# Math 43 (Spring 2012): Functions of a complex variable, Syllabus

Update (4/27/2012): By this point in the course, we are not really following the syllabus below very closely. The notes listed do not have that much resemblance to the corresponding descriptions anymore, so all the notes are listed at the bottom of this page with more descriptive naems.

The following tables lists the proposed schedule for the class. All chapter sections refer to Complex Analysis by Stein and Shakarchi.

Week Date Chapter Topic Notes
1 3/26 1.1 Introduction Class 1
3/28 1.1 Arithmetic with complex numbers Class 2
3/30 1.1 Topology and limits of complex numbers. Class 3
2 4/2 Real analysis/calculus review: differentiability classes of functions, derivatives as linear transformations Classes 4, 5, 6, 7
4/4 1.2 Examples of complex-valued functions
4/6 1.2 The derivative of a complex function, the Cauchy-Riemann equations
3 4/9 1.2 The Cauchy-Riemann equations and related ideas
4/11 1.3 Power series Class 8, 9
4/13 1.3 Contour integration Class 10, 11
4 4/16 2.1 Goursat's Theorem Class 12, 13, 14
4/18 2.2 Cauchy's Theorem Class 15
4/20 2.4 The Cauchy Integral Formula
5 4/23 2.3, 2.4 Consequences of the Cauchy Integral Formula Class 16, 17, 18, 19
4/25 2.4, 2.5.1 More consequences of the Cauchy Integral Formula
4/27 2.5.1, 2.5.2, 2.5.4 Even more consequences of the Cauchy Integral Formula (Morera's Theorem, the Schwarz Reflection Principle) Class 20, 21, 22, 23
6 4/30 3.1 Zeros and singularities Class 24, 25
5/2 3.2 The Residue Formula
5/4 3.2 Applications of the Residue Formula
7 5/7 3.2, 3.3 More applications of the Residue Formula
5/9 3.3 Types of singularities: removable, poles, essential
5/11 3.3 More on singularities, the extended complex plane
8 5/14 3.4 The argument principle, Rouche's Theorem
5/16 3.4 Applications of the argument principle: the open mapping theorem, maximum modulus principle, and a quantitative fundamental theorem of algebra
5/18 3.5 Topology: simply connected domains and homotopy
9 5/21 3.5 Holomorphic functions in simply connected domains, the complex logarithm
5/23 5.1 Jensen's formula, density of zeros of entire functions
5/25 5.2 Functions of finite order
10 5/28 5.3, 5.4 Infinite products
5/30 5.4, 5.5 The Hadamard Factorization Theorem