# TMA4175 Complex Analysis, Spring 2019

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

## Messages

#### Office hours i the advent of the exam.: Wednesday 15 May 15-16 and Friday 24 May 15-16 in Room 1152, SB II.

The first lecture will take place on Tuesday the 8th of January in S21, 8.15–10.00
The first exercises will take place on Monday the 14th of January in S21, 14.15–15.00

### See below about the Exam.

* The lectures are in English

2 Ch.1, Ch.2
3 Ch.2 Elementary series
4 Ch. 3 Poincare's model Poincare's Model
5 Ch. 4 Integrals
6 Ch.4 Not chains. Integrals Replaces Ch. 4.4 http://folk.ntnu.no/hanche/art/goursat/monthly648-652-a.pdf
7 Ch. 4 Integrals. Principle of Argument Schwarz's lemma. Rouche's theorem. Argument Principle
8 Ch. 4. Ch. 5 Integrals .Products Laurent expansion, Singularities, Residues. PRODUCTS Order 1
9 Ch. 5 Products. Hadamard's thorem. Jensen's formula, Caratheodory's lemma Weierstrass' Product. Gamma function
10 Riemann's Mapping Thm Conformal Mapping. Chapter 6.
11 Ch. 6 Schwarz-Christoffel, Harmonic Functions
12 Ch.6 Perron's Method, Scwarz-Christoffel
13 Ch. 7 Elliptic Functions Weierstrass' elliptic function, Elliptic integral
14 Riemann's Zeta Function Zeta Function
xxxxxx Zeta. Double connected domain. A Proof
xxxxxx Perron's Method Perron's Method Barriers

### Textbook

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

### Lectures

Tuesday 8–10, S21 (in SB II) Friday 8–10, R4 (in Realfagbyg.)

### Teacher

Peter Lindqvist (SB II room 1152)\\

### Exercises

Monday 14–15, 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)

3 Exercises 2019
4 Exercises Exercise 2019 1) i(z-2)+2.; 2) 3(z+1)/(iz+1). ;3)Concentric
5 Excercises 2019 Ex. 5 in Euclidean space. Conformal mapping
6 Exercises 2019 2w =z +1/z. About H(z) , Joukowski's profile 1)iy. 2)6.28i
7 Exercises, More examples A conformal mapping error: this is for the intersection Conv only on real axis
8 Exercises EX: Map a quadrant of a disc onto a disc. Quater-circle 6 roots
9 Exercises In ex. 3 the 3rd deriv. Some solutions
10 Exercises 4.III.2019 Solutions
11 Exercises
12 Exercises Poisson Integral
13 Exercises Airy fct, see Stein -Shakarchi pp.328-329.
14 Exercises ! .IV.2019 circle-polygon
15 Exercises Last exercises 2019 Ex 1, - ln(2) < y < ln(2). Ex 4, k = 2 suffices. Ex 7, Phragmen-Lindelof
XXXX Last exercises 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 Functional Eqn for the Zeta function

 (notes above, 6 pages)

## 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.
L.-S. Hahn & B. Epstein: Classical Complex Analysis, Jones & Bartlett Publishers. (Explicit, good examples; printed on demand).

## Examination

29.V.2019

### Permitted aids: one A4-sized sheet of yellow paper stamped by the Department of Math. Sciences (on it you may write whatever you want)!

Exam 2018 Text. Solutions
Exam 2019Text. Solutions
Exam cont 2019 Text Solutions

### EXTRA NOTES

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

Joukowski,s example Joukowski