A reliable source of definitions, theorems, and formulas, this authoritative handbook provides convenient access to information from every area of mathematics. Coverage includes Fourier transforms, Z transforms, linear and nonlinear programming, calculus of variations, random-process theory, special functions, combinatorial analysis, numerical methods, game theory, and much more.
Convenient access to information from every area of mathematics: Fourier transforms, Z transforms, linear and nonlinear programming, calculus of variations, random-process theory, special functions, combinatorial analysis, game theory, much more.
Convenient access to information from every area of mathematics: Fourier transforms, Z transforms, linear and nonlinear programming, calculus of variations, random-process theory, special functions, combinatorial analysis, game theory, much more.
Preface
Chapter 1. Real and Complex Numbers. Elementary Algebra.
1.1. Introduction. The Real-number System
1.2. "Powers, Roots, Logarithms, and Factorials. Sum and Product Notation"
1.3. Complex Numbers
1.4. Miscellaneous Formulas
1.5. Determinants
1.6. Algebraic Equations: General Theorems
1.7. Factoring of Polynomials and Quotients of Polynomials. Partial Fractions
1.8. "Linear, Quadratic, Cubic, and Quartic Equations"
1.9. Systems of Simultaneous Equations
1.10. "Related Topics, References, and Bibliography"
Chapter 2. Plane Analytic Geometry
2.1. Introduction and Basic Concepts
2.2. The Straight Line
2.3. Relations Involving Points and Straight Lines
2.4. Second-order Curves (Conic Sections)
2.5. "Properties of Circles, Ellipses, Hyperbolas, and Parabolas"
2.6. Higher Plane Curves
2.7. "Related Topics, References, and Bibliography"
Chapter 3. Solid Analytic Geometry
3.1. Introduction and Basic Concepts
3.2. The Plane
3.3. The Straight Line
3.4. "Relations Involving Points, Planes, and Straight Lines"
3.5. Quadric Surfaces
3.6. "Related Topics, References, and Bibliography"
Chapter 4. Functions and Limits. Differential and Integral Calculus
4.1. Introduction
4.2. Functions
4.3. "Point Sets, Intervals, and Regions"
4.4. "Limits, Continuous Functions, and Related Topics"
4.5. Differential Calculus
4.6. Integrals and Integration
4.7. Mean-value Theorems. Values of Indeterminate Forms. Weierstrass's Approximation Theorems.
4.8. "Infinite Series, Infinite Products, and Continued Fractions"
4.9. Tests for the Convergence and Uniform Convergence of Infinite Series and Improper Integrals
4.10. Representation of Functions by Infinite Series and Integrals. Power Series and Taylor's Expansion
4.11. Fourier Series and Fourier Integrals
4.12. "Related Topics, References, and Bibliography"
Chapter 5. Vector Analysis
5.1. Introduction
5.2. Vector Algebra
5.3. Vector Calculus: Functions of Scalar Parameter
5.4. Scalar and Vector Fields
5.5. Differential Operators
5.6. Integral Theorems
5.7. Specification of a Vector Field in Terms of Its Curl and Divergence
5.8. "Related Topics, References, and Bibliography"
Chapter 6. Curvilinear Coordinate Systems
6.1. Introduction
6.2. Curvilinear Coordinate Systems
6.3. Representation of Vectors in Terms of Components
6.4. Orthogonal Coordinate Systems. Vector Relations in Terms of Orthogonal Components
6.5. Formulas Relating to Special Orthogonal Coordinate Systems
6.6. "Related Topics, References, and Bibliography"
Chapter 7. Functions of a Complex Variable
7.1. Introduction
7.2. Functions of a Complex Variable. Regions of the Complex-number Plane
7.3. "Analytic (Regular, Holomorphic) Functions"
7.4. Treatment of Multiple-valued Functions
7.5. Integral Theorems and Series Expansions
7.6. Zeros and Isolated Singularities
7.7. Residues and Contour Integration
7.8. Analytic Continuation
7.9. Conformal Mapping
7.10. Functions Mapping Specified Regions onto the Unit Circle
7.11. "Related Topics, References, and Bibliography"
Chapter 8. The Laplace Transformation and Other Functional Transformations
8.1. Introduction
8.2. The Laplace Transformation
8.3. Correspondence between Operations on Object and Result Functions
8.4. Table of Laplace-transform Pairs and Computation of Inverse Laplace Transforms
8.5. "Formal" Laplace Transformation of Impulse-function Terms"
8.6. Some Other Integral Transformations
8.7. "Finite Integral Transforms, Generating Functions, and z Transforms"
8.8. "Related Topics, References, and Bibliography"
Chapter 9. Ordinary Differential Equations
9.1. Introduction
9.2. First-order Equations
9.3. Linear Differential Equations
9.4. Linear Differential Equations with Constant Coefficients
9.5. Nonlinear Second-order Equations
9.6. Pfaffian Differential Equations
9.7. "Related Topics, References, and Bibliography"
Chapter 10. Partial Differential Equations
10.1. Introduction and Survey
10.2. Partial Differential Equations of the First Order
10.3. "Hyperbolic, Parabolic, and Elliptic Partial Differential Equations. Characteristics."
10.4. Linear Partial Differential Equations of Physics. Particular Solutions.
10.5. Integral-transform Methods
10.6. "Related Topics, References, and Bibliography"
Chapter 11. Maxima and Minima and Optimization Problems
11.1. Introduction
11.2. Maxima and Minima of Functions of One Real Variable
11.3. Maxima and Minima of Functions of Two or More Real Variables
11.4. "Linear Programming, Games, and Related Topics"
11.5. Calculus of Variations. Maxima and Minima of Definite Integrals
11.6. Extremals as Solutions of Differential Equations: Classical Theory
11.7. Solution of Variation Problems by Direct Methods
11.8. Control Problems and the Maximum Principle
11.9. Stepwise-control Problems and Dynamic Programming
11.10. "Related Topics, References, and Bibliography"
Chapter 12. Definition of Mathematical Models: Modern (Abstract) Algebra and Abstract Spaces
12.1. Introduction
12.2. Algebra of Models with a Single Defining Operation: Groups
12.3. "Algebra of Models with Two Defining Operations: Rings, Fields, and Integral Domains"
12.4. Models Involving More Than One Class of Mathematical Objects: Linear Vector Spaces and Linear Algebras
12.5. Models Permitting the Definition of Limiting Processes: Topological Spaces
12.6. Order
12.7. "Combination of Models: Direct Products, Product Spaces, and Direct Sums"
12.8. Boolean Algebras
12.9. "Related Topics, References, and Bibliography"
Chapter 13. Matrices. Quadratic and Hermitian Forms
13.1. Introduction
13.2. Matrix Algebra and Matrix Calculus
13.3. Matrices with Special Symmetry Properties
13.4. "Equivalent Matrices. Eigenvalues, Diagonalization, and Related Topics"
13.5. Quadratic and Hermitian Forms
13.6. Matrix Notation for Systems of Differential Equations (State Equations). Perturbations and Lyapunov Stability Theory
13.7. "Related Topics, References, and Bibliography"
Chapter 14. Linear Vector Spaces and Linear Transformations (Linear Operators). Representation of Mathematical Models in Terms of Matrices
14.1. Introduction. Reference Systems and Coordinate Transformations
14.2. Linear Vector Spaces
14.3. Linear Transformations (Linear Operators)
14.4. Linear Transformations of a Normed or Unitary Vector Space into Itself. Hermitian and Unitary Transformations (Operators)
14.5. Matrix Representation of Vectors and Linear Transformations (Operators)
14.6. Change of Reference System
14.7. Representation of Inner Products. Orthonormal Bases
14.8. Eigenvectors and Eigenvalues of Linear operators
14.9. Group Representations and Related Topics
14.10. Mathematical Description of Rotations
14.11. "Related Topics, References, and Bibliography"
"Chapter 15. Linear Integral Equations, Boundary-value Problems, and Eigenvalue Problems"
15.1. Introduction. Functional Analysis
15.2. Functions as Vectors. Expansions in Terms of Orthogonal Functions
15.3. Linear Integral Transformations and Linear Integral Equations
15.4. Linear Boundary-value Problems and Eigenvalue Problems Involving Differential Equations
15.5. Green's Functions. Relation of Boundary-value Problems and Eigenvalue Problems to Integral Equations
15.6. Potential Theory
15.7. "Related Topics, References, and Bibliography"
Chapter 16. Representation of Mathematical Models: Tensor Algebra and Analysis
16.1. Introduction
16.2. Absolute and Relative Tensors
16.3. Tensor Algebra: Definition of Basic Operators
16.4. Tensor Algebra: Invariance of Tensor Equations
16.5. Symmetric and Skew-Symmetric Tensors
16.6. Local Systems of Base Vectors
16.7. Tensors Defined on Riemann Spaces. Associated Tensors
16.8. Scalar Products and Related Topics
16.9. Tensors of Rank Two (Dyadics) Defined on Riemann Spaces
16.10. The Absolute Differential Calculus. Covariant Differentiation
16.11. "Related Topics, References, and Bibliography"
Chapter 17. Differential Geometry
17.1. Curves in the Euclidean Plane
17.2. Curves in the Three-dimensional Euclidean Space
17.3. Surfaces in Three-dimensional Euclidean Space
17.4. Curved Spaces
17.5. "Related Topics, References, and Bibliography"
Chapter 18. Probability Theory and Random Processes
18.1. Introduction
18.2. Definition and Representation of Probability Models
18.3. One-dimensional Probability Distributions
18.4. Multidimensional Probability Distributions
18.5. Functions of Random Variables. Change of Variables
18.6. Convergence in Probability and Limit Theorems
18.7. Special Techniques for Solving Probability Theorems
18.8. Special Probability Distributions
18.9. Mathematical Description of Random Processes
18.10. Stationary Random Processes. Correlation Functions and Spectral Densities
18.11. Special Classes of Random Processes. Examples
18.12. Operations on Random Processes
18.13. "Related Topics, References, and Bibliography"
Chapter 19. Mathematical Statistics
19.1. Introduction to Statistical Methods
19.2. Statistical Description. Definition and Computation of Random-sample Statistics
19.3. General-purpose Probability Distributions
19.4. Classical Parameter Estimation
19.5. Sampling Distributions
19.6. Classical Statistical Tests
19.7. "Some Statistics, Sampling Distributions, and Tests for Multivariate Distributions"
19.8. Random-process Statistics and Measurements
19.9. Testing and Estimation with Random Parameters
19.10. "Related Topics, References, and Bibliography"
Chapter 20. Numerical Calculations and Finite Differences
20.1. Introduction
20.2. Numerical Solution of Equations
20.3. "Linear Simultaneous Equations, Matrix Inversion, and Matrix Eigenvalue Problems"
20.4. Finite Differences and Difference Equations
20.5. Approximation of Functions by Interpolation
20.6. "Approximation by Orthogonal Polynomials, Truncated Fourier Series, and Other Methods"
20.7. Numerical Differentiation and Integration
20.8. Numerical Solution of Ordinary Differential Equations
20.9. "Numerical Solution of Boundary-value Problems, Partial Differential Equations, and Integral Equations"
20.10. Monte-Carlo Techniques
20.11. "Related Topics, References, and Bibliography"
Chapter 21. Special Functions
21.1. Introduction
21.2. The Elementary Transcendental Functions
21.3. Some Functions Defined by Transcendental Integrals
21.4. The Gamma Function and Related Functions
21.5. Binomial Coefficients and Factorial Polynomials. Bernoulli Polynomials and Bernoulli Numbers.
21.6. "Elliptic Functions, Elliptic Integrals, and Related Functions"
21.7. Orthogonal Polynomials
21.8. "Cylinder Functions, Associated Legendre Functions, and Spherical Harmonics"
21.9. Step Functions and Symbolic Impulse Functions
21.10. References and Bibliography
Appendix A. Formulas Describing Plane Figures and Solids
Appendix B. Plane and Spherical Trigonometry
"Appendix C. Permutations, Combinations, and Related Topics"
Appendix D. Tables of Fourier Expansions and Laplace-transform Pairs
"Appendix E. Integrals, Sums, Infinite Series and Products, and Continued Fractions"
Appendix F. Numerical Tables
Squares
Logarithms
Trigonometric Functions
Exponential and Hyperbolic Functions
Natural Logarithms
Sine Integral
Cosine Integral
Exponential and Related Integrals
Complete Elliptic Integrals
Factorials and Their Reciprocals
Binomial Coefficients
Gamma and Factorial Functions
Bessel Functions
Legendre Polynomials
Error Function
Normal-distribution Areas
Normal-curve Ordinates
Distribution of t
Distribution of x²
Distribution of F
Random Numbers
Normal Random Numbers
sin x/x
Chebyshev Polynomials
Glossary of Symbols and Notations
Index