School of Mathematics
Course 414 -
Complex Analysis 1999-2000 (Optional JS & SS Mathematics, SS Two-subject Moderatorship
Lecturer: Branka Pavlovi\'c
Duration: 21 weeks.
Number of lectures per week: 3
Assessment: Regular assignments.
End-of-year Examination: One 3-hour examination
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.
- 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.
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
- Metric space structures on H(G) and C(G).
Boundedness and compactness in H(G).
- Normal families; metric space structure of M(G).
- Continuous linear operators and dual spaces.
Hahn-Banach theorem (without proof) and applications.
- Hurwitz's theorem, the Schwarz lemma, the Riemann mapping theorem.
Oct 10, 1999
L. V. Ahlfors, Complex Analysis, Third Edition, McGraw-Hill, New
John B. Conway, Functions of One Complex Variable, Second
Edition, Graduate Texts in Mathematics 11, Springer-Verlag, New
Reinhold Remmert, Theory of Complex Functions, Graduate Texts
in Mathematics 122, Springer-Verlag, New York, 1991.
W. Rudin, Real and Complex Analysis, Second
Edition, McGraw-Hill, New York, 1974.