Problem Sheets in PDF: Sheet 1 Sheet 2

Solutions to the problems: Solutions to Sheet 1 Solutions to Sheet 2

Conformal 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. Power series expansion of holomoprhic functions. 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 maping theorem.

Spaces of holomorphic functions. Seminorms and relation with compact convergence. Montel's theorem. Biholomphic maps between open sets. The Riemann mapping theorem (see the last page in the Thorsten Wedhorn's Lecture Notes).

- 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

See also Polya's famous book

How to give a Mini-talk in Mathematics

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)

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 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.