# Complex AnalysisAutumn 2019

Lecturer
Prof. Dr. Paul Biran
Coordinator
Younghan Bae

## Auxiliary Materials

The following material will be distributed during the final exam and you are not allowed to use your printed copy of this material. Also, it is not allowed to use your own auxiliary materials.

Auxiliary

## Content

We will discuss the following subjects: Complex functions of one variable, Cauchy-Riemann equations, Cauchy theorem and integral formula, singularities, residue theorem, index of closed curves, analytic continuation, special functions, conformal mappings, Riemann mapping theorem.

week material covered
1 Basic topology of the complex plane. Definitions from topology (open, closed, compact, path connected, continuity of a function).
2 Complex derivative functions and the Cauchy-Riemann equation. Comparing linear algebra over the real numbers and the complex numbers. The first order approximation of a complex function.
3 Consequences of the Cauchy-Riemann equation. A definition of path integrals.
4 Definition of a primitive function. Cauchy theorem for a holomorphic function on an unit circle.
5 Function of winding numbers. Global Cauchy theorem.
6 Cauchy estimate theorem. Maximum principle. Lemma of Schwarz.
7 A criterion when a continuous function has primitives. Homotopy between curves.
8 Convergence of a sequence of functions.
9 Taylor expansion. Radius of convergence. Convergence of a sequence of functions.
10 Laurent expansion.
11 Zeros of holomorphic functions. Singularities of holomorphic functions. Riemann extension theorem for removable singularity
12 Meromorphic functions. Casorati–Weierstrass theorem. Residue theorem. Argument principle.
13 Rouche's theorem. Branched covering. Open mapping theorem. Hurwitz theorem

## Exercises

The new exercises will be posted here on every weekend. We expect you to look at the problems and to prepare questions for the exercise class on Tuesday.

Please hand in your solutions by the following Friday at 13:00 in your assistant's box in HG J68. Your solutions will usually be corrected and returned in the following exercise class or, if not collected, returned to the box in HG J68.

exercise sheet due by solutions
Exercise sheet 1 September 20 Solution 1
Exercise sheet 2 September 27 Solution 2
Exercise sheet 3 October 4 Solution 3
Exercise sheet 4 October 11 Solution 4
Exercise sheet 5 October 18 Solution 5
Exercise sheet 6 October 25 Solution 6
Exercise sheet 7 November 1 Solution 7
Exercise sheet 8 November 8 Solution 8
Exercise sheet 9 November 15 Solution 9
Exercise sheet 10 November 22 Solution 10
Exercise sheet 11 November 29 Solution 11
Exercise sheet 12 December 06 Solution 12
Exercise sheet 13 December 13 Solution 13
Exercise sheet 14 December Solution 14

## Exercise classes

timeroomassistantlanguage
Tu 13-15HG D 5.2Subhajit Janaen
Tu 13-15HG D 7.2Niclas Küpperen
Tu 13-15HG G 26.3Levie Bringmansen
Tu 13-15LEE D 101Rush Brownen
Tu 13-15LEE D 105Daniel Hainschinken
Tu 13-15LFW C 11Valeria Ambrosio en
Tu 13-15ML J 34.3Fabian Jinen
Tu 13-15NO C 6Filippo Bertaen
Tu 15-17NO C 44Paula Truölen

## Literature

• "An introduction to complex function theory" by B. Palka (Undergraduate Texts in Mathematics. Springer-Verlag, 1991.)