Lengths and the estimation lemma. Chapter 2: Exercise 10, Problem 4. And there are specialist texts on individual topics which also cover much more than we shall: some of these will be mentioned in the course as we go along. Furthermore, since u is bounded by hypothesis and Pu is bounded by construction on D 0;r , we have that g is bounded on D 0;r. Properties of analytic and harmonic functions This is the first half of a year-long course which forms the basis for the Ph. I shall also try to sort out some interesting web references, for example relevant articles in such as and. The claim follows immediately from the M-test and completeness.
If there is not a student volunteer I will do the problem. For a point of view based in formal and convergent power series convenient for locally computing composition inverses and solutions of differential equations you can consult Henri Cartan, ; Addison-Wesley Serge Lang, ; Springer. Course description: This will be a standard first year graduate class in complex analysis and it will prepare students for the complex analysis half of the analysis prelim. Automorphisms of the Riemann sphere Moebius transformations. It will be replica of the complex analysis half of the analysis prelim. To learn more, see our.
Chapter 6: Exercises 1, 3, 5, 7, 10, 12, 15, 17. Conformal maps between subsets of the plane Riemann Mapping Theorem. Shakarchi Other good books are: Ahlfors's Complex Analysis, and Conway's Functions of one complex variable. A standard source of information of functions of one complex variable, this text has retained its wide popularity in this field by being consistently rigorous without becoming needlessly concerned with advanced or overspecialized material. This is a corrected version.
Chapter 3: Exercises 18, 21, 22. Prerequisites: Math 4111, 4171 and 4181, or permission of instructor. Green and Krantz postpone analytic continuation to Chapter 10, which is something we do not want to do. First, there are many other excellent standard texts, including John B. We want to get rid of these assumptions.
A detailed syllabus will appear bit by bit as the course progresses. McMullen has a discussion on the Uniformization Theorem. Holomorphic maps of the unit disc Schwarz's Lemma. Make sure you can do the problems I don't assign. The hope is that this makes taking notes optional. Solutions will also be posted and will include students' work.
For example, jaguar speed -car Search for an exact match Put a word or phrase inside quotes. While the author presented the solution in p. The proof of the preceding lemma where M depends only on z0. Homework assignments will appear on this page approximately every week. Homework and examination grades will be regularly updated on.
Grades: Your grade will be determined by your homework scores and the final. Consequences of Cauchy's Theorem: 10. Do ten from the following problems. Exercises 1, 2, 4, 5. Chapter 5: Exercises 3,5, and Problem 1. Contents of the course: Complex numbers, holomorphic functions, Cauchy-Riemann equations, power series, complex integration.
Some of the material can also be found in Hyperbolic Geometry from a local viewpoint, L. Priestley, 'Introduction to Complex Analysis' Oxford University Press 1990 John M. At the end of week 11 we had completed Section V. Application of Residue Theorem to integrals Whittaker and Watson chapter 6. For most of the course we will probably follow the text but in some places I will take a different approach. Roughly speaking if I think you have a good chance of passing the prelim you will get an A. The exponential and logarithmic functions.
Since the annulus is bounded, f has finitely many zeroes in the region. Let f z be an entire function of genus h. Then n converges uniformly on Proof. Finally, Harvey Cohn, ; Dover leads with lots of beautiful pictures and physical intuition into Riemann surfaces and complex algebraic geometry. Chapter 8: Exercises, 20, 24 a,b. There is a short section on the Riemann zeta function, showing the use of residues in a more exciting situation than in the computation of definite integrals. Complex iteration: Julia sets and the Mandelbrot set.
The statements about conformality and continuity follow from a general theorem about the group of linear fractional transformations of the Riemann sphere Ahlfors p. Grading Your grade will be based on several homework assignments 30% , one Midterm 30% and a Final exam 40%. I expect to cover the material in Ahlfors in the first semester. Provide details and share your research! His office is in 381-B, and he will hold office hours on Wednesdays 1-3. Both exams are in Rm.