Kompleks Analyse

KomAn (Complex Analysis) Blok 3A Spring 2010

Quick access to Homepage for studies in mathemtics

The principal mathematician in the course

Augustin-Louis Cauchy (1789-1857)

If you want to know more about Cauchy or any other mathematician, then look at MacTutor History of Mathematics Archive

The responsible for the course is Christian Berg . < berg@math.ku.dk> Information concerning the course will be given from this homepage. The course and exam will be given in English.

Officiel description of the course

Schedule

Lectures: Tuesday 10.15-12.00. Thursday 9.15-10.00 and 13.15-15.00 in Auditorium 4.
First lecture on February 2. Lecturer: Christian Berg

Exercices: First exercise class on February 2.
Class 1: Tuesday 08.15-10.00, Thursday 10.15-12.00 in N037(DIKU). Teaching Assistant: Andreas N. Aaserud < andreas.naes.aaserud@gmail.com>
Class 2: Tuesday 08.15-10.00, Thursday 10.15-12.00 in N004(DIKU). Teaching Assistant: Flemming B. von Essen < von.Essen@math.ku.dk>

Teaching material

Christian Berg: Complex Analysis, Matematisk Afdeling 2010.

The book is for sale in Polyteknisk Boghandel, Science section at the Biocentre, and is almost a reprint of the edition of last year: A few misprints have been corrected and there are small additions a few places, but the issue of last year can be used without too much trouble.
Additional literature for the interested student

A translation to Danish of the principal terminology of the course

Maple
If you want to do some experiments in Maple to illustrate parts of the theory, you can download the following Maple worksheet and run it using maple. koman1.mw (You shall rightclick on the icon with the mouse. Then choose: "Save target as" and you save it where you want and call it koman1.mw. It is important that it gets the extension mw. The name itself is unimportant.)

Obligatory work
During the course there will be assigned 3 obligatory sets of exercises which shall be handed to the teaching assistant by Thursday at 17.00 in the weeks 6,8,11. Two of these have to be passed in order to be admitted for the written exam.
Exercise set for 11.2.2010: Exercises 1.7 and 1.10
Exercise set for 25.2.2010: Exercises 4.14 and 4.19
Exercise set for 18.3.2010: problem3.

Exam
The exam takes place Thursday April 15, 2010. It is a 3-hour written exam with grades. The first 90 minutes are without access to books, notes and calculators, the last 90 minutes with access to these things. Reexam: Same conditions as the ordinary evaluation.
Exam April 15
Solutions.
Grades for the exam April 15, 2010.

Guide to the coming exam on April 15, 2010 as well as the reexam on September 2, 2010 guide.

Lectures
I do not have time to cover all proofs in details. This does not mean that the proofs are not important: You are supposed to study them yourself. It is important to learn the main concepts and to understand how they are related. In the first 90 minutes of the exam you will be asked to prove simple statements from the course material.

Reading the book
Every section starts with a small introduction telling about the main results of the section. Therefore, it can happen that you do not understand every word in this introduction because you have to read the whole section to get the full understanding. On the other hand, as soon as you start reading the main text of a section, you are supposed to understand every statement, and if this is not the case, you may have to go back to previous sections or refresh what you have learned in previous courses. Please ask me or the teaching assistents if there is something you do not understand. Mathematics is learning by doing. So when you think you understand something: Close the book and write it down for yourself on a piece of paper or explain it to another student.

Overall plan for the course

Week 5: § 1 - Details: Week 5 Comments: I mentioned the deep topological result: Given a continuous injective function f:U -> C, where U is an open subset of C. Then the image f(U) is open in C and the inverse function f^{\circ -1}: f(U) -> U is again continuous.

Week 6: § 2, 3.1, 3.2 - Details: Week 6 I almost finished Section 3.2, except for Example 3.11. Litterary question: What is the relation between the proof of Goursat's Lemma and the author Thomas Mann?

Week 7: § 3.2, 4 - Details: Week 7 You can find a lot of historical comments about complex analysis in Reinhold Remmert: Theory of Complex functions, Graduate texts in Mathematics vol 122, Springer 1991. The given proof of Goursat's Lemma is due to Alfred Pringsheim (Transactions of Amer. Math. Soc. 2 (1901), 413-421)-he was father in law of Thomas Mann.

Week 8: § 5 - Details: Week 8

Week 9: No teaching

Week 10: § 6 - Details: Week 10

Week 11: § 7 - Details: Week 11 Remember that the evaluation of the course is open from Tuesday 16 to Tuesday 23 of March. You find the form on Absalon.

Week 12: § 8,9 - Details: Week 12

Week 13-15: The course is over. Special sessions with Andreas/Flemming with old exams:
Andreas Aaserud: Thursday April 8, 10-12 in NO37: Problems from the written exams: April 2006-August 2007
Flemming B. von Essen: Tuesday April 13, 13-15 in Aud. 10: Problems from the written exams: April 2008-August 2009.
If you have questions about specific problems in these exams, then send an email to Andreas or Flemming so they can be prepared.

Week 15: Question session with CB: Tuesday April 13, 10-12 in Aud 4. I will answer questions about proofs and concepts from the notes.

Old exams

Exams in Koman (Complex analysis)

Koman April 2006

Koman August 2006

Koman April 2007 Solution Solution

Koman August 2007 Solution Solution

Koman April 2008 Solution solution

Koman August 2008 Solution solution

Koman March 2009 Solution Solutions

Koman August 2009

Exams (in Danish) for the previous course 2KF: