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 14-16 | ETZ E 6 | ||
| Tu 14-16 | HG E 33.1 | C. Bortolotto | English |
| Tu 14-16 | HG G 26.3 | L. Malli | English |
| Tu 14-16 | IFW A 32.1 | H. Liang | English |
| Tu 14-16 | LEE C 104 | S. Abramyan | English |
| Tu 14-16 | LEE D 101 | C. Nussbaumer | |
| Tu 14-16 | LEE D 105 | L. Pastor Pérez | English |
| Tu 14-16 | LFW C 11 | M. Gong | English |
| Tu 14-16 | ML F 38 | E. Rothlin | English |
| Tu 14-16 | ML J 34.3 | E. Mazzoni | English |
| Tu 14-16 | NO C 44 | A. Theorin Johansson | English |
| Tu 14-16 | 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 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 |