Registration for Spring term opens Thursday, February 23, 2016.
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The course will be an introduction to complex analysis. The following topics will be covered: Holomorphic functions and the Cauchy-Riemann equations, Cauchy's theorem and Cauchy's integral formula, Taylor expansions, entire functions and Liouville's theorem, zeros of holomorphic functions, isolated singularities and Laurent expansions, meromorphic functions, the Residue Theorem, the Maximum Modulus Principle, conformal mappings and linear fractional transformations, harmonic functions, infinite products and the Weierstrass Factorization Theorem, the Gamma function, and the prime number theorem.
Based on homework (70%), and final exam (30%).
Collaboration for solving the homework problems is allowed, however the students should write up the solutions individually. The homework sets should be handed in with a cover sheet that only has the student's first and last names on it, due to privacy policies.
Elias M. Stein and Rami Shakarchi, Complex Analysis (Princeton Lecture Series in Analysis II), ISBN-13: 978-0-691-11385-2.
|Mon April 3rd||Holomorphic functions, Cauchy-Riemann equations|
|Wed April 5th||Holomorphicity from the Cauchy-Riemann equations, power series|
|Fri April 7th||Radius of convergence, complex differentiabilty of power series, smooth curves|
|Monday April 10th||Integral of complex functions over curves, primitives, integral of functions with perimitives|
|Wed April 12th||Examples of integrals of curves, logarithm of complex numbers and its branches, statmenet of Goursat's theorem|
|Fri April 14th||Proof of Goursat's theorem, idea of existence of primitives for holomorphic functions on a disk|
|Mon April 17th||Existence of primitives for holomorphic functions on the interior of toy contours, Cauchy's theorem|
|Wed April 19th||Evaluation of some integrals using Cauchy's theorem, statment of the Cauchy integral formula|
|Fri April 21st||Cauchy's integral formula, Cauchy inequalitites, Liouville's theorem, fundamental theorem of algebra|
|Mon April 24th|
|Wed April 26th|
|Fri April 28th|
|Mon May 1st|
|Wed May 3rd|
|Fri May 5th|
|Mon May 8th|
|Wed May 10th|
|Fri May 12th|
|Mon May 15th|
|Wed May 17th|
|Fri May 19th|
|Mon May 22nd|
|Wed May 24th|
|Fri May 26th|
|Mon May 29th|
|Wed May 31st|
|Fri June 2nd|
|Mon June 5th|
|Wed June 7th|
|Fri June 9th|
|4 p.m., Thursday April 13th||Chapter 1 of the textbook, exercises 7, 8, 9, 12, 16, 19|
|4 p.m., Thursday April 20th||Chpater 1: exercises 10, 11, 25; Chapter 2: exercise 5 and problem 1 (textbook)|
|4 p.m., Thursday April 27th||Chapter 2: exercises 1, 2, 6, 7, 11, 12|