Course 414 - Complex Analysis

Lecturer: Richard M. Timoney

Date: 1995-96

Groups: Optional JS and SS Mathematics, SS Two-subject Moderatorship

Prerequisites: 221

Duration: 21 weeks.

Lectures per week: 3

Assessment: Regular assignments.

Examinations: One 3-hour examination

  1. Review of the definition of analytic functions, the Cauchy-Riemann equations, differentiation of power series, and contour integrals. Various forms of Cauchy's theorem and the Cauchy integral formula; winding numbers and homotopy.
  2. Logarithms, simple connectedness and antiderivatives.
  3. Identity theorem for analytic functions, maximum modulus theorem.
  4. Open mapping theorem, argument principle, inverses of analytic functions and Rouché's theorem. Removable singularities, Casorati-Weierstrass theorem, Residue theorem.
  5. Metric space structures on H(G) and C(G). Boundedness and compactness in H(G); normal families.
  6. Continuous linear operators and dual spaces. Hahn-Banach theorem (without proof) and applications. Runge's theorem.
  7. Hurwitz's theorem, the Schwarz lemma, the Riemann mapping theorem.
  8. [If time allows] Infinite products, interpolation theorems. H(G) as an algebra.

Objectives: This course will build on material covered in 221. Initially it will cover some familiar material in greater detail and then continue on to cover basic material in complex analysis. Some functional analytic techniques will be developed and applied to prove results in complex analysis.

Textbooks:

[1] John B. Conway, Functions of One Complex Variable, Springer-Verlag Graduate Texts in Mathematics.

[2] D.H. Luecking and L.A. Rubel, Complex Analysis - a Functional Analysis Approach, Springer-Verlag Universitext (1984).