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Complex Analysis (Summer 2015)


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 CauchyRiemann
differential equations for $f$. So with the equivalences from the
tutorial we end up with (for smooth $f$ and simplyconnected $U$): $$
\begin{align*} f ~\text{holomorphic} ~&\Leftrightarrow~ f
~\text{has a holomorphic antiderivative}~F, \text{i.e.}~ F' = f\\
&\Leftrightarrow~ \int_\gamma f(z)\mathrm{d}z = 0 \quad
\text{for any closed curve}~ \gamma ~\text{in}~ U \end{align*} $$ 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 antiderivative.
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.
Exercise sheets
 exercise
sheet 01, due May 05
 exercise
sheet 02, due May 05
 exercise
sheet 03, due May 12
 exercise
sheet 04, due May 15
 exercise
sheet 05, due May 22
 exercise
sheet 06, due May 29
 exercise
sheet 07, due June 05
 exercise
sheet 08, due June 12
 exercise
sheet 09, due June 19
 exercise
sheet 10, due June 30
 exercise
sheet 11, due July 07
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
