# TMA4175 Kompleks analyse, våren 2019

# TMA4175 Complex Analysis, Spring 2019

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

## Messages

#### The lectures on Fridays 8-10 will take place in R4

#### 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!

#### 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

Week | Section | Comments | |
---|---|---|---|

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 | |

15 | Repetition and minor comments | ||

xxxxxx | Zeta. Double connected domain. | A Proof | |

xxxxxx | Perron's Method | Perron's Method | Barriers |

## Information about the course

### 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)

Week | Problems | Comments | ||
---|---|---|---|---|

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

#### ALL THE EXERCISES (Øvinger)

#### Chapter 1

#### Chapter 2

#### Chapter 3

#### 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)!

## NOTES, LINKS

### EXTRA NOTES

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

Joukowski,s example Joukowski