# Math 322 – Complex Analysis, Spring 2018

Instructor: Anatolii Grinshpan
Office hours: Mon 2-3 and Wed 2-4, Learning Terrace.

Apr 2.  Introduction. The field of complex numbers. Polar form. Conjugation.

Apr 4.  Triangle inequality. De Moivre’s formula. Roots. Geometry in the complex plane.

Reading and exercises: 1.1, 1.1.1, 1.2. Homework 1.

Apr 9.   Subsets of the complex plane. Worksheet. Lines and circles. Mappings of the complex plane.

Apr 11. Examples of mappings. Worksheet. Exponential, hyperbolic, and trigonometric functions.

Reading and exercises: 1.3, 1.4, 1.5. Homework 2.

Apr 16. Logarithms, roots and powers. Branches of the argument. Examples.

Apr 18. Curves in the plane. Integration of curves. Line integrals.

Reading and exercises: 1.5, 1.6. Homework 3

Apr 23. Plane vector fields. Circulation and flux. Green's theorem. Green's theorem for rectangles.

Apr 25. Complex differentiability. Cauchy-Riemann equations. Physical interpretation.

Reading and exercises: 1.6, 2.1, 2.1.1. Homework 4.

Apr 30. Midterm 1 (Chapter 1).

May  2. Holomorphic functions. Conformality. Worksheet. Supplemental notes.

Reading and exercises: 2.1, 2.1.1. Homework 5.

May 7. Series of complex numbers. Power series. Ratio test. Disk of convergence.

May 9. Geometric series. Uniform convergence. Example. Cauchy-Hadamard formula.

Reading and exercises: 2.2. Homework 6.

May 14. Differentiation of power series. Abel’s theorems. Circle of convergence. Example. Multiplication and division.

May 16. Cauchy vs Green. Cauchy’s theorem for a triangle. Cauchy’s theorem for a convex region. Circular averages.

Reading and exercises: 2.3, 2.3.1. Homework 7.

May 21. Cauchy's formula for a circle. Local power series representation. Mean value property. Maximum principle.

May 23. Cauchy estimates. Identity principle. Morera's theorem. Liouville's theorem. The fundamental theorem of algebra.

Reading and exercises: 2.4. Homework 8.

May 28. Memorial day.

May 30. Midterm 2 (2.1-2.4).

June 4. Zeros of holomorphic functions. Isolated singularities. Casorati-Weierstrass theorem. Residues.

June 6. Definite integrals via Residue Theorem. Jordan’s lemma. Laurent expansions and isolated singularities.

Reading and exercises: 2.5, 2.6. Homework 9.

June 11. Questions session.

June 14. Final exam: 8-10 AM, Curtis 250A.