Math 322 – Complex Analysis, Spring 2019

Instructor: Anatolii Grinshpan
Office hours: Tue and Thu 11-2, Korman 207.

Course information    Academic calendar   

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).

Reading: Chapter 7.

 

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).