School of Mathematics
Course 414 -
Complex Analysis 1999-2000 (Optional JS & SS Mathematics, SS Two-subject Moderatorship
)
Lecturer: Richard M. Timoney
Requirements/prerequisites: 221
Duration: 21 weeks.
Number of lectures per week: 3
Assessment: Regular assignments.
End-of-year Examination: One 3-hour examination
Description:
- 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.
- Logarithms,
simple connectedness and antiderivatives.
- Identity theorem for analytic functions, maximum modulus theorem.
- Open mapping theorem, argument principle, inverses of
analytic functions and Rouché's theorem. Removable
singularities, Casorati-Weierstrass theorem, Residue
theorem.
- Metric space structures on H(G) and C(G).
Boundedness and compactness in H(G); normal families.
- Continuous linear operators and dual spaces.
Hahn-Banach theorem (without proof) and applications.
Runge's theorem.
- Hurwitz's theorem, the Schwarz lemma, the Riemann mapping theorem.
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).
May 7, 1999
File translated from TEX by tth 1.46