# MMG700, Analytiska funktioner, Autumn 18

## Latest news

Welcome to the course! The schedule for the course can be found in TimeEdit.

Here is the "formelblad" that will be attached to the exam

and here are solutions of the exam problems, Jan 15, 2019

## Teachers

Course coordinator: Håkan Samuelsson Kalm, hasam"at"chalmers.se, phone extension 3568, room H5025

Teaching assistants:

Lab supervisor:

## Course literature

See the literature list.

## Program

#### Lectures (always in room MVF21, P=Priestley's book, LN=Lecture notes)

Day
Chapter Content
Mon, Nov 5
10.00-11.45
P: 1, 2
LN: App. A, Ch. 1
Intro and repetition; complex arithmetics, polynomials, de Moivre,
lines, circles, sectors etc in the plane; Möbius transformations
Tue, Nov 6
10.00-11.45
P: 2
LN: 1
Möbius transformations, circlines, the Riemann sphere
Thu, Nov 8
10.00-11.45
P: 3, 10
LN: App. B, Ch. 2
Open and closed sets in the plane and on the Riemann sphere, continuity and limits, two-variable calculus in complex notation
Mon, Nov 12
10.00-11.45
P: 5
LN: 3
Holomorphicity, Green's formula, the Cauchy-Riemann equations, elementary properties of holomorphic functions
Tue, Nov 13
10.00-11.45
P: 5, 13
LN: 3
Cauchy's theorem and formula
Thu, Nov 15
10.00-11.45
P: 6
LN: 4
complex series, convergences tests, functions defined by series, power series
Mon, Nov 19
10.00-11.45
P: 6, 14
LN: 4
power series, Taylor's formula, representation of holomorphic functions
Tue, Nov 20
10.00-11.45
P: 7
LN: 5
holomorphic functions defined by power series, exp, trig, log, etc.
Thu, Nov 22
10.00-11.45
P: 13
LN: 6
Liouville's theorem, the fundamental theorem of algebra
Mon, Nov 26
10.00-11.45
P: 8
LN: 7
conformal mappings, Möbius transformations again
Tue, Nov 27
10.00-11.45
P: 13
LN:8
Morera's and Goursat's theorems
Thu, Nov 29
10.00-11.45
P: 15
LN: 9
zeroes and their orders, identity theorem, analytic continuation
Mon, Dec 3
10.00-11.45
P:15
LN: 9
counting zeroes, Rouche's theorem
Tue, Dec 4
10.00-11.45
P: 12
LN: 10
homotopy, homology, Cauchy's theorem again, winding numbers, simply connected domains
Thu, Dec 6
10.00-11.45
P: 12, 16
LN: 10, 11
cont. of the above, Maximum principle
Mon, Dec 10
10.00-11.45
P: 16
LN: 11
Schwarz' lemma, mapping theorems
Tue, Dec 11
10.00-11.45
P: 17
LN:12
singularities, Laurent series and expansions
Thu, Dec 13
10.00-11.45
P: 18, 19
LN:12, 13
residues and the Residue theorem, real integral with complex analytic methods
Mon, Jan 7
10.00-11.45
P: 19, 20
LN:13
cont. of the above
Tue, Jan 8
10.00-11.45
P: 23
LN: 14
something about harmonic functions, the Dirichlet problem, and physical applications
Thu, Jan 10
10.00-11.45

repetition and old exams

#### Recommended exercises

Week
Exercises (Priestley's book)
35 1.1, 1.2, 1.4, 1.7, 1.9, 1.10;
3.1; 2.1, 2.3, 2.11, 2.15i
36
5.4, 5.5a, 5.6, 5.10;
6.2, 6.4, 6.7
37
7.1, 7.2, 7.10b, 7.11i, iii, 7.12, 7.13, 7.15
8.2, 8.3, 8.4, 8.5, 8.9, 8.10;
10.1, 10.3, 10.5, 10.7
38
11.1, 11.3;
13.1, 13.2, 13.3, 13.4, 13.6, 13.7, 13.8, 13.11, 13.12
14.1, 14.2, 14.3, 14.4, 14.5, 14.7, 14.8
39
15.1, 15.3, 15.6, 15.7, 15.10, 15.11, 15.12, 15.13
40
16.2, 16.6, 16.7
17.1, 17.5, 17.9, 17.15, 17.18
41 18.2, 18.3, 18.4, 18.6, 18.9;
20.1, 20.2, 20.4, 20.8, 20.9, 20.13, 20.16, 20.20
42 Repetition and old exams

## Computer labs

Here are some simple MATLAB exercises that help illustrate the theory. They are not part of the examination and shouldn't be reported.

LAB 1 a mapping game
LAB 2 the Argument principle
LAB 3 a potential problem

#### Reference literature:

Learning MATLAB, Tobin A. Driscoll ISBN: 978-0-898716-83-2 (The book is published by SIAM).

## Course requirements

The learning goals of the course can be found in the course plan.

## Examination

The examination consists of a written exam containing three theory exercises and five problems; at least two of the theory exercises are proofs from this list.

Maximum score: 24; at least 12 to pass (G) and at least 18 to pass with distinction (VG). The exam is scheduled on Tuesday, Jan 15, 8.30-12.30. Two re-exams will be given.

Student representatives:

Rahim Nkunzimana (gusnkura "at" student.gu.se)

Peter Ryberg (gusrybpe "at" student.gu.se)

## Examination procedures

In Chalmers Student Portal you can read about when exams are given and what rules apply on exams at Chalmers. In addition to that, there is a schedule when exams are given for courses at University of Gothenburg.

Before the exam, it is important that you sign up for the examination. If you study at Chalmers, you will do this by the Chalmers Student Portal, and if you study at University of Gothenburg, you sign up via GU's Student Portal, where you also can read about what rules apply to examination at University of Gothenburg.

At the exam, you should be able to show valid identification.