Математика. Том 3. Избранные главы для магистерских программ. Mathematics. Volume 3. Selected chapters for the Master's Degree: Учебник для вузов

Книга адресована студентам, которые рискуют обогатиться знаниями по дополнительным разделам математики, предусмотренным магистерскими программами как в болонской, так и в англо-американской системах высшего профессионального образования. Выбор дополнительных разделов и отдельных глав неявно предусмотрен для потребительской аудитории технологических и инженерных направлений. Книга также будет полезна преподавателям не только инженерно-технических, но и гуманитарных специальностей. При изложении материала, в отличие от большинства подобных учебников, автор стремился удовлетворить требования диалектической теории познания, получившей обновленное название сингапурской математики. Отличительной особенностью предлагаемой книги является то, что изложение сопровождается множеством примеров, вопросов для самоконтроля и заданий для упражнений. Еще одной отличительной чертой книги является отсутствие абстракций из истории математики, которыми часто напичканы учебники по математике в ущерб строгости формулировок теорем, красоте их доказательств и мысленных построений. Наконец, третья особенность нашей книги — полное отсутствие каких-либо компьютерных приложений и упоминаний о новейших разделах математики, таких как нечеткая логика, искусственные нейронные сети и анализ данных. Мы разместили этот материал в предыдущем томе, где читатель найдет множество актуальных компьютерных приложений. The book is addressed to students who risk enriching themselves with knowledge of additional sections of mathematics provided by master's programs both in the Bologna and in the Anglo-American systems of higher professional education. The choice of additional sections and individual chapters is implicitly provided for the consumer audience of technological and engineering areas. The book will also be useful to the teaching staff not only in engineering and technology, but also in the humanities. When presenting the material, unlike most similar textbooks, the author sought to meet the requirements of the dialectical theory of knowledge, which received the updated name of Singaporean mathematics. A distinctive feature of the proposed book is that the presentation is accompanied by many examples, questions for self-control and tasks for exercises. Another distinguishing feature of the book is the absence of abstractions from the history of mathematics, with which mathematics textbooks are often stuffed to the detriment of the rigor of the formulation of theorems, the beauty of their proofs, and mental constructions. Finally, the third feature of our book is the complete absence of any computer applications and mentions of the latest branches of mathematics, such as fuzzy logic, articial neural networks and data analysis. We placed this material in the previous volume, where the reader will find many relevant computer applications. Abstract Preface BASIC NOTATION Part I. Theory of functions of a complex variable Chapter 1. Complex numbers and series with complex terms 1.1. Complex numbers and their writing forms 1.1.1. Equality of complex numbers 1.1.2. Conjugate complex numbers 1.1.3. Complex plane. Infinity on the set of complex numbers 1.2. Operations on complex numbers 1.2.1. Addition and subtraction of complex numbers 1.2.2. Multiplication and division of complex numbers 1.2.3. Exponentiation and root extraction operations 1.2.4. Properties of complex conjugation operation 1.3. Complex numbers in trigonometric and show forms 1.3.1. Trigonometric form of a complex number 1.3.2. Exponential form of a complex number 1.3.3. Complex number argument 1.3.4. Principal value of the complex number argument 1.3.5. Series of operations on numbers in trigonometric form 1.4. Sequences of complex numbers 1.5. Series with complex terms 1.5.1. Tests for the convergence of series with complex terms 1.5.2. Tests for absolute convergence of series with complex terms QUESTIONS AND TASKS FOR SELF-TEST References Chapter 2. Functions of a complex variable 2.1. Basic concepts of functions of a complex variable 2.2. Mappings on the complex plane 2.3. Inverse and multivalued functions of a complex variable 2.3.1. Functions inverse to non-leaf. Allocation of unambiguous branches 2.3.2. Argument function Arg(z) 2.4. Limit of a function of a complex variable 2.5. Continuity at a point of a function of a complex variable 2.6. Derivative of a function of a complex variable 2.7. Elementary functions of a complex variable 2.7.1. The exponential function of a complex variable 2.7.2. Trigonometric and hyperbolic complex functions 2.7.3. Complex logarithm 2.7.4. Logarithmic function of a complex variable 2.7.5. Inverse trigonometric and hyperbolic complex functions 2.8. Differentiation of functions of a complex variable 2.8.1. Rules of differentiation 2.8.2. Cauchy - Riemann conditions for function differentiability 2.8.3. Cauchy  Riemann conditions in polar coordinates 2.8.4. Geometric meaning of the module and argument of the derivative QUESTIONS AND TASKS FOR SELF-TEST References Chapter 3. Analytic functions and their properties 3.1. Definition of an analytic function 3.2. Properties of analytic functions 3.3. Infinite di ̇erentiability of an analytic function 3.4. Relationship between analytic and harmonic functions 3.5. Conjugate harmonic functions 3.6. Recovery of an analytic function from its real or imaginary part 3.7. Application of analytic functions 3.7.1. Analytic functions in field theory problems 3.8. Using conformal mapping QUESTIONS AND TASKS FOR SELF-TEST References Chapter 4. Integration of functions of a complex variable 4.1. Introductory concepts 4.2. Calculation of integrals of functions of a complex variable 4.2.1. Methods for calculating integrals f(z) dz 4.3. Basic Cauchy theorem for a simple contour 4.4. Cauchy's theorem for a complex contour 4.5. Cauchy integral formula 4.6. Calculation of integrals over a closed contour 4.7. Cauchy-type integrals 4.7.1. Principal value of the Cauchy-type integral 4.7.2. Existence theorem for the principal value of the integral 4.8. Boundary values of the Cauchy integral 4.9. Methods for calculating Cauchy-type integrals QUESTIONS AND TASKS FOR SELF-TEST References Chapter 5. Function series in the complex domain 5.1. Function sequences 5.1.1. Definition of a functional sequence 5.1.2. Uniform convergence of a functional sequence 5.2. Functional series in the complex domain 5.2.1. Region of convergence and uniform convergence of series 5.2.2. Weierstrass test and uniform convergence 5.2.3. Weierstrass' theorem for series of analytic functions 5.2.4. Finding the convergence region of the series 5.2.5. Examples of studying the convergence of series with complex terms 5.3. Power series with complex terms and their properties 5.3.1. Circle of convergence of a power series 5.4. Properties of power series 5.4.1. Generalization of properties of power series 5.5. Series with complex terms in integer powers 5.6. Expansion of functions into power series. Taylor Series QUESTIONS AND TASKS FOR SELF-TEST References Chapter 6. Laurent series and expansion of functions in integer powers 6.1. Zeros of analytic functions 6.1.1. An algorithm for finding zeros of analytic functions 6.2. Laurent series and expansion of functions in integer powers 6.3. Examples of expansion of functions in Laurent series QUESTIONS AND TASKS FOR SELF-TEST References Chapter 7. Special points. Residues 7.1. Isolated singular points and poles 7.1.1. Classification of singular points 7.2. Types of singular points of a function 7.3. Theorems of Sochocki and Picard 7.3.1. Laurent series in the neighborhood of a singular point 7.4. Deductions and their application 7.4.1. Deduction definition 7.5. Theorems on residues 7.5.1. Computation of residues at a pole and a removable singular point 7.5.2. Algorithm for calculating the residue of a function 7.6. Computing integrals using residues 7.6.1. Calculation of contour integrals using residues 7.6.2. Integrals of the form R(cos x, sin x) dx 7.6.3. Computing improper integrals using residues QUESTIONS AND TASKS FOR SELF-TEST References Part II. Calculus of variations Chapter 8. Introductory concepts and definitions 8.1. Introduction 8.2. Dido's task 8.3. The line of the fastest descent 8.4. Surface of revolution of least area 8.5. Equilibrium of a deformed membrane 8.6. Extrema of functionals QUESTIONS AND TASKS FOR SELF-TEST References Chapter 9. Variation method in problems with fixed boundaries 9.1. Basic information about functionals 9.2. Functional spaces 9.3. Variations of comparison curves and functionals 9.4. Euler equation QUESTIONS AND TASKS FOR SELF-TEST References Chapter 10. Variational problems with moving boundaries 10.1. Moving end problem 10.2. Transversality conditions 10.3. Extremals with corner points 10.3.1. The problem of reflection of extremals 10.3.2. The problem of refraction of extremals 10.4. Spatial problems with moving boundaries QUESTIONS AND TASKS FOR SELF-TEST References Chapter 11. Variational problems for a conditional extremum 11.1. Links of the form φ(x, y1, y2, . . . , yn) = 0 11.2. Links like φ(x, y1, y2, . . . , yn, y′1, y′2 , . . . , y′n) = 0 11.3. Isoperimetric problems QUESTIONS AND TASKS FOR SELF-TEST References Part III. Equations of Mathematical Physics Introduction Chapter 12. Classification of partial differential equations 12.1. Classification of partial differential equations of the 2nd order 12.1.1. Differential equations with two independent variables 12.1.2. Canonical form of equations 12.2. Classification of 2nd order equations with many variables 12.3. Canonical forms of linear equations with constant coefficients QUESTIONS AND TASKS FOR SELF-TEST References Chapter 13. The first concentr of tasks of mathematical physics 13.1. Derivation of the heat equation 13.1.1. Fourier's law 13.1.2. Newton's law 13.1.3. Heat balance equation 13.2. Steady-state thermal process 13.3. Initial and boundary conditions 13.4. Boundary conditions for the heat equation 13.4.1. Third boundary value problem 13.4.2. The second boundary value problem for the heate quation 13.4.3. First boundary value problem 13.5. Diffusion equation 13.5.1. Boundary value problems for the diffusion equation 13.6. Equation of fluid motion 13.7. Membrane equations 13.8. Wave equation 13.8.1. String vibrations 13.9. The problem of longitudinal vibrations of a rod QUESTIONS AND TASKS FOR SELF-TEST References Chapter 14. Fourier method for solving boundary value problems 14.1. Fourier method in solving heat conduction problems 14.2. Solution of the second boundary value problem for the heate quation by the Fourier method 14.3. Application of the Fourier method to solving inhomogeneous boundary value problems 14.4. General Fourier method and properties of eigenfunctions 14.5. Uniqueness theorem for solving boundary value problems for the wave equation 14.6. Application of the Fourier method to solving boundary value problems for the Laplace equation 14.6.1. Solution of the problem for a circular area 14.6.2. Solution of the Dirichlet problem for a rectangular area QUESTIONS AND TASKS FOR SELF-TEST References Chapter 15. Cauchy problem and Laplace equation 15.1. Continuous dependence of the solution on the initial conditions 15.2. The Cauchy problem for partial differential equations 15.2.1. Formulation of the Cauchy problem for the wave equation 15.3. The Cauchy problem for the heat equation 15.4. Solving boundary value problems for the Laplace equation using the source function 15.5. Application of the source function to the solution of inhomogeneous and nonlinear Laplace equations QUESTIONS AND TASKS FOR SELF-TEST References Chapter 16. Potential theory 16.1. Volumetric potential 16.2. Logarithmic potential 16.3. Improper integrals 16.4. Convergence theorems for the improper integral 16.5. Continuity and differentiability of volumetric and logarithmic potentials 16.6. Existence of second derivatives of the volume potential 16.7. Double layer potential 16.8. Limiting properties of the double layer potential 16.8.1. Using properties of the double layer potential in solving the Dirichlet problem for the Laplace equation 16.9. Simple layer potential 16.9.1. Solution of the second boundary value problem using the potential of a simple layer 16.10. Proof of the existence of a solution to the Fredholm integral equation of the second kind QUESTIONS AND TASKS FOR SELF-TEST References Part IV. Numerical Methods Introductory remarks Chapter 17. Numerical methodsof linear algebra 17.1. Matrix norm 17.2. Numerical methods for solving systems of linear algebraic equations (SLAE) 17.2.1. Condition number 17.2.2. Numerical schemes for implementing the Gauss method 17.3. Algorithm of numerical Gauss method 17.3.1. Sweep method for solving SLAE 17.3.2. Algorithm for solving systems of equations by the sweep method 17.4. Solution of SLAE by square roots method 17.4.1. Square Root Algorithm 17.5. Method of simple iterations for solving SLAE 17.5.1. Algorithm of the method of simple iterations 17.6. Solution of SLAE by Seidel's method 17.6.1. Seidel's method algorithm 17.7. Schultz iterative method of finding the inverse matrix 17.7.1. Algorithm of the Schultz iterative method 17.8. Problems about own values and matrix vectors 17.8.1. Direct expansion method 17.8.2. Direct Deployment Method Algorithm 17.8.3. Iteration method for finding eigenvalues and vectors 17.8.4. Iteration method algorithm QUESTIONS AND TASKS FOR SELF-TEST References Chapter 18. Methods for solving ordinary differential equations 18.1. Numerical methods for solving the Cauchy problem 18.1.1. Basic concepts and definitions 18.2. Discrete and continuous-discrete methods 18.3. Local and global errors 18.4. Stability of methods for solving the Cauchy problem 18.5. Numerical methods for solving boundary value problems 18.5.1. Problem Statement and Main Provisions 18.6. Finite element method 18.6.1. Algorithm for applying the finite element method QUESTIONS AND TASKS FOR SELF-TEST References Chapter 19. Methods for solving partial differential equations 19.1. Numerical methods for solving equations of mathematical physics with two variables 19.1.1. Statement of the problem and main provisions 19.1.2. Problem statements for first order equations 19.1.3. Statements of problems for equations of parabolic type 19.1.4. Statements of problems for equations of hyperbolic type 19.1.5. Statements of problems for equations of elliptic type 19.2. Principles of constructing di ̇erence schemes for partial differential equations 19.3. Difference schemes for solving equations in partial derivatives of the 1st order 19.4. Numerical methods for solving partial differential equations 19.4.1. Line method 19.4.2. Characteristic method 19.4.3. Godunov's method QUESTIONS AND TASKS FOR SELF-TEST References Afterword Index