JS & SS Mathematics, SS Two-subject Moderatorship
The format will be the same as in the year 2015/2016 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:
Solutions to the problems:
Solutions to Sheet 1
Solutions to Sheet 2
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.
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).
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
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.
How to give a Mini-talk in Mathematics
Various Lecture Notes:
Course 214 - Complex Variable by David Wilkins
Complex Analysis Lecture Notes (includes material on Homotopy and Homology) by
Advanced Complex Analysis
with Course Notes
by Curtis T McMullen (Fields Medal 1998)
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 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
from where you can also send me anonymous messages.