School of Mathematics, Trinity College

Course MA3423/4 - Topics in Complex Analysis 2011-12 (click for more information)

JS & SS Mathematics, SS Two-subject Moderatorship

Lecturer Dmitri Zaitsev

The exam will have 2 sections with 4 questions each. Credit will be given for the best 3 questions from each section.

Problem Sheets in PDF: Sheet 1 Sheet 2 Sheet 3 Sheet 4

Course outline:

Real and complex differentiability. Holomorphic functions. Branches of multi-valued functions. Branches of logarithm and of the nth root. Conformal mappings.

Complex integration along piecewise smooth paths. Antiderivatives. Calculating integrals using antiderivatives. Elements of homology and homological version of Cauchy's theorem.

Cauchy-Green's Formula and relation to Cauchy's Integral formula. Power series expansion of holomoprhic functions. Radius and disk of convergence of power series. Cauchy-Hadamard formula. Theorem of Morera. Cauchy's estimates. Liouville's theorem. Application to the Fundamental Theorem of Algebra. Compact convergence and Weierstrass theorem.

Order of zeroes. The identity principle. Laurent series expansion in a ring. Isolated singularities. Removable singularities, poles, essential singularities. Riemann extension theorem. Meromorphic functions. Casorati-Weierstrass theorem.

The argument principle. Rouché's theorem. Open mapping theorem. The univalence theorem (local injectivity criterion). Inverse function theorem.

Spaces of holomorphic functions. Seminorms. Montel's theorem. Biholomphic maps between open sets. The Riemann mapping theorem.

Möbius transformations. Riemann sphere (extended complex plane). Stereographic projection. Rationality of meromorphic functions on the Riemann sphere. Automorphisms of the Riemann sphere and the complex plane. Schwarz Lemma. Automorphisms of the disk. Cayley transform. Automorphisms of the upper half-plane. Homogeneity of the Riemann sphere, complex plane and disk.

Schwarz Reflection Principle. Mittag-Leffler's theorem.

[1] L. V. Ahlfors, Complex Analysis, Third Edition, McGraw-Hill, New York, 1978.
[2] J. B. Conway, Functions of One Complex Variable, Second Edition, Graduate Texts in Mathematics 11, Springer-Verlag, New York, 1978.
[3] R. Remmert, Theory of Complex Functions, Graduate Texts in Mathematics 122, Springer-Verlag, New York, 1991.
[4] R. V. Churchill, J. W. Brown, Complex Variables and Applications, Fourth edition. McGraw-Hill Book Co., New York, 1984.
[5] B. P. Palka, An Introduction to Complex Function Theory, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1991.

Problem book: George Polya and Gabor Szego, "Problems and theorems in analysis: Series, integral calculus, theory of functions". In Google Books
See also Polya's famous book "How to Solve It". In Google Books

Some links.
Course 214 - Complex Variable by David Wilkins
Complex Analysis Project by John H. Mathews.
Graphics for Complex Analysis by Douglas N. Arnold.
A Complex Function Viewer by The University of British Columbia SunSITE.
Wolfram Mathworld Pages on Complex Analysis
Wikipedia Pages on Complex Analysis
Conformal Projections in Cartography by Carlos A. Furuti

Old Complex Analysis course web pages.
Old 3423/4 web page for 2009-10 with Problem Sheets
Old 414 web page for 2007-08 with Problem Sheets and Solutions
Old 414 web page for 2005-06 with Problem Sheets
Old 414 web page for 2003-04 by Richard M. Timoney with Lecture Notes and Problem Sheets.

For exam-related problems look in TCD past examination papers and Mathematics department examination papers.

I will appreciate any (also critical) suggestions that you may have for the course. Let me know your opinion, what can/should be improved, avoided etc. and I will do my best to follow them. Feel free to come and see me if and when you have a question about anything in this course. Or use the feedback form from where you can also send me anonymous messages.