### Complex Analysis

#### Course Description

Functions of complex variables. Analytic functions. Multiple-valued functions. Elementary functions. Conformal mappings. Integral of functions of complex variable.Taylor and Laurent series. Zeroes and singular points. Poles of analytic functions. Residue theorem and applications. Inverse of Laplace transform. Orthogonal polynomials. Generating functions, differential equations. Special functions. Gamma and beta functions.

#### General Competencies

Learning advanced complex calculus and it’s use in various applications: conformal mappings, integral transforms, orthogonal polynomials and special functions.

#### Learning Outcomes

1. perform basic operations with complex numbers in various representations
2. understand the concept of elemenary function and their interconnection
3. apply the technique of conformal mappings in analyze of simple models
4. use the technique of power series expansion in the calculus with elementary functions
5. apply the technique of residuum, especially in the inverse integral transformation
6. understand the concept of orthogonal systems and expansion into series of orthogonal polynomials
7. use the properties of gamma function in various situations

#### Forms of Teaching

Lectures

Lectures are organized through two cycles. The first cycle consists of 7 weeks of classes and mid-term exam, a second cycle of 6 weeks of classes and final exam. Classes are conducted through a total of 15 weeks with a weekly load of 3 hours.

Exams

Mid-term exam in the 8th week of classes and final exam in the 15th week of classes.

Consultations

Consultations are held one hour weekly according to arrangement with students.

Continuous Assessment Exam
Quizzes 0 % 20 % 0 % 20 %
Mid Term Exam: Written 0 % 40 % 0 %
Final Exam: Written 0 % 40 %
Exam: Written 0 % 80 %
##### Comment:

If student hasn't attended quizzes or wants to improve obtained results, he can take a written exam for 100% of grade.

#### Week by Week Schedule

1. Complex plane. Regions. Contours. Sequences and series, Convergence.
2. Functions of complex variables. Differentiability. Analytic functions. Cauchy-Riemann conditions. Harmonic functions.
3. Conformal mappings. Multiple-valued functions. The power and root functions. Branch point.
4. Exponential and logarithmic function. Trigonometric and other elementary functions.
5. Bilinear transformation. Conform Mappings. Examples of region mappings.
6. Integral of functions of complex variable. Independence of path.
7. Cauchy theorem. Cauchy integral formulas. Applications.
8. Exam
9. Taylor series. Zeroes of analytic functions
10. Laurent series. Singular points and poles of analytic functions.
11. Residue theorem and applications. Inverse Laplace transform. Applications.
12. Special functions. Gamma and beta functions.
13. Bernoulli polynomials. Generating functions. Orthogonal polynomials and corresponding differential equations.
14. Legendre polynomials. Hermite and Čebišev polynomials. Applications.
15. Exam.

#### Study Programmes

Control Engineering and Automation -> Electrical Engineering and Information Technology (Profile)

Electrical Engineering Systems and Technologies -> Electrical Engineering and Information Technology (Profile)

Electrical Power Engineering -> Electrical Engineering and Information Technology (Profile)

Electronic and Computer Engineering -> Electrical Engineering and Information Technology (Profile)

Electronics -> Electrical Engineering and Information Technology (Profile)

Information Processing -> Information and Communication Technology (Profile)

Telecommunication and Informatics -> Information and Communication Technology (Profile)

Wireless Technologies -> Information and Communication Technology (Profile)

Software Engineering and Information Systems -> Computing (Profile)

Computer Engineering -> Computing (Profile)

Computer Science -> Computing (Profile)

#### Literature

N. Elezović (2010.), Funkcije kompleksne varijable, Element
D. G. Zill, P. D. Shanahan (2003.), A First Course in Complex Analysis with Applications, Jones and Bartlett
A. D. Wunsch (1994.), Complex variables with Applications, Addison-Wesley

#### Lecturers in Charge

4 ECTS
L1 English Level
L1 e-Learning
45 Lecturers
0 Exercises
0 Laboratory exercises