School of Mathematics, Trinity College

MAU34205 - Topics in Complex Analysis I 2019

JS & SS Mathematics , SS Two-subject Moderatorship

Lecturer Dmitri Zaitsev

Annual Examinations: The format will be the same as in the year 2017/2018 exam, which can be considered a sample paper. Credit will be given for the best 3 questions out of total 4 questions.

Problem Sheets in PDF: Sheet 1 Sheet 2

Most solutions to the exercises are similar to ones from previous years, see the links below.

Course outline:

Conformal mappings (definition, examples, relation between conformal and holomorphic mappings).

Complex integration of holomorphic functions along arbitrary continuous paths. Homotopy version of Cauchy's theorem. Elements of homology and homological version of Cauchy's theorem.

Cauchy-Green's Formula and relation to Cauchy's Integral formula. Theorem of Morera. Liouville's theorem. Application to the Fundamental Theorem of Algebra. Compact convergence and Weierstrass theorem.

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

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

Montel's theorem. Biholomphic maps between open sets. The Riemann mapping theorem (see the last page in the Thorsten Wedhorn's Lecture Notes).

Textbooks (some books are available online, just copy-paste and search):

- Ahlfors, L.V., Complex Analysis, Third Edition, McGraw-Hill, New York, 1978. (scanned copy in pdf)
- Conway, J.B., Functions of One Complex Variable, Second Edition, Graduate Texts in Mathematics 11, Springer-Verlag, New York, 1978. In Google Books
- Remmert, R., Theory of Complex Functions, Graduate Texts in Mathematics 122, Springer-Verlag, New York, 1991. In Google Books
- Remmert, R., Classical topics in complex function theory. Translated from the German by Leslie Kay. Graduate Texts in Mathematics, 172. Springer-Verlag, New York, 1998. In Google Books
- Churchill, R.V., Brown, J.W., Complex Variables and Applications, Fourth edition. McGraw-Hill Book Co., New York, 1984.
- Palka, P.B., An Introduction to Complex Function Theory, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1991.

The following book of Fulton is quite remarkable in that it includes many topological and homological aspects of Complex Analysis on a deeper but still elementary level:
- Fulton, W., Algebraic topology. A first course. Graduate Texts in Mathematics, 153. Springer-Verlag, New York, 1995. In Google Books

Problem book: George Polya and Gábor Szegő, "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

Mini-Talks topics:

My blog post How to give a Mini-talk in Mathematics
All topics around Complex Analysis at large are welcome, e.g. please have a look at the Advanced Complex Analysis by C. McMullen, where I had highlighted several topics of interest, pick few topics and try to search web and/or textbooks for it, then let me know the topic you liked and I will provide more sources if needed.

Various Lecture Notes:

NEW: Collaboration Repository on Gitlab -- please join and help us create great and useful lecture notes together!

Course 214 - Complex Variable by David Wilkins
Complex Analysis Lecture Notes (includes material on Homotopy and Homology) by Torsten Wedhorn
Advanced Complex Analysis with Course Notes by Curtis T McMullen (Fields Medal 1998)

Miscelaneous Links:

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
Free Lecture Notes in Complex Analysis on the Github Awesome Math list
Differentiating power series and An experiment concerning mathematical writing by Timothy Gowers
Ask yourself dumb questions – and answer them! by Terence Tao
A Virtual Mathematics Library by Lee Stemkoski

Old Complex Analysis course web pages.
Old 3423 web page for 2017 with Problem Sheets and Solutions
Old 3423 web page for 2015 with Problem Sheets
Old 3423/4 web page for 2013-14 with Problem Sheets
Old 3423/4 web page for 2011-12 with Problem Sheets
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.