Course detail

# Functions of a Complex Variable

Functions of a complex variable. Elementary functions. Limit of a function of a complex variable. Continuity of a function of a complex variable. Derivative of a function of a complex variable. Integral of a function of a complex variable. Power series and Taylor series. Laurent series. Isolated singularities. Residues. Conform mapping. Entire functions. Principle of maximum of the modulus. Meromorphic functions.

Learning outcomes of the course unit

Students get acquainted with elementary functions in complex domain. They learn to look for their limits and to investigate their continuity. They are introduced to the derivative of a function of a complex variable and to Cauchy-Riemann conditions. A next topic is the integral of a function of a complex variable, Cauchy´s integral theorem and Cauchy's integral formula. As an application of residues, some integrals are computed with aid of the residue theorem. By means of linear, bilinear, power, exponential, and logarithmic functions, participants of the course will learn to map regions in Gaussian plane bijectively and conformally.

Prerequisites

Differential and integral calculus.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Bajpai, A.C., Mustoe, L.R., Walker, D.: Advanced Engineering Mathematics. John Wiley & Sons, Chichester, 1990.
Druckmuller, M., Ženíšek, A.: Funkce komplexní proměnné. PC-DIR real, Brno, 2001.
Černý, I._: Anylýza v komplexním oboru. Academia, Praha, 1983.
Noguchi, J.: Introduction to Complex Anylysis. AMS, Providence, 1997.
Druckmuller, M., Svoboda, K.: Vybrané statě z matematiky I. FS VUT, Brno, 1986.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline.

Assesment methods and criteria linked to learning outcomes

The course is finished by an oral examination. The examiner verifies the knowledge of definitions, theorems, and algorithms and the ability of their use in concrete applications.

Language of instruction

Czech, English

Work placements

Not applicable.

Aims

The course aims to acquaint the students with the theory of functions of a complex variable. The explanation points via some indispensable concepts to the study of holomorphic functions. A next important notion is the integral of a function of a complex variable including the solution of the problem of independence of the path. Holomorphic functions can be developed, depending on the type of region, into Taylor series or, more generally, into Laurent series. Then, the classification of isolated singularities and residues with the residue theorem follow. The topic of conformal mapping gives a geometric application of holomorphic functions. The final part of the subject studies entire and meromorphic functions. As an application, a part concerning Hurwitz polynomials is attached.

Specification of controlled education, way of implementation and compensation for absences

Attendance at lectures is recommended. The lessons are planned on the basis of a weekly schedule. It is possible to study individually according to the recommended literature with the use of consultations.

Classification of course in study plans

• Programme D-APM-K Doctoral, 1. year of study, summer semester, 0 credits, recommended

• Programme D4P-P Doctoral

branch D-APM , 1. year of study, summer semester, 0 credits, recommended

#### Type of course unit

Lecture

20 hours, optionally

Teacher / Lecturer

Syllabus

Week 1: Functions of a complex variable.
Week 2: Elementary functions.
Week 3: Limit and continuity of a function of a complex variable.
Week 4: Derivative of a function of a complex variable, holomorphic functions.
Week 5: Integral of a function of a complex variable.
Week 6: Cauchy´s integral theorem and Cauchy´s integral formula.
Week 7: Power series.
Week 8: Taylor series, Laurent series.
Week 9: Isolated singularities, residues.
Week 10: Conformal mapping.
Week 11: Entire functions.
Week 12: Priciple of maximum of the modulus.
Week 13: Meromorphic functions.