# Course Details

#### Name:

**Complex analysis / Complex analysis**

### Basic information

##### Credits:

*Master Degree in Mathematical Engineering:* 6 Ects (b)

##### Degree(s):

Elective 1^{st} year Master Degree in Mathematical Engineering curriculum Comune

##### Language:

English

### Course Objectives

Knowledge of basic topics of complex analysis: elementary functions of complex variable, differentiation, integration and main theorems on analytic functions . Ability to use such knowledge in solving problems and exercises

### Course Content

- Complex numbers. Sequences. Elementary functions of complex numbers. Limits, continuity. Differentiation. Analytic functions. Armonic functions
- Contour integrals. Cauchy's Theorem. Cauchy's integral formula. Maximum modulus theorem. Liouville's theorem. Morera theorem.
- Series representation of analytic functions. Taylor's theorem. Laurent's series and classification of singularities
- Calculus of residues. The residue theorem. Application in evaluation of integrals on the real line and Principal Value. The logarithmic residue, Rouche's theorem.
- Fourier transform for L^1 functions. Applications. Fourier transform for L^2 functions. Plancherel theorem.
- Laplace transform and applications.

### Prerequisites and Learning Activities

Knowledge of all topics treated the Mathematical Analysis courses in the first and second year : real function of real variables, limits, differentiation, integration; sequences and series of funcions; ordinary differential equations

### Teaching Methods

**Language**: English

theoretical lectures and exercises

### Assessment Methods and Criteria

Written exam and oral exam

### Textbooks

- J.E. Marsden, M.J. Hoffman,
**Basic complex analysis**. Freeman New York. * *
- W. Rudin,
**Real and complex analysis**. Mc Graw Hill. * *

### Course page updates

This course page is available (with possible updates) also for the following academic years:

*Course information last updated on: 14 settembre 2017, 18:18*