School of Mathematics
School of Mathematics
Course 414 
Complex Analysis 200102 (Optional JS & SS Mathematics, SS Twosubject Moderatorship
)
Lecturer: Dr. Richard M. Timoney
Requirements/prerequisites: 221
Duration: 21 weeks.
Number of lectures per week: 3
Assessment: Regular assignments.
Endofyear Examination: One 3hour examination
Description:
This syllabus is provisional. A revised one is planned which may
deal with iteration of polynomial functions.

Review of the definition of analytic functions, the CauchyRiemann
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, CasoratiWeierstrass theorem, Residue
theorem.
 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.
HahnBanach theorem (without proof) and applications.
Runge's theorem.
 Hurwitz's theorem, the Schwarz lemma, the Riemann mapping theorem.
Further detailed information about the course will become available
via the web site for the course at http://www.maths.tcd.ie/~richardt/414
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]
L. V. Ahlfors, Complex Analysis, Third Edition, McGrawHill, New
York, 1978.
 [2]
John B. Conway, Functions of One Complex Variable, Second
Edition, Graduate Texts in Mathematics 11, SpringerVerlag, New
York, 1978.
 [3]
Reinhold Remmert, Theory of Complex Functions, Graduate Texts
in Mathematics 122, SpringerVerlag, New York, 1991.
 [4]
W. Rudin, Real and Complex Analysis, Second
Edition, McGrawHill, New York, 1974.
Oct 6, 2001
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