# Math 322 – Complex Analysis, Spring 2019 Instructor: Anatolii Grinshpan
Office hours: Tue and Thu 11-2, Korman 207.

Apr 1.  Introduction to the course. The field of complex numbers.

Apr 3.  Geometry and topology of the complex plane.  Worksheet. Lines and circles.

Reading: Chapter 1. Homework 1.

Apr 8.   Complex mappings.  Examples of mappings. Differentiability and Cauchy-Riemann equations.

Apr 10. Real and complex differentiability. Physical interpretation. Holomorphic functions.

Reading: Chapter 2. Homework 2.

Apr 15. Complex exponential. Curves in the plane. Conformal mapping. Worksheet. Notes.

Apr 17. Linear-fractional transformations. Worksheet. Stereographic projection.

Reading: Chapters 2 and 3. Homework 3.

Apr 22. Exponential, hyperbolic, and trigonometric functions. Example.

Apr 24. Midterm 1 (Chapters 1-3).

Apr 29. Branches of the argument. Logarithms, powers, and roots. Worksheet.

May 1.  Branch points. Series of complex numbers. Geometric series.

Reading: Chapters 4 and 5. Homework 4.

May 6. Uniform convergence. Power series. Ratio test. Disk of convergence. Example. Cauchy-Hadamard formula.

May 8. Operations with power series. Differentiation of power series. Example.

Reading: Chapter 5. Homework 5.

May 13. Abel’s theorem. Integration of curves.

May 15. Line integrals. Plane vector fields: circulation and flux. Green's theorem for rectangles.

Reading: Chapter 6. Homework 6.

May 20. Cauchy’s theorem for a triangle. Cauchy’s theorem for a convex region. Circular averages.

May 22. Midterm 2 (Chapters 4-6).

May 27. Memorial day.

May 29. Cauchy’s formula for a circle. Local power series representation. Consequences of Cauchy’s formula.

Reading: Chapter 7. Homework 7.

June 3. Zeros of holomorphic functions. Identity principle. Isolated singularities.

June 5. Isolated singularities and Laurent expansions. Residue theorem.

Reading: Chapters 7 and 8.

June 11. Office hours, 11AM (Korman 228).

June 12. Final exam: 1-3PM, Curtis 453 (all lectures). 