Math 322 – Complex Analysis, Spring 2017


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

Course information    Academic calendar    Tutoring center  

Apr  3.  Introduction. The field of complex numbers.  Conjugation.

Apr  5.  Geometry of complex numbers. Subsets of C. Stereographic projection, Riemann sphere.

Reading: Chapter 1. Homework 1.

 

Apr 10. Mappings of C. Complex differentiability. Cauchy-Riemann equations. Physical interpretation.

Apr 12. Conformality. Harmonic functions. Lines and circles. Möbius transformations.

Reading: Chapters 2, 3. Homework 2.

 

Apr 17. Möbius transformations: mapping properties. Example.

Apr 19. Elementary complex functions.

Reading: Chapters 3, 4. Homework 3.

 

Apr 24. Logarithms, roots, and powers. Notes.

Apr 26. Midterm1: Chapters 1-4. Optional homework.

 

May 1. Sequences and series. Sequences and series of functions. Uniform convergence.

May 3. Power series. Example. Disk of  convergence. Differentiation. Example.

Reading: Chapter 5. Homework 4.

 

May   8.  Properties of power series. Cauchy-Hadamard formula.

May 10.  Complex integration. Example.

Reading: Chapters 5, 6. Homework 5.

 

May 15. Interpretation of complex integration. Cauchy’s theorem for a triangle.

May 17. Cauchy’s theorem for a convex region. Cauchy’s formula for a circle. Cauchy integral.

Reading: Chapter 7. Homework 6.

 

May 22. Liouville’s theorem. Zeros. Identity principle. Maximum principle. Schwarz’s lemma.

May 24. Midterm 2: Chapters 5-7.

 

May 29. Memorial day.

May 31. Morera’s theorem. Local mapping. Automorphisms of the unit disk.

Reading: Chapter 7. Homework 7. Osserman’s paper.

 

June  5.  Cauchy’s formula for an annulus. Laurent series. Isolated singularities. Residues.

June  7.  Casorati-Weierstrass theorem. Residue theorem. Conclusion of the course.

Reading: Chapter 8.

 

June 13. Final exam: 1-3PM, Curtis 456.