
	Test information First midterm Syllabus Algebra of complex numbers (sections 1.1-1.2 of M-H): definition of complex numbers and their product; cartesian and polar forms; complex conjugates, and algebraic manipulations involving the conjugate and modulus-square; de Moivre's theorem; exponential form; computing roots. Standard functions of a complex variable (section 1.3): exponential, logarithm, sine, cosine, complex powers; solving equations involving these functions. Analysis: Sequences (as in class notes): boundedness; convergence to a finite or infinite limit (precise definitions). The triangle inequality and its corollaries. Functions (1.4): limits (precise definition), continuity. Other topics in 1.4 are not on the syllabus. Holomorphic [analytic] functions (1.5): definition; standard rules for complex derivatives. The Cauchy-Riemann equations (1.5): the equations and their derivation; vanishing of the partial derivative with respect to z; harmonic functions; computing harmonic conjugates. The most recent material, on sufficiency of real differentiability and the C-R equations for holomorphy, and on conformality, will not be tested this time. What to expect I will set questions modeled on homework questions from the first four homework sets. Homework solutions to sets 1-3 are available on Blackboard, and the same will be true for set 4 soon after the submission deadline, and you should review them carefully. Always show your working for calculations, and explain your reasoning for other points. Typically, I break the question into parts. There might be a part asking you for a precise definition, so one of your priorities should be to make sure that you can reproduce these definitions accurately - giving the general idea is not good enough! There may be another part asking you to state a theorem or relevant technique, and possibly to prove it (providing that the proof is short and straightforward). After that, you may be asked to solve the problem using the definitions and theorems/techniques that have been set out. It's this last part that will resemble a homework question. I will take what you write literally, so read it through and make sure it's literally correct. If you write an equals sign, I will take that as an assertion that the two sides are equal. Look out for type errors. For example, "My favorite color is Thursday" is a type error: it's grammatical, but Thursday belongs to the type "days of the week" instead of the expected type "colors", so the sentence doesn't make sense. Mathematical types include "function", "complex number", "sequence of complex numbers", and so on. A mathematical type error might be an assertion that some sequence of complex numbers is equal to a certain complex number: this doesn't make sense. I'm not going to be testing for the ability to solve problems at high speed. Moderate speed, arising from reasonable fluency in the material, should be sufficient. No reference materials or electronic devices are permitted.