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## COMPLEX ANALYSIS

CODE 84039 2017/2018 7 credits during the 3nd year of 8760 Mathematics (L-35) GENOVA MAT/05 GENOVA (Mathematics) 1° Semester

## OVERVIEW

Language: Italian (unless otherwise requested)

This course consists of 60 hours of lectures, including 12 hours of exercises.

The course consists essentially of two parts. The first, whch is the main part, is a standard introduction to complex analysis, from first definitions to the residue theorem. The second part contains a selection of topics, including the theory of Euler's Gamma function.

## AIMS AND CONTENT

### LEARNING OUTCOMES

The couse aims at giving to students the bases of complex analysis. In particular, at the end of the course the students will understand the basic concepts of complex analysis and will be able to use them to solve some problems.

### TEACHING METHODS

The course consists of traditional lectures and exercises sessions. Moreover, exercises will be given to students for independent training.

### SYLLABUS/CONTENT

Power series: formal power series; convergent power series; analytic functions. Complex differentiation: holomorphic functions; Cauchy-Riemann equations; conformal transformations; elementary functions. Complex integration: integration along paths; primitives; Cauchy's theorem. Consequences of Cauchy's theorem: Cauchy's integral formula; holomorphic functions are analytic; further consequences (theorems by Morera and Liouville, mean value, maximum modulus, Weierstrass' convergence theorems). Singularities and residues: Laurent series; behavior near singularities; residue theorem and applications. Miscellanea: zeros and poles of meromorphic functions; Euler's Gamma function; analytic continuation.

V.Villani - Funzioni di una variabile complessa - ES Genova 1971.

H.Cartan - Elementary theory of analytic functions of one or several complex variables - Dover 1995.

R.Remmert - Theory of complex functions - Springer 1989.

S.Lang - Complex analysis - Springer 1999.

C.Presilla - Elementi di analisi complessa, 2a edizione - Springer 2014 (for exercises).

M.R.Spiegel - Variabili complesse - ETAS Libri 1974 (for exercises).

## LESSONS

### TEACHING METHODS

The course consists of traditional lectures and exercises sessions. Moreover, exercises will be given to students for independent training.

### LESSONS START

The class will start according to the academic calendar.

### Class schedule

All class schedules are posted on the EasyAcademy portal.

## EXAMS

### EXAM DESCRIPTION

Traditional: written and oral examination.

### ASSESSMENT METHODS

Evaluation of written and oral parts; during the oral examination, students may present a topic, chosen from an ad hoc list, prepared in advance.

### Exam schedule

Date Time Location Type Notes
11/01/2018 09:00 GENOVA Scritto
11/01/2018 14:30 GENOVA Orale
08/02/2018 09:00 GENOVA Scritto
08/02/2018 14:30 GENOVA Orale
12/06/2018 14:30 GENOVA Scritto
13/06/2018 14:30 GENOVA Orale
10/07/2018 14:30 GENOVA Scritto
11/07/2018 14:30 GENOVA Orale
11/09/2018 14:30 GENOVA Scritto
12/09/2018 14:30 GENOVA Orale

### FURTHER INFORMATION

Teaching style: in presence