Handbook of Mathematics for Engineers and Scientists ♦ Andrei D. Polyanin - Alexander V. Manzhirov

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Table of Contents
Part I. Definitions, Formulas, Methods, and Theorems
1. Arithmetic and Elementary Algebra
1.1. Real Numbers
1.2. Equalities and Inequalities. Arithmetic Operations. Absolute Value
1.3. Powers and Logarithms
1.4. Binomial Theoremand Related Formulas
1.5. Arithmetic and Geometric Progressions. Finite Sums and Products
1.6. Mean Values and Inequalities of General Form
1.7. Some Mathematical Methods
References for Chapter 1
2. Elementary Functions
2.1. Power, Exponential, and Logarithmic Functions
2.2. Trigonometric Functions
2.3. Inverse Trigonometric Functions
2.4. Hyperbolic Function
2.5. InverseHyperbolic Functions
References for Chapter 2
3. Elementary Geometry
3.1. Plane Geometry
3.2. Solid Geometry
3.3. Spherical Trigonometry
References for Chapter 3
4. Analytic Geometry
4.1. Points, Segments, and Coordinates on Line and Plane
4.2. Curves on Plane
4.3. Straight Lines and Points on Plane
4.4. Second-Order Curves
4.5. Coordinates, Vectors, Curves, and Surfaces in Space
4.6. Line and Plane in Space
4.7. Quadric Surfaces (Quadrics)
References for Chapter 4
5. Algebra
5.1. Polynomials and Algebraic Equations
5.2. Matrices and Determinants
5.3. Linear Spaces
5.4. Euclidean Spaces
5.5. Systems of Linear Algebraic Equations
5.6. LinearOperators
5.7. Bilinear and Quadratic Forms
5.8. Some Facts fromGroup Theory
References for Chapter 5
6. Limits and Derivatives
6.1. Basic Concepts ofMathematicalAnalysis
6.2. DifferentialCalculus for Functions of a SingleVariable
6.3. Functions of SeveralVariables. PartialDerivatives
References for Chapter 6
7. Integrals
7.1. Indefinite Integral
7.2. Definite Integral
7.3. Double and Triple Integrals
7.4. Line and Surface Integrals
References for Chapter 7
8. Series
8.1. Numerical Series and Infinite Products
8.2. Functional Series
8.3. Power Series
8.4. Fourier Series
8.5. Asymptotic Series
References for Chapter 8
9. Differential Geometry
9.1. Theory of Curves
9.2. Theory of Surfaces
References for Chapter 9
10. Functions of Complex Variable
10.1. Basic Notions
10.2. Main Applications
References for Chapter 10
11. Integral Transforms
11.1. General Formof Integral Transforms. Some Formulas
11.2. Laplace Transform
11.3. Mellin Transform
11.4. Various Forms of the Fourier Transform
11.5. Other Integral Transforms
References for Chapter 11
12. Ordinary Differential Equations
12.1. First-Order Differential Equations
12.2. Second-Order Linear Differential Equations
12.3. Second-Order Nonlinear Differential Equations
12.4. Linear Equations of ArbitraryOrder
12.5. Nonlinear Equations of ArbitraryOrder
12.6. Linear Systems of OrdinaryDifferential Equations
12.7. Nonlinear Systems of OrdinaryDifferential Equations
References for Chapter 12
13. First-Order Partial Differential Equations
13.1. Linear and Quasilinear Equations
13.2. Nonlinear Equations
References for Chapter 13
14. Linear Partial Differential Equations
14.1. Classification of Second-Order Partial Differential Equations
14.2. Basic Problems ofMathematical Physics
14.3. Properties and Exact Solutions of Linear Equations
14.4. Method of Separation of Variables (FourierMethod)
14.5. Integral TransformsMethod
14.6. Representation of the Solution of the Cauchy Problem via the Fundamental Solution
14.7. Boundary Value Problems for Parabolic Equations with One Space Variable. Green’s Function
14.8. Boundary Value Problems for Hyperbolic Equations with One Space Variable. Green’s Function. Goursat Problem
14.9. Boundary Value Problems for Elliptic Equations with Two Space Variables
14.10. Boundary Value Problems with Many Space Variables. Representation of Solutions via theGreen’s Function
14.11. Construction of the Green’s Functions. General Formulas and Relations
14.12. Duhamel’s Principles in Nonstationary Problems
14.13. Transformations Simplifying Initial and Boundary Conditions
References for Chapter 14
15. Nonlinear Partial Differential Equations
15.1. Classification of Second-Order Nonlinear Equations
15.2. Transformations of Equations of Mathematical Physics
15.3. Traveling-Wave Solutions, Self-Similar Solutions, and Some Other Simple Solutions. SimilarityMethod
15.4. Exact Solutionswith Simple Separation of Variables
15.5. Method of Generalized Separation of Variables
15.6. Method of Functional Separation of Variables
15.7. Direct Method of Symmetry Reductions of Nonlinear Equations
15.8. Classical Method of Studying Symmetries of Differential Equations
15.9. NonclassicalMethod of Symmetry Reductions
15.10. Differential ConstraintsMethod
15.11. Painlev´e Test for Nonlinear Equations of Mathematical Physics
15.12. Methods of the Inverse Scattering Problem (Soliton Theory)
15.13. Conservation Laws and Integrals ofMotion
15.14. Nonlinear Systems of PartialDifferential Equations
References for Chapter 15
16. Integral Equations
16.1. Linear Integral Equations of the First Kind with Variable Integration Limit
16.2. Linear Integral Equations of the Second Kind with Variable Integration Limit
16.3. Linear Integral Equations of the First Kind with Constant Limits of Integration
16.4. Linear Integral Equations of the Second Kind with Constant Limits of Integration
16.5. Nonlinear Integral Equations
References for Chapter 16
17. Difference Equations and Other Functional Equations
17.1. Difference Equations of Integer Argument
17.2. Linear Difference Equations with a Single Continuous Variable
17.3. Linear Functional Equations
17.4. Nonlinear Difference and Functional Equations with a Single Variable
17.5. Functional Equationswith SeveralVariables
References for Chapter 17
18. Special Functions and Their Properties
18.1. Some Coefficients, Symbols, and Numbers
18.2. Error Functions. Exponential and Logarithmic Integrals
18.3. Sine Integral and Cosine Integral. Fresnel Integrals
18.4. Gamma Function, Psi Function, and Beta Function
18.5. IncompleteGamma and Beta Functions
18.6. Bessel Functions (Cylindrical Functions)
18.7. Modified Bessel Functions
18.8. Airy Functions
18.9. Degenerate Hypergeometric Functions (Kummer Functions)
18.10. Hypergeometric Functions
18.11. Legendre Polynomials, Legendre Functions, and Associated Legendre Functions
18.12. ParabolicCylinder Functions
18.13. Elliptic Integrals
18.14. Elliptic Functions
18.15. Jacobi Theta Functions
18.16. Mathieu Functions and ModifiedMathieu Functions
18.17. Orthogonal Polynomials
18.18. Nonorthogonal Polynomials
References for Chapter 18
19. Calculus of Variations and Optimization
19.1. Calculus of Variations and Optimal Control
19.2. Mathematical Programming
References for Chapter 19
20. Probability Theory
20.1. Simplest Probabilistic Models
20.2. Random Variables and Their Characteristics
20.3. Limit Theorems
20.4. Stochastic Processes
References for Chapter 20
21. Mathematical Statistics
21.1. Introduction to Mathematical Statistics
21.2. Statistical Estimation
21.3. Statistical Hypothesis Testing
References for Chapter 21
Part II. Mathematical Tables
T1. Finite Sums and Infinite Series
T1.1. Finite Sums
T1.2. Infinite Series
References for Chapter T1
T2. Integrals
T2.1. Indefinite Integrals
T2.2. Tables of Definite Integrals
References for Chapter T2
T3. Integral Transforms
T3.1. Tables of Laplace Transforms
T3.2. Tables of Inverse Laplace Transforms
T3.3. Tables of Fourier Cosine Transforms
T3.4. Tables of Fourier Sine Transforms
T3.5. Tables of Mellin Transforms
T3.6. Tables of Inverse Mellin Transforms
References for Chapter T3
T4. Orthogonal Curvilinear Systems of Coordinate
T4.1. Arbitrary Curvilinear Coordinate Systems
T4.2. Special Curvilinear Coordinate Systems
References for Chapter T4
T5. Ordinary Differential Equations
T5.1. First-Order Equations
T5.2. Second-Order Linear Equations
References for Chapter T5
T6. Systems of Ordinary Differential Equations
T6.1. Linear Systems of Two Equations
T6.2. Linear Systems of Three and More Equations
T6.3. Nonlinear Systems of Two Equations
T6.4. Nonlinear Systems of Three or More Equations
References for Chapter T6
T7. First-Order Partial Differential Equations
T7.1. Linear Equations
T7.2. Quasilinear Equations
T7.3. Nonlinear Equations
T8. Linear Equations and Problems of Mathematical Physics
T8.1. Parabolic Equations
T8.2. Hyperbolic Equations
T8.3. Elliptic Equations
T8.4. Fourth-Order Linear Equations
References for Chapter T8
T9. Nonlinear Mathematical Physics Equations
T9.1. Parabolic Equations
T9.2. Hyperbolic Equations
T9.3. Elliptic Equations
T9.4. Other Second-Order Equations
T9.5. Higher-Order Equations
References for Chapter T9
T10. Systems of Partial Differential Equations
References for Chapter T10
T11. Integral Equations
References for Chapter T11
T12. Functional Equations
References for Chapter T12
Supplement. Some Useful ElectronicMathematical Resources
Index

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