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
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 


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 
time  room  assistant  language 

Tu 1416  ETZ E 6  
Tu 1416  HG E 33.1  C. Bortolotto  English 
Tu 1416  HG G 26.3  L. Malli  English 
Tu 1416  IFW A 32.1  H. Liang  English 
Tu 1416  LEE C 104  S. Abramyan  English 
Tu 1416  LEE D 101  C. Nussbaumer  
Tu 1416  LEE D 105  L. Pastor Pérez  English 
Tu 1416  LFW C 11  M. Gong  English 
Tu 1416  ML F 38  E. Rothlin  English 
Tu 1416  ML J 34.3  E. Mazzoni  English 
Tu 1416  NO C 44  A. Theorin Johansson  English 
Tu 1416  NO C 6  J. Huber  German 
Day  Content 

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 CauchyRiemann
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: simplyconnected 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 