Fakultät II
Institut für Mathematik

# Complex Analysis

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# Complex Analysis (Summer 2015)

 Lectures Tutorials Alexander Bobenko Tue 12-14 MA 144 Fri 10-12 MA 141 Jan Techter Wed 14-16 MA 848 Thu 16-18 MA 851

### Contents

Analysis of functions of one complex variable: holomorphic functions, Möbius transformations, power series, Cauchy's integral theorem, zeros and singularieties, meromorphic functions, Laurent series, analytic continuation, residues, Riemann mapping theorem, partial fraction decomposition.

This is a course of the Berlin Mathematical School held in english or deutsch (depending on the audience).

### News

[02.09.2015]
Examination dates for October are free for registration at the office of Mathias Kall, MA873:
• Wednesday, October 14
• Friday, October 16
[17.07.2015]
The exact examination dates of October 2015 will be announced until September.
[17.07.2015]
The certificates are now available at the secretariat (Mathias Kall, MA873).
We will also generate corresponding Qispos entries in the next few days.
Attention: Those of you who have not submitted their complete data of Name, Mattrikelnummer, Studiengang (e.g. by registering on the list during the first lecture) please contact me, or Mathias Kall. We cannot make your certificate, nor your Qispos entry without it.
[07.07.2015]
Slight change in plan: I cannot be there on Wednesday, July 08.
You will get back all remaining exercise sheets in the tutorials of the next week.
All people who still needed some points should already be informed about their current score.
[01.07.2015]
No tutorials on Wednesday/Thursday, July 08/09.
(I will be in MA848 on Wednesday, July 08 at 14:15 for a short time to hand back exercise sheets.)
No lectures on Friday, July 10, and Tuesday, July 14.
The last tutorials will be on Wednesday/Thursday, July 15/16,
and the last lecture on Friday, July 17.
[25.06.2015]
Exercise Sheet 11 is now online. It is the last one for this semester.
You should be able to solve Exercises 1 to 3 by now. Exercise 4 is about compact convergence, which will be covered in more detail during the next lectures and the tutorials of the next week.
The deadline is due Tuesday, July 07.
[22.06.2015]
Exercise 3 of Exercise Sheet 10 is about computing certain real integrals using the theorem of residues. Since this subject came up very recently in the lecture and will be picked up again not until the tutorials this week the deadline of the exercise sheet has been extended until Tuesday, June 30.
[22.06.2015]
I am sorry I had to cancel the last thursday tutorial.
The tutorial on wednesday was solely about winding numbers. I will repeat the relevant results briefly at the beginnig of the upcoming thurday tutorial.
[04.06.2015]
Clarification on Exercise Sheet 7, Exercise 4:
It is now stated that $\mathrm{Ord}(f - f(z_0), z_0) = \min\{k \in \mathbb{N} | f^{(k)}(z_0) \neq 0\}$, which is the order of the zero of $f - f(z_0)$ at the point $z_0$.
Recall that for $n = 1$, $f$ is locally biholomorphic. For $n \geq 2$ you might want to use another local representation in terms of a biholomorphic function.
What can you say about intersection angles of curves under a biholomorphic map?
[03.06.2015]
There will be 11 Exercise Sheets.
To get the certificate you need a total of 110 points.
[03.06.2015]
Remark (or actually "no Remark") on Exercise 2 of Exercise Sheet 7: It is exactly formulated as it should be.
[27.05.2015]
Title of Exercise 2 on Exercise Sheet 6 changed.
The old title might have been misleading since there seems to be no need for applying Cauchy's integral formula.
[13.05.2015]
There will be no office hour on Monday, May 18 (Jan Techter).
During my absence in the week May 18 - 22 questions can be directed to Isabella Thiessen, MA 866.
The tutorials will take place.
[13.05.2015]
Supplement, 4th tutorial:
To see more clearly how to obtain Cauchy's integral theorem from the vector field interpretation of complex integration, it might help to consider the following additional line of thought:
Suppose $U$ is simply connected. Then the vector fields $W_1$ and $W_2$ have a potential if and only if $\mathrm{curl}(W_1) = \mathrm{curl}(W_2) = 0$, which in turn are the Cauchy-Riemann differential equations for $f$. So with the equivalences from the tutorial we end up with (for smooth $f$ and simply-connected $U$): \begin{align*} f ~\text{holomorphic} ~&\Leftrightarrow~ f ~\text{has a holomorphic anti-derivative}~F, \text{i.e.}~ F' = f\\ &\Leftrightarrow~ \int_\gamma f(z)\mathrm{d}z = 0 \quad \text{for any closed curve}~ \gamma ~\text{in}~ U \end{align*} Note that vanishing integrals over closed curves is equivalent to the intergral between two points being invariant of the path, which is equivalent to having a complex anti-derivative.
Also try to rewrite the complex integral only in terms of the vector field $W_1 = \bar{f}$ and apply Gauss' and Stokes' theorem for real vector analysis in 2 dimensions to obtain the same result.
Additionally/Alternatively try to reformulate everything in terms of differential forms and apply Stokes' theorem for differential forms.
[13.05.2015]
Correction on Exercise Sheet 4. Exercise 1 (iii): Since the '$a$' in that part of the exercise has nothing to do with the '$a$' in the rest of the exercise it got renamed to '$q$'. It is also explicitly stated that $q \in \mathbb{C}$ with $|q| < 1$ to ensure that $g$ is actually an automorphism of the unit disc.
[05.05.2015]
There will be no lecture on Friday, May 08.
The deadline of Exercise Sheet 4 will therefore be extended until Tuesday, May 12, before the lecture.
[04.05.2015]
Slight corrections in exercise 2 and 4 on sheet 3.
[29.04.2015]
From now on the lectures on Tuesdays will take place in MA 144.
[21.04.2015]
The deadline of the first exercise sheet will be extended by one week.
Since Friday, May 01, is a holiday, the deadline for the first and the second exercise sheet will be on Tuesday, May 05, before the lecture.
[21.04.2015]
First tutorial on Wednesday, April 22.
[10.04.2015]
First lecture on Tuesday, April 14.

### Homework policy

• To get a certificate for the tutorial, you need to satisfactorily complete 50% of the homework assignments.
• The exercises are to be solved in groups of two people.
• The homeworks are due weekly at the beginning of the lecture on Friday.

### Literature

• Alexander Bobenko, lecture notes
• Dirk Ferus, lecture notes
• Jänich, Funktionentheorie. Eine Einführung.
• Ahlfors, Complex Analysis.
• Lang, Complex Analysis.
• Needham, Visual Complex Analysis (also translated as: Anschauliche Funktionentheorie.)

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

### Office hours

 Alexander Bobenko Thu 13-14 MA 881 Jan Techter Mon 12-13 MA 866

 Jan Techter . 02.09.2015.