# TMA4175 KOMPLEKS ANALYSE (COMPLEX ANALYSIS)

#### MISCELLANEOUS COMMENTS

** The exam. is the 4th (fourth!) of June.**** The same concerns the exam in August**

**Hjelpemiddel under eksamen: Ett A5-ark stemplet fra Instituttet med valgfri paaskrift av studenten.** **During the exam. you are allowed to use a piece of paper of size A5, stamped in advance at the Institute. On it you may write your own notes before the exam.***Not A4 but A5*

**Midterm 14.III.2008 in aud F2, 14-16 o'clock.**Midterm results
The Riemann Mapping TheoremThe Zeta Function of Riemann

The proof that the integral representing the winding number yields an *integer* is more "reliable" in Ahlfors's book, and will be presented in that form.

The proof in §9.16 does not presuppose that w(z) is *conformal* and so the analyticity of the inverse mapping is not immediately available. For conformal mappings it is simpler, because Schwarz' lemma applies also for the inverse mapping in this case. (Recall dw/dz = 1/(dz/dw)).

**The exercises are essential**

The book available at "Tapir".

ISBN 978-0-8218-4399-4

Some graphics related to conformal mappingshttp://www.ima.umn.edu/~arnold/complex.html

## Spring 2008

*First lecture on Monday the 7th of January 2008.* The first exercises on Tuesday the 15th.

### About the course

“Complex analysis is a splendid realm within the world of mathematics, unmatched for its beauty and power. It has varifold elegant and oftentimes unexpected applications to virtually every part of mathematics. It is broadly applicable beyond mathematics, and in particular it provides powerful tools for the sciences and engineering.”

### Book

R. Nevanlinna & V. Paatero: *Introduction to Complex Analysis* (*Einführung in die Funktionentheorie*)

### Syllabus

**The exercises.** Important!

Chapter 1. .. 1.8 - 1.17

Chapter 2. . .In toto

Chapter 3. .. 3.1 - 3.13

Chapter 5. .. In toto

Chapter 6. .. 6.1 - 6.13

Chapter 7. .. In toto

Chapter 8. .. 8.1 - 8.17, 8.20

Chapter 9. .. 9.1 - 9.20, 9.24, 9.25, 9.27, 9.28 (= Hurwitz' theorem), 9.32 - 9.34

Chapter 10. .. In toto

Chapter 11. .. 11.1, 11.8 - 11.16, 11.22 - 11.24.

Chapter 12. .. 12.10

Chapter 13. .. (Not the Mittag-Leffler Theorem in 13.11)

Chapter 14. .. Only some basic facts and §3.

Chapter 15. .. In toto. The proof of Stirling's Formula can be omitted.

Chapter 16. .. 16.1 - 16.4, (16.12).

### Examination

Written examination 4.6.2008 (counts 80%) and midterm 14.3.2008 (counts 20%).
**Please, notice the new date.**

### Lectures

- Mondays 12:15–14:00 in R60
- Fridays 14:15–16:00 in F2

### Exercises

Tuesdays 17:15–18:00 in F6

- EX.15/1 Chapter 1: ex. 10, 11, 12, 14, 32, 36, 40. Solutions
- EX.22/1 Chapter 1, ex. 35, Chapter 2, ex. 1,5, Chapter 3, ex.3,6,16,18. Solutions
- EX.29/1 Chapter 3: ex.21, 29. Chapter 5: ex 2, 4, 6, 11, 20. Chapter 6: ex.2. Solutions
- EX.05/2 Chapter 7: ex.4, 8, 9, 11, 12, 15. Chapter 8, ex.6, 7.Solutions
- EX.12/2 Chapter 8:ex.13, 9. Chapter 9: ex. 1, 3, 6, 7, 8, 11.Solutions
- EX.19/2 Chapter 9: ex.15. Chapter 10, ex. 2, 3, 4, 5, 6.
*The calculus of residues is regarded as known*Solutions - EX 11/3 Chapter 3: ex.22, Chapter 5: ex.19, Chapter 9: ex. 12 ? , Chapter 10: ex. 7, 12.Solutions
- EX.01/4 Chapter 10: ex. 16, 17, 18, 19. Chapter 11: ex. 3,4.Solutions
- EX.08/4 Chapter 11: ex. 9, 10, 21, 26.Solutions
- EX 15/4 Chapter 17: ex. 16, 18. Chapter 13: ex. 1, 2, 3.Solutions
- EX.22/4 Chapter 13: ex. 4, 6, 10, 12. Chapter 15: ex. 1.Solutions
- EX.29/4 Chapter 15: ex. 2, 4. Chapter 16: ex 2, 3. Chapter 17: ex. 6.Solutions
- 6.5.2008 Only one example.
**The End**

### Teacher

Peter Lindqvist, 1152 SB II, NTNU

### Other references

- L. V. Ahlfors:
*Complex Analysis*can also be consulted.Advanced. - Z. Nehari:
*Introduction to Complex Analysis*. – This book is elementary with good exercises. - R. Boas:
*Invitation to Complex Analysis*. – A good exposition, easy to read. - Th. Gamelin:
*Complex Analysis*. - Elias Stein & Rami Shakarchi:
*Complex Analysis*, Princeton