logo PRZ
Item card
logo WYDZ

Complex Analysis


Some basic information about the module

Cycle of education:
2019/2020
The name of the faculty organization unit:
The faculty Mathematics and Applied Physics
The name of the field of study:
Mathematics
The area of study:
sciences
The profile of studing:
The level of study:
second degree study
Type of study:
full time
discipline specialities :
Applications of Mathematics in Economics
The degree after graduating from university:
master
The name of the module department :
Departament of Discrete Mathematics
The code of the module:
1486
The module status:
mandatory for teaching programme Applications of Mathematics in Economics
The position in the studies teaching programme:
sem: 1 / W30 C45 / 6 ECTS / E
The language of the lecture:
Polish
The name of the coordinator:
Lucyna Trojnar-Spelina, PhD
office hours of the coordinator:
piątek, 9:30-11:00
semester 1:
Krzysztof Piejko, PhD

The aim of studying and bibliography

The main aim of study:
Get aequitancedwith the theory of complex functions, undestanding some analogues and differences with the theory of real functions of one and two variable. Make the easy using the cocepts of complex analysis.

The general information about the module:
The scope of the material concerns the introduction of the concept of the complex function of a real variable and its interpretation and application, complex function of a complex variable, its limits, continuity and derivatives and the concept of holomorphic function and meromorphic function and their properties

Bibliography required to complete the module
Bibliography used during lectures
1 Franciszek Leja Funkcje zespolone PWN Warszawa. 2008
2 J. Chądzyński Wstęp do analizy zespolonej Wydawnictwo Naukowe PWN. 2000
Bibliography used during classes/laboratories/others
1 J. Krzyż Zbiór zadań z funkcji analitycznych Wydawnictwo Naukowe PWN. 2005
2 Bolesław Szafnicki Zadania z funkcji zespolonych PWN Warszawa, Kraków. 1971
Bibliography to self-study
1 A. Ganczar Analiza Matematyczna w zadaniach Wydawnictwo Naukowe PWN. 2010

Basic requirements in category knowledge/skills/social competences

Formal requirements:
The student satisfies the formal requirements set out in the study regulations

Basic requirements in category knowledge:
Basic knowledge of differential calculus of functions of real variables. Basic knowledge of algebra and geometry.

Basic requirements in category skills:
Ability to calculate derivatives and integrals of real functions of one and several variables.

Basic requirements in category social competences:
Awareness of the level of knowledge and skills in mathematics, particularly in complex analysis and the tendency of its widening.

Module outcomes

MEK The student who completed the module Types of classes / teaching methods leading to achieving a given outcome of teaching Methods of verifying every mentioned outcome of teaching Relationships with KEK Relationships with PRK
MEK01 Knows the basic transformation of the complex plane, can describe conics on the complex plane. Knows concept of the sequence and series of complex numbers, including power series. Student can compute limit of the sequence and examine the convergence of the series lectures, tutorials, written test K-W01+
K-W04++
K-U08+
K-U14+
K-K07+
 P7S-KK
P7S-KO
P7S-KR
P7S-UW
P7S-WG
MEK02 knows the concept of derivative and formal derivative of complex functions, can provide some elementary functions in a power series lectures, exercises written test K-W02++
K-W07+
K-U01+
K-U14+
K-K04+
 P7S-KO
P7S-KR
P7S-UW
P7S-WG
P7S-WK
MEK03 Knows the concept of analytic function and holomorphic function and knows their basic properties. It can determine whether the function is analytic or holomorphic lectures, exercises written or oral exam K-W01+
K-W05++
K-U02+
K-U03+
K-U04+
K-U05+
K-U13+
K-U14+
K-K02+
 P7S-KK
P7S-KO
P7S-UK
P7S-UO
P7S-UW
P7S-WG
MEK04 Knows the notion of a meromorphic function, and singular points and residues of functions. Can determine a type of singularity of the selected point. lectures, exercises written or oral exam K-W01+
K-W03+
K-U05+
K-U14+
K-U15+
K-K01+
 P7S-KK
P7S-UK
P7S-UO
P7S-UU
P7S-UW
P7S-WG

The syllabus of the module

Sem. TK The content realized in MEK
1 TK01 Sequences and series of complex numbers. Properties of the complex plane, open sets, closed, consistent. Transformations on the complex plane, translation, rotation, jednokładność, simple and symmetry of the circle. Leaning on płaszcyźnie complex, smooth curve, curve Jordan contour. Equations of staight line, circle, ellipse, hyperbola. W01-W15, C01-C15 MEK01
1 TK02 Sequences and series of complex numbers. Complex functions of a real variable and its derivative. Complex functions. Limit, continuity, real and imaginary part of the complex function. The complex inverse function. Functional sequences and series, power series and Cauchy theorem - Hadamard, Abel, Tauber. Examples of complex function exp z, trigonometric, logarithmic, exponential. Derivatives of complex functions of a complex variable, Cauchy - Riemann derivatives formal interpretation of the derivative, analytic functions (regular), equiangular projection, conformal and homographic W01-W15, C01-C15 MEK02
1 TK03 Integration in the complex domain, ordinary integral, curves integral , the original function. Cauchy's integral theorem and its generalizations, Cauchy's integral formula and its consequences. Expansions of holomorphic function in a power series, zeros of holomorphic functions . Morera theorem. Cauchy inequality.Complete functions, Liouville theorem. The principle of maximum module minimumm principle, Schwarz's Lemma. Fundamental Theorem of Algebra. W01-W15, C01-C15 MEK03
1 TK04 Meromorphic functions. Singularities, residue of the function, the assertion of the residuum, residua of logarithmic derivative. Weierstrass theorem about the distribution of the total functions for the infinite product. Little Theorem of Picard. Distribution of meromorphic functions, Mittag-Leffler theorem. W01-W15, C01-C15 MEK04

The student's effort

The type of classes The work before classes The participation in classes The work after classes
Lecture (sem. 1) The preparation for a test: 10.00 hours/sem.
 contact hours: 30.00 hours/sem.
 complementing/reading through notes: 10.00 hours/sem.
Studying the recommended bibliography: 15.00 hours/sem.
Class (sem. 1) The preparation for a Class: 10.00 hours/sem.
The preparation for a test: 12.00 hours/sem.
 contact hours: 45.00 hours/sem.
 Finishing/Studying tasks: 15.00 hours/sem.
Others: 10.00 hours/sem.
Advice (sem. 1) The preparation for Advice: 3.00 hours/sem.
 The participation in Advice: 5.00 hours/sem.
Exam (sem. 1) The preparation for an Exam: 12.00 hours/sem.
 The written exam: 3.00 hours/sem.

The way of giving the component module grades and the final grade

The type of classes The way of giving the final grade
Lecture The basis for passing the lecture is attendance.
Class To pass classes student should participate in exercise classes. The grade of classes is the arithmetic mean of positive marks of two written tests. It can be increased by 1/2 degree after taking into account the activity of the classes. Each of the tests includes basic tasks, associated with the effect of modular training (I test - MEK01, II test - MEK02) and additional tasks that can be associated with other educational content, implemented in the classroom. The solution of additional tasks allows to get higher than 3.0 mark of the written test.
The final grade

Sample problems

Required during the exam/when receiving the credit
(-)

Realized during classes/laboratories/projects
(-)

Others
(-)

Can a student use any teaching aids during the exam/when receiving the credit : no

The contents of the module are associated with the research profile yes

1 K. Piejko; L. Trojnar-Spelina On (k_1A_1, k_2A_2, k_3A_3)-edge colourings in graphs and generalized Jacobsthal numbers 2025
2 L. Trojnar-Spelina Coefficient Estimates in a Class of Close-to-Convex Functions 2025
3 L. Trojnar-Spelina Estimates of Some Coefficient Functionals for Close-to-Convex Functions 2024
4 M. Nunokawa; J. Sokół; L. Trojnar-Spelina Some results on p-valent functions 2023
5 R. Kargar; L. Trojnar-Spelina Starlike functions associated with the generalized Koebe function 2021
6 M. Nunokawa; J. Sokół; L. Trojnar-Spelina On a sufficient condition for function to be p-valent close-to-convex 2020