Funktionentheorie/Complex Analysis Autumn 2022

Emmanuel Kowalski
Cynthia Bortolotto
Lectures (in English)
Tuesdays, 10 to 12, in HF F 7 (starting from 27.9).
Wednesdays, 8 to 9 in ML D 28.
Exercise classes (in English and German): Tuesdays, 14:00 to 16:00.


Introduction to complex analysis and its applications.


A DMATH forum is available for discussion and to ask/answer questions about the course, see here.


Details of the weekly organization for the exercises will be posted later.
There will be a bonus system, which can lead to a 0.25 point bonus in the final exam. The requirement is to hand back the Exercise Sheets with at least significant work on the marked exercises ("Sinnvoll bearbeitet"). Please upload your solutions in SAM-up.

exercise sheet due by solutions Bonus points exercise
Exercise sheet 1 (with Exercise 4 corrected) September 28 at 16h Exercise sheet 1 solutions Exercise 4
Exercise sheet 2 Octorber 5 at 16h Exercise sheet 2 solutions Exercises 3 and 5
Exercise sheet 3 October 12 at 16h Exercise sheet 3 solutions Exercises 3
Exercise sheet 4 October 19 at 16h Exercise sheet 4 solutions Exercises 2 and 4
Exercise sheet 5 October 26 at 16h Exercise sheet 5 solutions Exercise 1
Exercise sheet 6 November 2 at 16h Exercise sheet 6 solutions Exercise 5
Exercise sheet 7 November 9 at 16h Exercise sheet 7 solutions Exercise 3
No exercise sheet to handle on November 16th
Exercise sheet 8 November 23 at 16h Exercise sheet 8 solutions Exercise 1
Exercise sheet 9 November 30 at 16h Exercise sheet 9 solutions Exercise 2
Exercise sheet 10 December 7 at 16h Exercise sheet 10 solutions Exercise 5
Exercise sheet 11 December 14 at 16h Exercise sheet 11 solutions Exercise 2
Exercise sheet 12 December 21 at 16h Exercise sheet 12 solutions Exercise 5

Exercise classes

Tu 14-16ETZ E 6
Tu 14-16HG E 33.1 C. BortolottoEnglish
Tu 14-16HG G 26.3L. MalliEnglish
Tu 14-16IFW A 32.1H. LiangEnglish
Tu 14-16LEE C 104S. AbramyanEnglish
Tu 14-16LEE D 101C. Nussbaumer
Tu 14-16LEE D 105L. Pastor PérezEnglish
Tu 14-16LFW C 11M. GongEnglish
Tu 14-16ML F 38E. RothlinEnglish
Tu 14-16ML J 34.3E. Mazzoni English
Tu 14-16NO C 44A. Theorin JohanssonEnglish
Tu 14-16NO C 6J. HuberGerman

Summary of the lectures

We indicate here the topics discussed in each lecture, with references to the literature where applicable, and with links to the lecture notes.
The video recordings of the lectures are available on the ETH Video portal.
20.9.2022 Introduction to the course, examples of applications, definition of holomorphic functions, algebraic stability properties of holomorphic functions.
Notes for Chapter 1
Notes for Chapter 2
21.9.2022 Convergent power series are holomorphic. Examples and counterexample (the complex conjugate function).
Notes for Chapter 2
27.9.2022 Holomorphy and differentiability; the Cauchy-Riemann equations. Line integrals.
Notes for Chapter 2
28.9.2022 Line integrals and primitives.
Notes for Chapter 2
4.10.2022 Chapter 3: Cauchy's Theorem. Goursat's Theorem, existence of primitives in a circle, Cauchy's Integral Formula.
Notes for Chapter 3
5.10.2022 Chapter 3: proof of Goursat's Theorem.
Notes for Chapter 3 (with some corrections and clarifications)
11.10.2022 Chapter 4: applications of Cauchy's Theorem and integral formula: analyticity, Cauchy's inequalities for derivaties, Liouville's Theorem.
Notes for Chapter 4
12.10.2022 Chapter 4: zeros of holomorphic functions, analytic continuation.
Notes for Chapter 4
18.10.2022 Chapter 4: proof of the principle of analytic continuation. Limits of holomorphic functions, Morera's theorem.
Notes for Chapter 4
19.10.2022 Chapter 4: holomorphic functions defined by integrals
Notes for Chapter 4
25.10.2022 Chapter 5: singularities and meromorphic functions, residue theorem.
Notes for Chapter 5 (beginning)
26.10.2022 Chapter 5: residue theorem and examples.
Notes for Chapter 5 (beginning)
1.11.2022 Chapter 5: meromorphic functions, counting zeros, open image and maximum modulus principle.
Notes for Chapter 5 (beginning)
8.11.2022 Mock exam
9.11.2022 Solution of the mock exam
15.11.2022 Chapter 6: Eta, THeta, Zeta (a long example). Definitions of the functions, infinite products..
Notes for Chapter 6 (beginning)
16.11.2022 Chapter 6: Eta, THeta, Zeta. Analytic continuation of the zeta function, application to prime numbers.
Notes for Chapter 6 (beginning)
22.11.2022 Chapter 6: Eta, THeta, Zeta (a long example). Sketch of Riemann's approach to counting primes; the Riemann Hypothesis.
Notes for Chapter 6 (beginning)
23.11.2022 Chapter 7: Homotopy and applications. Definition and statement of Cauchy's Theorem for homotopic curves.
Notes for Chapter 7 (beginning; corrected statement)
29.11.2022 Chapter 7: Proof of Cauchy's Theorem for homotopic curves.
Notes for Chapter 7 (beginning)
30.11.2022 Chapter 7: simply-connected open sets, existence of primitives. The complex logarithm.
Notes for Chapter 7 (beginning)
6.12.2022 Chapter 7: The residue theorem and homotopy; winding numbers.
Notes for Chapter 7
7.12.2022 Chapter 8: conformal mapping (definition, first examples).
Notes for Chapter 8 (beginning)
13.12.2022 Chapter 8 (conformal mapping): more examples, statement of Riemann's mapping theorem. Outline of the proof. Schwarz Lemma, automorphisms of the disc.
Notes for Chapter 8 (beginning)
14.12.2022 Chapter 8 (conformal mapping); reduction of Riemann's Theorem to the existence of an extremum.
Notes for Chapter 8 (beginning)
20.12.2022 Chapter 8 (conformal mapping): end of the proof of Riemann's Theorem; Montel's Theorem. Final remarks.
Notes for Chapter 8
21.12.2022 Review of the course, questions
Summary of the main definitions and results