I recommend a video made a couple of years ago by Etienne Ghys, of the Ecole Normale Superieure de Lyon. I will place one copy of each of these books on reserve at the Olin Library. This is strongly encouraged but not mandatory nor is attendence to the problem session mandatory. Solutions will also be posted and will include students' work. This will involve us first covering some of the same ground again for example Cauchy's Theorem , but with more precise statements and detailed proofs. Problems 1 a , 2.
Clearly, ζN is the sum of holomorphic functions on the region Ω. First, there are many other excellent standard texts, including John B. Open sets and closed sets. Holomorphic maps of the unit disc Schwarz's Lemma. I will not accept late assignments. Chapter 6: Exercises 1, 3, 5, 7, 10, 12, 15, 17. Course began in the week beginning 10th January 2011 and finishes in the week ending 1st April 2011.
We proved in Problem Set 1 Ahlfors Section 4. For example, jaguar speed -car Search for an exact match Put a word or phrase inside quotes. Green and Krantz postpone analytic continuation to Chapter 10, which is something we do not want to do. Assessment: 90% final examination May 2011 and 10% based on weekly exercise sheets. Late homework will not be accepted. I will drop the lowest two grades you receive on homework.
Grades: Your grade will be determined by your homework scores and the final. Although the book is quite expensive you can almost surely find a much cheaper used copy on Amazon or abe. Evaluating real integrals and sums of series using the Residue Theorem. Shakarchi Other good books are: Ahlfors's Complex Analysis, and Conway's Functions of one complex variable. The official course description on is quite flexible as regards the contents of the second half of the course, and I would like to take your interests into account in deciding exactly what we cover.
According to Ahlfors, the numbers Bk are called Bernoulli numbers, the values of which one can look up. Conformal maps between subsets of the plane Riemann Mapping Theorem. The main novel feature of the course will be that we will use 2-dimensional hyperbolic geometry to prove some of the standard compactness results for holomorphic maps. This both assumed Green's theorem and the Jordan Curve Theorem. McMullen has a discussion on the Uniformization Theorem. Application of Residue Theorem to integrals Whittaker and Watson chapter 6.
Class Schedule: Lectures are on Monday, Wednesday and Friday from 2-3 in Cupples I Rm. Provide details and share your research! We shall encounter many famous results Picard's Theorem, the Riemann Mapping Theorem, the Prime Number Theorem,. Math grad students must do option b. 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. A more detailed version of this objective is available on the School of Mathematical Sciences webpage. If there is not a student volunteer I will do the problem.
Homework is accepted until the solutions have been posted. Mean Value Theorem Maximum Modulus Corollary 1. The Zeta function: Riemann's Hypothesis. Some subset of each assignment will be graded. Exercises 2, 5, 7, 8, 13. Roughly speaking if I think you have a good chance of passing the prelim you will get an A.
This says that if integrals around closed loops are zero, then you are analytic it is a converse of Cauchy's Theorem. The student who prepared a particular problem will have the opportunity to present their problem at the board. Learning Outcomes A rigorous understanding of classical complex analysis. Priestley, 'Introduction to Complex Analysis' Oxford University Press 1990 John M. Homework: There will be weekly homework assignments.
Schwarz reflection principle, zeros, poles and residues, Calculus of residues, applications to definite integrals. It seems to me that this is related to the Riemann surface defined by the square root. Chapter 6: Problems 2 and 3. The exponential and logarithmic functions. 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. Contents of the course: Complex numbers, holomorphic functions, Cauchy-Riemann equations, power series, complex integration.
Finally, Harvey Cohn, ; Dover leads with lots of beautiful pictures and physical intuition into Riemann surfaces and complex algebraic geometry. Chapter 3: Exercises 1, 14, 15, 16, 17. The list of headings below is from the 2009-2010 course, and may change a little in 2010-11. This is a corrected version. While the author presented the solution in p.