## 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.

## Assignments

## 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.

After the exam has been graded, you can see your results in Ladok by logging on to your Student portal.

**At the annual (regular) examination: **

When it is practical, a separate review is arranged. The date of the
review will be announced here on the course homepage. Anyone who can not
participate in the review may thereafter retrieve and review their exam
at the Mathematical
Sciences Student office. Check that you have the right grades and
score. Any complaints about the marking must be submitted in writing at
the office, where there is a form to fill out.

** At re-examination: **

Exams are reviewed and retrieved at the Mathematical
Sciences Student office. Check that you have the right grades and
score. Any complaints about the marking must be submitted in writing at
the office, where there is a form to fill out.

## Old exams

Exam,
Aug 24, 2018 Solutions

Exam,
Jan 3, 2018 Solutions

Exam,
Oct 27, 2017 Solutions

Exam,
Aug 18, 2017 Solutions

Exam,
Jan 3, 2017 Solutions

Exam,
Oct 28, 2016 Solutions

Exam,
Aug 19, 2016 Solutions

Exam,
Jan 5, 2016 Solutions