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
01	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
02	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
03	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
04	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

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
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	Final mark, after completing all the MEKs is the arithmetic mean of marks of the exercices and exam rounded to the nearest mark permitted by the regulations of studies. Written or oral exam includes basic tasks related to MEK03 and MEK04 and additional tasks that can be associated with other educational content, implemented in the classroom. Solutions of the additional tasks is the basis for obtaining the mark of the exam, higher than 3.0.

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; J. Sokół; L. Trojnar-Spelina	Some results on p-valent functions	2023
2	R. Kargar; L. Trojnar-Spelina	Starlike functions associated with the generalized Koebe function	2021
3	M. Nunokawa; J. Sokół; L. Trojnar-Spelina	On a sufficient condition for function to be p-valent close-to-convex	2020
4	A. Ebadian; R. Kargar; L. Trojnar-Spelina	Further results for starlike functions related with Booth lemniscate	2019
5	L. Trojnar-Spelina; I. Włoch	On a new type of the companion Pell numbers	2019
6	L. Trojnar-Spelina; I. Włoch	On generalized Pell and Pell-Lucas numbers	2019
7	R. Kargar; L. Trojnar-Spelina	Some applications of differential subordination for certain starlike functions	2019