The aim of studying and bibliography

The main aim of study: Extending the concept of real numbers - the concept of complex numbers and algebraic operations on complex numbers, analogies and differences in terms of operations on real numbers. Extending the concept of complex functions of a complex variable functions. The ability to calculate derivatives oran borders. Interpretations of these concepts.

The general information about the module: The module consists of 30 hours of lectures and 15 hours of exercises. It ends with an exam

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	Bolesław Szafnicki	Zadania z funkcji zespolonych	PWN Warszawa, Kraków.	1971
2	J. Krzyż	Zbiór zadań z funkcji analitycznych	Wydawnictwo Naukowe PWN,.	2005

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 the concepts of limit, continuity and differentiation for real functions of one and two real variables. Basic information of analytic geometry in the plane,

Basic requirements in category skills: It can set the boundaries of sequences and functions. It can calculate the derivatives and partial derivatives of elementary functions.

Basic requirements in category social competences: Is aware of the level of their knowledge and skills in mathematics, particularly in complex analysis and the need to raise.

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
01	Familiar with the concept of a complex number and its relationship to real numbers. It can perform actions in the set of complex numbers, describe and represent collections on the complex plane. Familiar with the concept of numerical sequences and series (in the set of complex numbers)	lectures, tutorials	colloquium or exam	K_W01+ K_U01+ K_K01+	P6S_KK P6S_UK P6S_WK
02	Familiar with the basic functions of complex, in the fields of real and complex, familiar with the term limit and continuity of complex functions.	lectures, tutorials	colloquium or exam	K_W02+ K_W03+ K_U01+ K_K01+	P6S_KK P6S_UK P6S_WG P6S_WK
03	Familiar with the concept of a complex derivative and formal derivative of complex functions. It can set the derivatives of .basic complex functions.	lectures, toutorial	colloquium or exam	K_W04++ K_W05+ K_K01+	P6S_KK P6S_WG P6S_WK

Attention: Depending on the epidemic situation, verification of the achieved learning outcomes specified in the study program, in particular credits and examinations at the end of specific classes, can be implemented remotely (real-time meetings).

The syllabus of the module

Sem.	TK	The content	realized in	MEK
4	TK01	The body of complex numbers, complex numbers, Gauss' plane,the sets on the complex plane, conic equation in the form of complex, composite, sequences and series.	W01-W15, C01-C15	MEK01
4	TK02	Complex functions of a real variable, smooth curves, complex functions of a complex variable: complex polynomials, roots of polynomials, Fundamental Theorem of Algebra. Limits and continuity of functions of a complex, variables.	W01-W15, C01-C15	MEK02
4	TK03	Real and imaginary part of the function, examples of complex functions: Exponential, trigonometric, logarithmic and exponential. Differentiation of complex functions, derivative of a complex function Cauchy-Riemann conditions, formal derivatives. - Information on holomorphic functions.	W01-W15, C01-C15	MEK03

The student's effort

The type of classes	The work before classes	The participation in classes	The work after classes
Lecture (sem. 4)		contact hours: 30.00 hours/sem.	complementing/reading through notes: 5.00 hours/sem. Studying the recommended bibliography: 10.00 hours/sem.
Class (sem. 4)	The preparation for a Class: 5.00 hours/sem. The preparation for a test: 5.00 hours/sem.	contact hours: 15.00 hours/sem.	Finishing/Studying tasks: 4.00 hours/sem.
Advice (sem. 4)	The preparation for Advice: 1.00 hours/sem.	The participation in Advice: 1.00 hours/sem.
Exam (sem. 4)	The preparation for an Exam: 7.00 hours/sem.	The written exam: 2.00 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	written exam
Class	test, activity
The final grade	The mean of grades from exam and exercise

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	M. Nunokawa; K. Piejko; J. Sokół	Applications of Jack’s lemma	2024
2	K. Piejko; J. Sokół; K. Trąbka-Więcław	Coefficient bounds in the class of functions associated with Sakaguchi\'s functions	2023
3	K. Piejko; J. Sokół; K. Trąbka-Więcław	On q-starlike functions	2023
4	K. Piejko; J. Sokół	On convolution and q-calculus	2020
5	K. Piejko; J. Sokół; K. Trąbka Więcław	On q-Calculus and Starlike Functions	2019