# Funktionentheorie/Complex Analysis Autumn 2023

$$f(z)=\frac{1}{2\pi i}\int_{\gamma}\frac{f(w)}{w-z}\,dw$$
Lecturer
Özlem Imamoglu
Coordinator
Lauro Silini
Lectures
Tuesdays, 10 to 12, in HG F 7.
Wednesdays, 8 to 9 in ML D 28.

## Content

Introduction to Complex Analysis and its applications. Here the official Course Catalogue page.

## Summary of the lectures

Streaming under password protection in availabe HERE.
DayContentNotesChapters
19.09.23, 20.09.23 Introduction, examples, holomorphicity, properties of holomorphic functions. Cauchy-Riemann equations. N 1, N 2 Stein_1 Stein_2
26.09.23, 27.09.23 Power Series, line integrals. N 3, N 4 Stein_3
03.10.23, 04.10.23 Goursat and Cauchy Theorems. N 5, N 6 Stein_4
10.10.23, 11.10.23 Cauchy Theorem and Cauchy integral Formula, Liouville Theorem. N 7, N 8 Stein_5
17.10.23, 18.10.23 Analytic continuation, limit points, order of zeros. N 9 N 10
24.10.23, 25.10.23 Analytic continuation, sequences of holomorphic functions, zeta function, Morera's Theorem. N 11 N 12 Stein_6
31.10.23, 01.11.23 Holomorphic functions by integration, singularities, Riemann's Theorem of removable singularities, poles. N 13 N 14 Stein_7
07.11.23, 08.11.23 Residue Theorem, Laurent series for poles with finite order, Applications to real integrals. N 15 N 16
14.11.23, 15.11.23 Meromorphic functions, essential singularities, Casorati-Weierstrass, stereographic projection, Argument Principle. N 17 N 18
21.11.23, 22.11.23 Rouche' Theorem, example of application for the Fundamental Theorem of Algebra, open mapping Theorem, Maximum modulus Principle, Homotopy and simply connected domains. Homotopy Theorem. N 19 N 20 Stein_8
28.11.23, 29.11.23 The Homotopy Theorem, symply connectedness, primitives, Complex Logarithm, Principal branch. N 21 N 22
05.12.23, 06.12.23 Winding numbers, conformal maps, conformal equivalences. N 23 N 24 Update for N 24: continuity of $$f^{-1}$$ (p. 244) Stein_9
12.12.23, 13.12.23 Riemann Mapping Theorem N 25 N 26
19.12.23, 20.12.23 Riemann Mapping Theorem, Montel Theorem N 27 N 28 Zeta function

## Exercises

Upload your solutions before the corresponding deadline using the SAM-up tool. Only the exercises marked with a star will be graded and are eligible (in case of significant work on them) for bonus points.

Serie 1 Solutions 1 26.09.23
Serie 2 Solutions 2 10.10.23
Serie 3 Solutions 3 17.10.23 Small typo in 3.3, rescaled contour in the diagonal.
Serie 4 Solutions 4 24.10.23
Serie 5 Solutions 5 31.10.23
Serie 6 Solutions 6 07.11.23 Corrected typo in 6.3: the domain has to be connected
Serie 7 Solutions 7 14.11.23
Serie 8 Solutions 8 21.11.23
Serie 9 Solutions 9 28.11.23
Serie 10 Solutions 10 05.12.23
Serie 11 Solutions 11 12.12.23
Serie 12 Solutions 12 19.12.23
Mock Exam Solutions

## Exercise classes

timeroomassistant
Tu 14-16ETZ G 91U. Faure
Tu 14-16HG E 33.1J. Meng
Tu 14-16LEE D 101 L. Renkin
Tu 14-16LEE D 105 D. Schwenkreis