# Complex analysis and Riemann Surfaces

Course Code : | 1001WETCRO |

Study domain: | Mathematics |

Academic year: | 2017-2018 |

Semester: | 2nd semester |

Sequentiality: | Min. 8/20 for Calculus and Metric spaces and calculus. |

Contact hours: | 60 |

Credits: | 6 |

Study load (hours): | 168 |

Contract restrictions: | No contract restriction |

Language of instruction: | Dutch |

Exam period: | exam in the 2nd semester |

Lecturer(s) | David Eelbode |

### 1. Prerequisites *

an active knowledge of

- Dutch

The student should have some notions about he following topics: convergence of series, limit theorems, continuity, power series (all treated in the courses mentioned below).

*Sequentiality

Calculus and Sets (1BWIS-032) AND Metric spaces and differential calculus (1BWIS-042)

### 2. Learning outcomes *

- being able to reconstruct proofs from the course notes
- being able to understand a given proof for theorems which were not covered during the classes
- being able to solve exercises related to the theory

### 3. Course contents *

The following topics will be treated during the classes:

* the complex numbers and the Riemann sphere

* the notion of a holomorphic function and its properties (including the relation with conformal mappings and rotational invariance)

* elementary complex functions

* the integral representation formulae for complex holomorphic functions (Cauchy-Goursat and Cauchy)

* power series representations (Taylor and Laurent series)

* residue calculus and applications

* harmonic functions in the plane and their link with holomorphic functions

* elementary notions from the theory of Riemann surfaces

### 4 International dimension*

### 5. Teaching method and planned learning activities

Personal work

### 6. Assessment method and criteria

### 7. Study material *

#### 7.1 Required reading

1. Course text (in Dutch)

2. extra slides may be handed out during classes (possibly in English)

**7.2 Optional reading**

The following study material can be studied voluntarily :1.Rudin, Walter . Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. xiv+416 pp. ISBN: 0-07-054234-1

2. Forster, Otto . Lectures on Riemann surfaces. Translated from the 1977 German original by Bruce Gilligan. Reprint of the 1981 English translation.Graduate Texts in Mathematics, 81. Springer-Verlag, New York, 1991. viii+254 pp. ISBN 0-387-90617-7.

3. Jost, Jürgen . Compact Riemann surfaces. An introduction to contemporary mathematics.Third edition. Universitext. Springer-Verlag, Berlin, 2006. xviii+277 pp. ISBN: 978-3-540-33065-3.

4. Gunning, R. C. Lectures on Riemann surfaces. Princeton Mathematical Notes Princeton University Press, Princeton, N.J. 1966 iv+254 pp.

### 8. Contact information *

David Eelbode: e-mail: david.eelbode@ua.ac.be (G. 312)