# TMA4175 Complex Analysis, Spring 2018

"Everything should be made as simple as possible - but no simpler [than that]" A. EINSTEIN

## Messages

### Written and printed (textbook) aids are allowed at the exam!

This is NOT an internet course. Necessary details and calculations, missing from the text in the book, are lectured. The exercises are essential. If you cannot participate, you had better arrange so that somebody present provides you with notes!

Aud. R73 is not easy to find (4th floor, Realfag E, near main entrance)!!!!

The first lecture will take place on Monday the 8th of January in R73, 14.15–16.00
The first exercises will take place on Tuesday the 16th of January in S21, 16.15–17.00

### See below about the Exam.

* The lectures are in English

 Week Section Comments 2 Ch.1, Ch.2 3 Ch.2 Elementary series 4 Ch. 3 Poincare's model Link below 5 Ch. 4 Integrals 6 Ch.4 Integrals Replaces Ch. 4.4 Goursat's proof again 7 Ch. 4 Integrals. Schwarz's lemma. Rouche's theorem. Argument Principle 8 Ch. 4 Integrals Laurent expansion, Singularities, Residues 9 Riemann's Mapping Thm Conformal Mapping 10 Ch. 6 Schwarz-Cristoffel, Harmonic Functions 11 Gamma function, Products Jensen's Formula 12 Riemann's Zeta Function Zeta Function 13 Easter 14 Zeta. Double connected domain. A Proof 15 Perron's Method Perron's Method Barriers 16 Barriers. Repetition 17 Repetition 18

### Textbook

Lars Ahlfors: Complex Analysis (available on Amazon. Many copies in the library)

### Lectures

Monday 14–16, R73 Thursday 12–14, S24 (in SB II)

### Teacher

Peter Lindqvist (SB II room 1152)

### Exercises

Tuesday 16–17, S21 (in SB II)

"That never any knowledge was delivered in the same order it was invented, not in the mathematic,.." Sir Francis Bacon (1561-1626)

 Week Problems Comments 3 Exercises 4 Exercises 5 Excercises Ex. 5 in the plane. Conformal mapping 6 Exercises About H(z) 7 Exercises A conformal mapping error: this is for the intersection 8 Exercises 9 Exercises In ex. 3 the 3rd deriv. 10 Exercises 6.III.2018 11 Exercises 12 Exercises Poisson Integral 15 Exercises Airy fct, see Stein -Shakarchi pp.328-329. 16 Exercises Ex. 1, wrong cases Solutions 17 Last exercises Monday 23.IV.2018 Solutions

## Preliminary Syllabus (Pensum) 2018

#### Chapter 4

(not §4.4)

#### Chapter 5

(from §5.4 what is needed in §6.1)

#### Chapter 6

(not §6.5)

#### Chapter 7

(Only §7.3.1 - §7.3.3)

## The End

### Extra literature

Th. W. Gamelin: Complex Analysis, Springer. (Seems to cover the syllabus)
R. Boas: Invitation to Complex Analysis. (Easy to read).
E. Stein & R. Shakarchi: Complex Analysis (Princeton Lectures in Analysis II), Princeton

## Examination

24.V.2018

### Written and printed (textbooks) aids are allowed at the exam.!

Exam 2018 Text. Solutions

### EXTRA NOTES

Conformal mappings of the unit disc. Poincare's model.conformal2018.pdf

Joukowski,s example Joukowski