Complex Analysis I (Summer 2017) — Komplexe Analysis I

Contents

This is an Additional Basic Course of the Berlin Mathematical School held in English or German (depending on the audience). The content is analysis of one complex variable: holomorphic functions, Möbius transformations, contour integrals, Cauchy's integral theorem, singularities, residues, Laurent series, partial fraction decomposition, analytic continuation, Riemann mapping theorem, ...

Prerequisites are the contents of the courses Analysis I and II (single- and multivariable calculus) and Linear Algebra.

There is a summary of contents for your convenience.

News

The last lecture was be on Thursday, July 20. Note that Monday, May 01, Thursday, May 25 and Monday, June 05 were public holidays, so there were neither lectures nor tutorials on these days.

[2017, Aug 08]

Nice holidays to everyone!

[2017, Jul 11]

If you would like to take the exam in August, please register as soon as possible (in case you have not done so already).

[2017, Jul 11]

A missing assumption was spotted in Exercise 33 of Sheet 9.

[2017, Jun 19]

The tutorial on Tuesday, June 20, was exceptionally held by Isabella Retter.

[2017, Jun 06]

On Sheet 7, there is now a typo less in Exercise 23 and a weaker assumption in Exercise 25. For Exercise 25, it is ok if you prove the old version with the stronger assumption!

[2017, May 26]

Sheet 6 is now online. There is no bonus task, but feel invited to watch the short movie Möbius Transformations Revealed by Douglas Arnold and Jonathan Rogness.

[2017, Apr 24]

There was a typo in Exercise 2 of Sheet 1. Please check out the corrected version!

[2017, Feb 06]

First lecture on Thursday, April 20.
First tutorials in the second week.

Exercise sheets

• Exercise Sheet 13, not due.
• Exercise Sheet 12, due before the lecture on July 13
• Exercise Sheet 11, due before the lecture on July 06
• Exercise Sheet 10, due before the lecture on June 29
• Exercise Sheet 9, due before the lecture on June 22
• Exercise Sheet 8, due before the lecture on June 15
• Exercise Sheet 7, due before the lecture on June 08
• Exercise Sheet 6, due before the lecture on June 01
• Exercise Sheet 5, due before the lecture on May 29 (Monday). A solution to Exercise 20.
• Exercise Sheet 4, due before the lecture on May 18
• Exercise Sheet 3, due before the lecture on May 11
• Exercise Sheet 2, due before the lecture on May 04
• Exercise Sheet 1, due before the lecture on April 27

Growing collection of quiz questions from the tutorial sessions.

Homework policy

• To get a certificate for the tutorial, you need to satisfactorily complete 50% of the homework assignments. This is required for taking the final exam (Modulprüfung).
• The exercises are to be solved in groups of two people.
• The homework is due weekly at the beginning of the lecture on Thursday. Late homework is only accepted with a medical excuse.

Exams

The final examinations (Modulprüfung) for this course will be oral. Please check Boris Springborn's homepage for available dates and arrange a date for your exam with Mathias Kall, room MA 873.

Literature

• Summary of contents for this course.
• Jänich, Funktionentheorie – Eine Einführung. (in German)
• Ahlfors, Complex Analysis.
• Dirk Ferus' lecture notes (mostly in German, also contains a list of more good books).
Further references include:
• Alexander Bobenko's lecture notes (in German)
• Needham, Visual Complex Analysis, also translated as: Anschauliche Funktionentheorie.
• Wegert, Visual Complex Functions – An Introduction with Phase Portraits.
• Conway, Functions of One Complex Variable.
• Freitag/Busam, Funktionentheorie 1. (in German)
• Behnke/Sommer: Theorie der analytischen Funktionen einer komplexen Veränderlichen. (in German)
... and countless other good books. Ferus' lecture notes also mention the following writings on the history of the subject:
• Klein, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. (in German)
• Gaier, Über die Entwicklung der Funktionentheorie in Deutschland von 1890 bis 1950. In: Ein Jahrhundert Mathematik 1890-1990. Festschrift zum Jubiläum der DMV. (in German)

There is a book collection ("Semesterapparat") for this course in the Mathematics Library on the first floor.

Office hours