MMG700 Analytic Function Theory

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Course PM

Here are solutions to the exam on Aug 24.

Here are suggested solutions to the exam on June 11.

There will be a re-exam on June 11, in the morning. It will be a "home exam".

 

This page contains the program of the course: lectures, exercise sessions and computer labs. Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM.

We will use these lecture notes. Priestley's book is recommended as extra reading; we will also use it as a source of exercises.

Program

The schedule of the course is in TimeEdit.

Lectures

 

(Aways in MVF21, LN=lecture notes, P=Priestley's book)

Day Chapter Content
Mon, Nov 4
10.00-11.45

LN: App. A, Ch. 1

P: 1, 2

 Intro and repetition; complex arithmetics, polynomials, de Moivre,
lines, circles, sectors etc in the plane; Möbius transformations
Tue, Nov 5
10.00-11.45

LN: 1

P: 2

 Möbius transformations, circlines, the Riemann sphere
Thu, Nov 7
10.00-11.45

LN: App. B, Ch. 2

P: 3, 10

 Open and closed sets in the plane and on the Riemann sphere, continuity and limits, two-variable calculus in complex notation
Mon, Nov 11
10.00-11.45

LN: 3

P: 5

 Holomorphicity, Green's formula, the Cauchy-Riemann equations, elementary properties of holomorphic functions
Tue, Nov 12
10.00-11.45

LN: 3

P: 5, 13

 Cauchy's theorem and formula
Thu, Nov 14
10.00-11.45

LN: 4

P: 6

 complex series, convergences tests, functions defined by series, power series
Mon, Nov 18
10.00-11.45

LN: 4

P: 6, 14

 power series, Taylor's formula, representation of holomorphic functions
Tue, Nov 19
10.00-11.45

LN: 5

P: 7

 holomorphic functions defined by power series, exp, trig, log, etc.
Thu, Nov 21
10.00-11.45

LN: 6

P: 13

 Liouville's theorem, the fundamental theorem of algebra
Mon, Nov 25
10.00-11.45

LN: 7

P: 8

 conformal mappings, Möbius transformations again
Tue, Nov 26
10.00-11.45

LN:8

P: 13

 Morera's and Goursat's theorems
Thu, Nov 28
10.00-11.45

LN: 9

P: 15

 zeroes and their orders, identity theorem, analytic continuation
Mon, Dec 2
10.00-11.45

LN: 9

P:15

 counting zeroes, Rouche's theorem
Tue, Dec 3
10.00-11.45

LN: 10

P: 12

 homotopy, homology, Cauchy's theorem again, winding numbers, simply connected domains
Thu, Dec 5
10.00-11.45

LN: 10, 11

P: 12, 16

 cont. of the above, Maximum principle
Mon, Dec 9
10.00-11.45

LN: 11

P: 16

 Schwarz' lemma, mapping theorems
Tue, Dec 10
10.00-11.45

LN:12

P: 17

 singularities, Laurent series and expansions
Thu, Dec 12
10.00-11.45

LN:12, 13

P: 18, 19

 residues and the Residue theorem, real integral with complex analytic methods
Tue, Jan 7
10.00-11.45

LN:13

P: 19, 20

 cont. of the above
Thu, Jan 9
10.00-11.45

LN: 14

P: 23

 something about harmonic functions, the Dirichlet problem, and physical applications
Day Sections Content

 

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Recommended exercises

Week Exercises
45

P: 1.1, 1.2, 1.4, 1.7, 1.9, 1.10; 3.1;

2.1, 2.3, 2.11, 2.15i

LN: A.1, A.2, A.4, A.5; B.1, B.2; 1.9, 1.10, 1.11, 1.8 
46

P: 5.4, 5.5a, 5.6, 5.10;
6.2, 6.4, 6.7

LN: 2.2, 2.6, 2.9, 2.10, 2.11, 2.12;

3.5, 3.11, 3.12, 3.13, 3.14, 3.15;

4.2, 4.3, 4.4, 4.6
47

P: 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

LN: 5.3, 5.7, 5.8, 5.9, 5.10, 5.11, 5.12, 5.13, 5.14, 5.15;

6.2, 6.4, 6.5, 6.6;
48

P: 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

LN: 7.2, 7.6, 7.7, 7.8, 7.9, 7.10;

8.2, 8.3
49

P: 15.1, 15.3, 15.6, 15.7, 15.10, 15.11, 15.12, 15.13;

LN: 9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7
50

P: 16.2, 16.6, 16.7
17.1, 17.5, 17.9, 17.15, 17.18

LN: 11.1, 11.2, 11.3;

12.4, 12.5, 12.6, 12.7, 12.8, 12.9, 12.10, 12.11
2, 2020

P: 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

LN: 13.1, 13.2, 13.3, 13.4, 13.5, 13.6, 13.7,

13.9, 13.10, 13.11;

14.2, 14.3, 14.6, 14.7
Day Exercises

 

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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.

LAB1 a mapping game

LAB2 the Argument principle

LAB3 a potential problem

 

Reference literature:

  1. Learning MATLAB, Tobin A. Driscoll. Provides a brief introduction to Matlab to the one who already knows computer programming. Available as e-book from Chalmers library.
  2. Physical Modeling in MATLAB 3/E, Allen B. Downey
    The book is free to download from the web. The book gives an introduction for those who have not programmed before. It covers basic MATLAB programming with a focus on modeling and simulation of physical systems.

 

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Course summary:

Date Details Due