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Methods of mathematical physics 3 semester

ОК.16

Обов’язкова дисципліна для ОП

First

2023/2024

3 Semester

3

The student must know the basic concepts and methods of complex analysis, in particular, the theory of analytical functions, the theory of residues, methods of conformal mappings, basic concepts and methods of equations of mathematical physics, in particular, the method of separation of variables and the method of generalized functions.

Full-time form

Successful mastering of courses: "Calculus", "General algebra", "Differential equations".

1. Fundamentals of theory of functions of complex variable. Complex numbers, functions of a complex variable, analytical functions, integral of analytical functions, power series of analytical functions, singular points of analytical functions. 2. Application of the theory of functions of a complex variable. Residues and their derivation, calculation of integrals using residues, flat vector fields and physical content of analytical function, conformal mappings, Dirichlet problem, mapping of polygons. 3. Method of variable separation. Classification of equations of mathematical physics, diffusion equation, wave equation, Laplace equation, Helmholtz equation, spherical functions, cylindrical functions. 4. Method of generalized functions. Definitions and properties of generalized functions, fundamental solution of linear differential equations, fundamental solution of equations of mathematical physics, quaternions and Maxwell's equation.

[1] Д. Д. Шека. Комплексний аналіз (в прикладах і задачах). - К.: 2021 [2] M. A. Lavrentiev, B. V. Shabat. "Methods of the theory of complex variable functions." M.:Phismatgiz, 1973. [3] Є. Д. Бiлоколос, Л. Л. Зайцева, Д. Д. Шека, «Збiрник задач з комплексного аналiзу. Частина I. Функцiї комплексної змiнної». – К.: 2013. [4] Є. Д. Білоколос, Д. Д. Шека, «Збірник задач з комплексного аналізу». – К: 2004. [5] A. G. Sveshnikov, A. N. Tikhonov, "The Theory of Functions of a Complex Variable", 1982, URL: https://archive.org/details/SveshnikovTikhonovTheTheoryOfFunctionsOfAComplexVariable [6] B. A. Fuchs, B. V. Shabat, "Functions of a Complex Variable and Some of Their Applications", Pergamon, 2014; https://doi.org/10.1016/C2013-0-01663-5. [7] Wegert Elias. Visual Complex Functions: An Introduction with Phase Portraits. Springer, 201

Lectures, practical classes, individual independent work

2 written modular tests (MTs) in semester 3 (s3) on practical classes; 2 written MTs in s4. MT is credited if the student scored at least 5 points for the module. 25 points (max) is awarded for each MT. Activity in practical classes is estimated at 10 points (max). According to the results of the semester assessment, a student can 60 points (max). Conditions for admission to the test in s3: student must have passed all the MTs and score at least 36 points during the semester. The condition for admission to the test(s3)/exam(s4) is students receive not less than the critical-calculated minimum of 36 points/semester. Students who scored a total of less than 36 points/semester must write an additional test for the required threshold for admission. Final assessment in s3/s4 in the form of test/exam: written and oral (max 40 points). The exam consists of 3 questions: theoretical question (max 15 points), test example on this question (max 5 points); 2 practical tasks (max 10 points).

Ukrainian