
Instructor: Anatolii Grinshpan
Office hours: Mon 24 and Wed 23,
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. CauchyRiemann 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 14. 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. CauchyHadamard 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 57.
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. CasoratiWeierstrass theorem. Residue theorem. Conclusion of the course.
Reading: Chapter 8.
June 13. Final exam: 13PM, Curtis 456.