### Further Mathematics,Part 3

Major: Micro and Nanosystems of the Internet of Things
Code of subject: 6.153.03.O.018
Credits: 6.00
Department: Applied Mathematics
Lecturer: PhD. Andrij P. Senyk
Semester: 3 семестр
Mode of study: денна
Мета вивчення дисципліни: Mastering the basic methods of the theory of functions of a complex variable and operational calculus, which are necessary for studying the courses of mathematical physics equations, probability theory, optimization methods, mathematical methods of modeling complex systems, and others. Mastering the basic methods of probability theory and mathematical statistics, mastering the skills of their practical application for the study of mass random phenomena and processes
Завдання: The study of an educational discipline involves the formation of competencies in students of education: • general competencies: ZK2 – knowledge and understanding of the subject area and understanding of professional activity. ZK5 - skills in using information and communication technologies. ZK6.- the ability to learn and master modern knowledge. ZK7 - the ability to search, process and analyze information from various sources. • professional competences: FK1 – the ability to use knowledge and understanding of scientific facts, concepts, theories, principles and methods for the design and application of electronics devices, devices and systems. FC5 – the ability to apply appropriate mathematical, scientific and technical methods, modern information technologies and computer software, skills in working with computer networks, databases and Internet resources to solve engineering problems in the field of electronics.
Learning outcomes: • Know: basic concepts of the theory of complex variable, their differentiation and integration, Taylor series and Laurent, elements of the theory of residues, basic properties and original images and their applications; basic concepts and definitions of the theory of probability and mathematical statistics; Methods of mass random phenomena; methods of processing and analysis of statistical data; • be able to: apply the obtained knowledge for solving mathematical problems of general and special disciplines; apply acquired theoretical knowledge to practical research processes and phenomena of nature; • have an idea: the application of complex variable theory, and theory of probability and mathematical statistics for solving problems of general and special disciplines
Required prior and related subjects: Mathematical analysis
Summary of the subject: Complex numbers and operations on them. Functions of complex variable and their image. Cauchy-Riemann conditions. Geometric content module and the original argument. Integration of functions of complex variable. Several Laurent. The theory of residues. Laplace transform and its properties. Subject and basic concepts of probability theory. Classification of random events. The algebra of random events. Law of total probability. Bayes Formula. The scheme of independent trials. The problems of mathematical statistics.
Опис: Topic 1. Complex numbers and operations on them Lecture 1. Complex numbers: definitions and properties. Geometric content of complex numbers. Complex number argument. Trigonometric and exponential forms of a complex number. Operations with complex numbers. . Lecture 2. Operations on complex numbers in trigonometric and exponential forms. Moivre's formula. Polynomials Factoring. An infinitely distant point. Extended complex plane. The region and its border. Topic 2. Functions of a complex variable (FCV) Lecture 3. The concept of a function of a complex variable. FKZ border. Differentiation of FKZ. Properties of differentiated FCZ. Cauchy-Riemann conditions. Analytical FCZ and their properties... . Lecture 4. Concept of analytical function. The geometric content of the module and the argument of the derivative... . Examples of some elementary functions of a complex variable and their properties. The concept of conformal mapping. Lecture 5. Integral of a function of a complex variable. The concept of a complex integral. Cauchy's integral formula Lecture 6. Series of functions of a complex variable. Basic concepts about series with complex terms. Row of Laurent. Analytical continuation Topic 3. Special points and their classification. Leftovers Lecture 7. Zeros and isolated singular points. Special points and their classification. Classification and study of singular points of a unique analytic function. Lecture 8.. Surpluses and their application. Remainders of single-valued analytical functions. Basic theorems of the theory of remainders. Application of the theory of remainders to the calculation of integrals. Topic 4. Fundamentals of operational calculus Lecture 9. Operational calculus: Laplace transform and its properties. Differentiation and integration of originals and images. Laplace transform inverse formula. Basic formulas and theorems. Lecture 10. Finding the original by its image. Solving linear differential equations and systems by the method of operational calculus. Topic 5. Theory of probability Lecture 11. Subject and basic concepts of probability theory. Classification of random events. Algebra of random events. Classical, statistical and geometric definition of probability. Basic formulas and rules of combinatorics, their application in probability theory Lecture 12. The probability of the sum of incompatible and arbitrary random events. Conditional probabilities. The probability of the product of random events. Independent and dependent events. Formula of total probability. Bayes formulas. Lecture 13. Scheme of independent tests. Y. Bernoulli's formula and its consequences. Probability distributions in the Bernoulli scheme. Poisson's theorem, local and integral theorems of Moivre-Laplace. Topic 6. Fundamentals of mathematical statistics. Lecture 14. Random variables. Discrete random variables, their distributions and numerical characteristics. Continuous random variables. Lecture 15. Distribution function. Distributions and numerical characteristics of continuous random variables. Definition and properties of multidimensional distribution functions.
Assessment methods and criteria: Current control (30%): practical tasks, reports on laboratory works, oral presentations. Final control (70%): exam.
Критерії оцінювання результатів навчання: A 100-point national rating scale is used to evaluate educational achievements during the semester Current control (PC) - 30 Of them: Surveys at practical classes 10 Control work 1 10 Control work 2 10 Total for PC 30 Examination control 70 Total for discipline 100
Порядок та критерії виставляння балів та оцінок: 100–88 points – (“excellent”) is awarded for a high level of knowledge (some inaccuracies are allowed) of the educational material of the component contained in the main and additional recommended literary sources, the ability to analyze the phenomena being studied in their interrelationship and development, clearly, succinctly, logically, consistently answer the questions, the ability to apply theoretical provisions when solving practical problems; 87–71 points – (“good”) is awarded for a generally correct understanding of the educational material of the component, including calculations, reasoned answers to the questions posed, which, however, contain certain (insignificant) shortcomings, for the ability to apply theoretical provisions when solving practical tasks; 70 – 50 points – (“satisfactory”) awarded for weak knowledge of the component’s educational material, inaccurate or poorly reasoned answers, with a violation of the sequence of presentation, for weak application of theoretical provisions when solving practical problems; 49-26 points - ("not certified" with the possibility of retaking the semester control) is awarded for ignorance of a significant part of the educational material of the component, significant errors in answering questions, inability to apply theoretical provisions when solving practical problems; 25-00 points - ("unsatisfactory" with mandatory re-study) is awarded for ignorance of a significant part of the educational material of the component, significant errors in answering questions, inability to navigate when solving practical problems, ignorance of the main fundamental provisions.
Recommended books: 1. А. П. Сеник, В. В. Пабирівський, О. М. Уханська, Л. Д. Озірковський Комплексне числення в інфокомунікаціях та електронній інженерії: навчальний посібник / – Львів: "Растр-7", 2021. – 164 c 2. Рудавський Ю.К., Костробій П.П., Уханська Д.В. та ін. Теорія функцій комплексної змінної. Інтегральні перетворення Фур’є та Лапласа. – Львів. 2007. 3. Костробій П.П., Уханська Д.В., Сало Т.М., Уханська О.М., Маркович Б.М. Елементи теорії функцій комплексної змінної. Перетворення Фур’є та Лапласа. Збірник задач і вправ. Електронний навчальний підручник. – Львів. 2010.. 4. Рудавський Ю.К., Костробій П.П. та ін. Збірник задач з теорії ймовірностей. – Львів, 2001. 5. Електронний навчально-методичний комплекс «Вища математика. Частина 3» /укладачі – ., СеникА.П., Пабирівський В.В., Уханська О.М., Гладун В.Р. – Сертифікат №03968. Адреса розміщення: http://vns.lpnu.ua/course/view.php?id=5344, Номер і дата реєстрації:№ E41-141-349/2021 від 06.05.2021 р.
Уніфікований додаток: An important tool for the implementation of the inclusive educational policy at the University is the Program for improving the qualifications of scientific and pedagogical workers and educational and support staff in the field of social inclusion and inclusive education. Contact at: St. Karpinsky, 2/4, 1st floor, room 112 E-mail: nolimits@lpnu.ua Websites: https://lpnu.ua/nolimits https://lpnu.ua/integration
Академічна доброчесність: The policy regarding the academic integrity of the participants of the educational process is formed on the basis of compliance with the principles of academic integrity, taking into account the norms "Regulations on academic integrity at the Lviv Polytechnic National University" (approved by the academic council of the university on June 20, 2017, protocol No. 35).