# TMA 4175 - Complex Analysis - Spring 2017

### Course Information

Instructor:

Yurii Lyubarskii - yura [at] math [dot] ntnu [dot] no, Room 954 Sentralbygg II

- Lectures: Mondays 08:15-10:00, S24, Wednesdays 12:15-14:00 room 734 SB2
- Exercise sessions: Tuesday 17:15-18:00, R50
- Office hours: Monday 10:15-11:00, or by appointment
- Exam: May, 31.

#### Messages

07.06.2017 EXAM TEXT

31.05.2017 EXAM SOLUTIONS (updated)

14.05.2017. TRAINING SET FOR RESIDUES AND INTEGRALS TRAINING SET 1

15.05.2017. TRAINING SET FOR CONFORMAL MAPPINGS TRAINING SET 2 (more to follow)

15.05.2017. PREEXAM MEETINGS (all meetings in my office, SB2, room 954, all start at 13:00 )

19.05, 29.05, 30.05

17.05.2017 TEST EXAM TEST EXAM is on the web. No solutions will be published. You are welcome to come during preexam meetings or by appointment. Have good 17th Mai !

09.01. Mention new lecture schedule!

17.01. I was asked to put information regarding IEASTE jobs on the course webpage. Here is the link: www.facebook.com/INTrondheim

01.02. Additional lecture on Wednesday, 08.02 14:15-16:00. Room 922 SB2

20.02. Additional hour lecture on Wednesday 22.02. 1415-15:00. Room 922 SB2

06.03. Additional hour lecture on Wednesday 08.03. 1415-15:00. Room 922 SB2

#### What you are expected to know from Matte4K

This is a short list of preliminary knowledge, the most of stuff will be reminded briefly

### Course material:

Lars V. Ahlfors. Complex Analysis, 3rd edition. The book can be found on amazon.co.uk

Additional sources:

Theodore W. Gamelin: "Complex Analysis".

#### Lecture plans/notes

Lecture 02, 11.01. Compactness

Lecture 03, 13.01 Compactness (cont), Continuous functions

Lecture 04, 18.01 Analytic functions. Introduction

Lecture 05, 23.01 Isolated singularities, Plan of the lecture

Lecture 06, 25.01 Isolated singularities, Continuation

Lecture 07, 30.01 Behavior near zero, Domain preservation, Maximum principle

On lecture 7 we covered only first 5 pages of notes. This is mainly reminder with some more details of what you had
on Matte4K.

We will continue on the next lecture.

01.02. Lecture 08. Todays lecture includes the second part of lecture notes prepared for lecture 7. The material can be found in sections 3.3, 3.4 Ch.4 of Ahlfors book.

Plan of the lectures 10 and 11, 08.02 Conformal mappings (beginning), Linear-fractional mappings

Lecture 12, 13.02. Symmetry principle. Examples

Lectures 13 and 14, 15.02 and 20.02. Residus: lecture plan. Handouts

To Lectures 13: Lagrange interpolation

Lectures 14 and 15, 20.02 and 22.02 Applications of residus

Lectures 16, 17, and 18(beginning) from 01.03, 06.03, and 08.03. Harmonic functions

Lecture 18, 08.03: Hyperbolic metric - handouts

Lecture 19, 13.03: Normal families, we follow Ahlfors book Chapter 5, 4.1-4.4

Lecture 20, 15.03: Hurwitz theorem, Univalent functions, handouts (Gamelin Ch.VIII, Sec. 3,4)

Lectures 21 and 22, 20.03 and 22.03: Riemann mapping theorem, proof. Ahlfors Ch. 6, Sec.1.1 or Gamelin Ch. XI, Sec. 6.

(We did follow proof in Gamelin, but the proofs are not very different)

Dirichlet problem, Mapping of concentric annuli

Short notes: univalent functions, proof of Riemannmapping theorem

Lecture 23, 27.03. Meromorphic functions, Mittag-Leffler theorem (for the plane), handouts, Gamelin Ch. XIII, Sec.2

Lecture 24, 29.03. Entire functions, Infinite products, Weierstrass theorem (handouts), Gamelin Ch. XIII, Sec, 3,4

Lecture 25, 03.04: Plane stationary flow of ideal fluid. (handouts), Gamelin, Ch. XI, Sec. 4

Lecture 26, 05.04: Fluid dynamic continuation. calculating the lifting force (handouts) Lectures 25,26 notes

### Home exercises

* 11.01. Ahlfors, Ch.3: Sec 1.2 # 4, 7; Sec. 1.3: 1-4; Sec. 1.4: 2-5

* 25.01. Ahlfors, Ch.4: Sec.3.2 # 1 - 5

* 08.02 Gamelin, Ch. 2 (Printout distributed on the lecture) Sec. 6: # 4,5,6(this one may be difficult), Sec.7: 1a,g, 2,3,5,9

* 22.02 Gamelin, Ch.7 (Printouts distributed on the lecture) Sec.1: # 1g, 2b, 3d; Sec.2: # 4,7; Sec.3: # 3; Sec.4: #2

* 01.03 Gamelin, Ch.7 (Printouts distributed on the lecture) Sec.4: # 4, 5, 8; Sec. 5 # 1, 4; Sec. 6: 3, 4; Sec. 7: # 1,4; Sec. 8: #1 a,e,f; 3.

* 08.03 Exercise set

This set (as well as most of those to follow) is a bit more demanding. You are welcome to ask questions (problem hour or whatever)

* 15.03 Gamelin: Ch. IX.1 ## 1,2,4; Ch. IX.2 ## 1,2,5,9,11,12; Ch. IX.3 1,3,5,9.

Here are the corresponding texts Ch IX.1, Ch IX.2, Ch IX.3.

* 22.03 Gamelin: Ch. XI.5 ## 1,2,8; Ch. XI.6 ## 331; Ch. XIII.2: ## 1,3,4,5,6 \\Here come the corresponding texts Ch XI.5, Ch XI.6, Ch XIII.2

### Content of the course (This section is under construction)

Below I describe the content of the lectures and refer to the corresponding sections in
the books of Ahlfors and Gamelin,

where the corresponding material can be found.
Typically we cover just parts of the corresponding sections. Please check

with yours
notes or lecture notes on the web.

It is assumed that you know Complex analysis part from Matte4K (see what to know list)

Elementary point set topology:

Ahlfors, Ch. 3, Sections 1.2-1.5

Refreshing analytic functions (with a bit of new material):

Ahlfors: Ch.2: 2.1 - 2.3 and 3.1 -3.4; Ch.3: 2.1 -2.3; Ch.4: 2.1 -2.3 and 3.1 - 3.4; Ch.5: 1.2 -1.3

Gamelin: Ch.2, 1-3; Ch.3 1-2; Ch.4 1-4; Ch.5, 1-5

Residus calculus.

Gamelin, Ch.7 or Ahlfors Ch.4, Section 5.3,

see also lecture notes regarding Lagrange interpolation.

Harmonic functions.

Gamelin, Ch3, 1-5, Ch10, 1-2; Ahlfors Ch.4, Sec. 6.1-6.4

Schwarz lemma and hyperbolic geometry.

Gamelin, Ch 9.

Normal families.

Ahlfors, Ch 5, Sec.4