SYLLABUS
COURSE TITLE
FACULTY/INSTITUTE
COURSE CODE
DEGREE PROGRAMME
COMPLEX ANALYSIS
FACULTY OF MATHEMATICS AND NATURAL SCIENCES
FIELD OF STUDY MATHEMATICS
DEGREE LEVEL 2
FORMA
STUDIÓW/STUDY
MODE STATIONARY
MATHEMATICS
SECOND DEGREE
FULL-TIME
COURSE FORMAT
YEAR AND SEMESTER
NAME OF THE TEACHER
BASIC
1 YEAR, 1 SEMESTER
DR HAB. STANISLAWA KANAS
COURSE OBJECTIVES
THE MAIN AIM OF THE COURSE IS TO introduction to the basic theory of complex analytic
functions and some applications, in order to get acquainted with a number of methods and
techniques applicable to other parts of mathematics, engineering or economics. The aim of
the course is to teach the principal techniques and methods of analytic function theory.
PREREQUISITES
LEARNING OUTCOMES
Calculus I, II, III
KNOWLEDGE: DEFINE FUNDAMENTAL TOPOLOGICAL CONCEPTS IN
THE CONTEXT OF THE COMPLEX PLANE, AND DEFINE AND
CALCULATE LIMITS AND DERIVATIVES OF FUNCTIONS OF A
COMPLEX VARIABLE. Represent analytic functions as power
series on their domains. Define a branch of the complex
logarithm. Classify singularities and find Laurent series for
meromorphic functions. State and prove fundamental results,
including: Cauchy’s Theorem and Cauchy’s Integral Formula,
the Fundamental Theorem of Algebra. Use them to prove
related results. Define a contour integrals. Define definite
integrals on the real line using the Residue Theorem.
SKILLS: DETERMINE SEVERAL REPRESENTATION OF COMPLEX
NUMBER. CALCULATE THE LIMIT OF A SEQUENCE AND A COMPLEX
FUNCTION. PROVE THAT THE SEQUENCE IS DIVERGENT. FIND
COMPLEX DERIVATIVE AND A FORMAL DERIVATIVE. FIND AN
IMAGE OF A REGION UNDER A CONFORMAL MAPPING. KNOWS
FUNDAMENTAL FUNCTIONS OF ONE COMPLEX VARIABLE.
CALCULATE THE COMPLEX INTEGRAL ALONG THE PATH. APPLY
CAUCHY THEOREM AND CAUCHY FORMULA. DETERMINE THE
RESIDUES, APPLY THEM TO INTEGRAL CALCULUS.
FINAL COURSE OUTPUT - SOCIAL COMPETENCES KNOWS THE
LIMITATIONS OF THEIR KNOWLEDGE AND UNDERSTAND THE THE
NEED FOR FURTHER EDUCATION. ABILITY TO FIND RELEVANT
INFORMATION AND THEIR APPLICATIONS. ABILITY TO FIND THEIR
PLACE IN THE GROUP
COURSE ORGANISATION –LEARNING FORMAT AND NUMBER OF HOURS
LECTURES,CLASSES - 60 HOURS
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COMPLEX NUMBERS AND THEIR BASIC GEOMETRY. STEREOGRAPHIC PROJECTION (3)
SEQUENCES AND COMPLEX NUMBER SERIES (3)
COMPLEX AND FORMAL DERIVATIVES (4)
ANALYTIC FUNCTIONS, POWER SERIES, CONFORMAL MAPPING (6)
CAUCHY THEOREM, INTEGRAL FORMULA (3)
COMPLEX INTEGRATION, SINGULARITIES, RESIDUES (3)
 SERIES OF ANALYTIC FUNCTIONS, TAYLOR SERIES, LAURENT SERIES (4)
 APPLICATION OF RESIDUES TO THE CALCULATION OF INTEGRALS (2)
 HARMONIC FUNCTIONS.(2)
COURSE DESCRIPTION
The course will cover material: Complex numbers, analytic functions (limits, continuity,
derivatives, Cauchy-Riemann equations, analytic functions, harmonic functions), Elementary
functions (exponential, logarithm, complex exponents, trigs, hyperbolic functions), Integrals
(definite integrals, contour integrals, antiderivatives, Cauchy theorem, Cauchy integral
formula, Liouville's theorem, fundamental theorem of algebra, maximum modulus
principle), Series (sequences, convergence of series, Taylor series, Laurent series, absolute
and uniform convergence, power series techniques), Residues and poles (residues, Cauchy's
residue theorem, residue at infinity, zeros of analytic functions) .
METHODS OF INSTRUCTION
REQUIREMENTS AND ASSESSMENTS
ORAL COMUNICATION, SOLVING TASKS, INDIVIDUAL WORK
THE STUDENT KNOWS THE BASIC FUNCTIONS OF ONE
COMPLEX VARIABLE, STATE THE CAUCHY-RIEMMAN
EQUATION AND DETERMINE WHETHER THE FUNCTION IS
ANALYTIC, STATE A REGION OF CONVERGENCE OF A
SEQUENCE AND A GEOMETRIC SERIES. CALCULATE
CONTOUR INTEGRAL USING DIFFERENT METHODS.
CALCULATE THE RESIDUES, FIND THE TAYLOR AND
LAURENT SERIES OF A FUNCTION.
GRADING SYSTEM
GRADING SCORE – 3.0 FOR 50 - 60%, 3.5 FOR 61 - 70
%, 4.0 FOR 71 – 80%, 4.5 FOR 81 – 90%, 5.0 FOR 91 –
100 %
TOTAL STUDENT WORKLOAD
NEEDED TO ACHIEVE EXPECTED
LEARNING OUTCOMES EXPRESSED
IN TIME AND ECTS CREDIT POINTS
LANGUAGE OF INSTRUCTION
INTERNSHIP
175 HOURS – 7 ECTS
ENGLISH
MATERIALS
PRIMARY OR REQUIRED BOOKS/READINGS:
1. A guide to complex variables, Steven G. Krantz,
McGraw-Hill Science/Engineering/Math; 2007
2. Complex analysis, Lars Ahlfors, McGraw-Hill
Science/Engineering/Math; 3 edition, 1979.
3. Real and complex analysis, Walter Rudin, McGrawHill Science/Engineering/Math; 3 edition, 1986.
4. Function Theory of One Complex Variable, Robert
E. Greene, Steven G. Krantz, AMS, 3 edition, 2006.
SUPPLEMENTAL OR OPTIONAL BOOKS/READINGS:
1. Dynamics in One Complex Variable , J. Milnor, (3rd
ed.), Princeton U. Press. 2006.
2. Complex dynamics, L. Carlesson, Th. Gamelin,
Springer; Corrected edition 1996.