Изданието на руски език:
Справочник по математике для научных работников и инженеров (определения, теоремы, формулы)
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CONTENTS
Preface vii
Chapter 1. Real and Complex Numbers. Elementary Algebra. 1
1.1. Introduction. The Real-number System 2
1.2. Powers, Roots, Logarithms, and Factorials. Sum and Product Notation 4
1.3. Complex Numbers 7
1.4. Miscellaneous Formulas 10
1.5. Determinants 12
1.6. Algebraic Equations: General Theorems 15
1.7. Factoring of Polynomials and Quotients of Polynomials. Partial Fractions 19
1.8. Linear, Quadratic, Cubic, and Quartic Equations 22
1.9. Systems of Simultaneous Equations 24
1.10. Related Topics, References, and Bibliography 27
Chapter 2. Plane Analytic Geometry 29
2.1. Introduction and Basic Concepts 30
2.2. The Straight Line 37
2.3. Relations Involving Points and Straight Lines 39
2.4. Second-order Curves (Conic Sections) 41
2.5. Properties of Circles, Ellipses, Hyperbolas, and Parabolas 48
2.6. Higher Plane Curves 53
2.7. Related Topics, References, and Bibliography 56
Chapter 3. Solid Analytic Geometry 57
3.1. Introduction and Basic Concepts 58
3.2. The Plane 67
3.3. The Straight Line 69
3.4. Relations Involving Points, Planes, and Straight Lines 70
3.5. Quadric Surfaces 74
3.6. Related Topics, References, and Bibliography 82
Chapter 4. Functions and Limits. Differential and Integral Calculus 83
4.1. Introduction 85
4.2. Functions 85
4.3. Point Sets, Intervals, and Regions 87
4.4. Limits, Continuous Functions, and Related Topics 90
4.5. Differential Calculus 95
4.6. Integrals and Integration JQ2
4.7. Mean-value Theorems. Values of Indeterminate Forms. Weierstrass's Approximation Theorems JJC
4.8. Infinite Series, Infinite Products, and Continued Fractions. ... 121
4.9. Tests for the Convergence and Uniform Convergence of Infinite Series and Improper Integrals 127
4.10. Representation of Functions by Infinite Series and Integrals. Power Series and Taylor's Expansion 131
4.11. Fourier Series and Fourier Integrals 134
4.12. Related Topics, References, and Bibliography 144
Chapter 5. Vector Analysis 145
5.1. Introduction 14g
5.2. Vector Algebra 147
5.3. Vector Calculus: Functions of a Scalar Parameter 151
5.4. Scalar and Vector Fields 153
5.5. Differential Operators 157
5.6. Integral Theorems 162
5.7. Specification of a Vector Field in Terms of Its Curl and Divergence. . 164
5.8. Related Topics, References, and Bibliography 166
Chapter 6. Curvilinear Coordinate Systems 168
6.1. Introduction jgg
6.2. Curvilinear Coordinate Systems 169
6.3. Representation of Vectors in Terms of Components 171
6.4. Orthogonal Coordinate Systems. Vector Relations in Terms of Orthogonal Components 174
6.5. Formulas Relating to Special Orthogonal Coordinate Systems ... 177
6.6. Related Topics, References, and Bibliography 177
Chapter 7. Functions of a Complex Variable 187
7.1. Introduction Igg
7.2. Functions of a Complex Variable. Regions of the Complex-number Plane . . 188
7.3. Analytic (Regular, Holomorphic) Functions 192
7.4. Treatment of Multiple-valued Functions 193
7.5. Integral Theorems and Series Expansions 196
7.6. Zeros and Isolated Singularities 198
7.7. Residues and Contour Integration 202
7.8. Analytic Continuation 205
7.9. Conformal Mapping 206
7.10. Functions Mapping Specified Regions onto the Unit Circle .... 219
7.11. Related Topics, References, and Bibliography 219
Chapter 8. The Laplace Transformation and Other Functional Transformations 221
8.1. Introduction 222
8.2. The Laplace Transformation 222
8.3. Correspondence between Operations on Object and Result Functions . 225
8.4. Tables of Laplace-transform Pairs and Computation of Inverse Laplace Transforms 228
8.5. "Formal" Laplace Transformation of Impulse-function Terms . . . 233
8.6. Some Other Integral Transformations 233
8.7. Finite Integral Transforms, Generating Functions, and z Transforms . 237
8.8. Related Topics, References, and Bibliography 240
Chapter 9. Ordinary Differential Equations 243
9.1. Introduction 244
9.2. First-order Equations. 247
9.3. Linear Differential Equations 252
9.4. Linear Differential Equations with Constant Coefficients 266
9.5. Nonlinear Second-order Equations 277
9.6. Pfaffian Differential Equations 284
9.7. Related Topics, References, and Bibliography 286
Chapter 10. Partial Differential Equations 287
10.1. Introduction and Survey 288
10.2. Partial Differential Equations of the First Order 290
10.3. Hyperbolic, Parabolic, and Elliptic Partial Differential Equations. Characteristics 302
10.4. Linear Partial Differential Equations of Physics. Particular Solutions. 311
10.5. Integral-transform Methods 324
10.6. Related Topics, References, and Bibliography 328
Chapter 11. Maxima and Minima and Optimization Problems . 330
11.1. Introduction 321
11.2. Maxima and Minima of Functions of One Real Variable 332
11.3. Maxima and Minima of Functions of Two or More Real Variables. 333
11.4. Linear Programming, Games, and Related Topics 335
11.5. Calculus of Variations. Maxima and Minima of Definite Integrals. . 344
11.6. Extremals as Solutions of Differential Equations: Classical Theory. . 346
11.7. Solution of Variation Problems by Direct Methods 357
11.8. Control Problems and the Maximum Principle 358
11.9. Stepwise-control Problems and Dynamic Programming 369
11.10. Related Topics, References, and Bibliography 371
Chapter 12. Definition of Mathematical Models: Modern (Abstract) Algebra and Abstract Spaces 373
12.1. Introduction 374
12.2. Algebra of Models with a Single Defining Operation: Groups. . . . 378
12.3. Algebra of Models with Two Defining Operations: Rings, Fields, and Integral Domains 382
12.4. Models Involving More Than One Class of Mathematical Objects: Linear Vector Spaces and Linear Algebras 384
12.5. Models Permitting the Definition of Limiting Processes: Topological Spaces 386
12.6. Order 391
12.7. Combination of Models: Direct Products, Product Spaces, and Direct Sums 392
12.8. Boolean Algebras 393
12.9. Related Topics, References, and Bibliography 400
Chapter 13. Matrices. Quadratic and Hermitian Forms .... 402
13.1. Introduction 403
13.2. Matrix Algebra and Matrix Calculus 403
13.3. Matrices with Special Symmetry Properties 410
13.4. Equivalent Matrices. Eigenvalues, Diagonalization, and Related Topics 412
13.5. Quadratic and Hermitian Forms 416
13.6. Matrix Notation for Systems of Differential Equations (State Equations). Perturbations and Lyapunov Stability Theory 420
13.7. Related Topics, References, and Bibliography 430
Chapter 14. Linear Vector Spaces and Linear Transformations (Linear Operators). Representation of Mathematical Models in Terms of Matrices , 431
14.1. Introduction. Reference Systems and Coordinate Transformations . 433
14.2. Linear Vector Spaces 435
14.3. Linear Transformations (Linear Operators) 439
14.4. Linear Transformations of a Normed or Unitary Vector Space into Itself. Hermitian and Unitary Transformations (Operators) 441
14.5. Matrix Representation of Vectors and Linear Transformations (Operators) 447
14.6. Change of Reference System 449
14.7. Representation of Inner Products. Orthonormal Bases 452
14.8. Eigenvectors and Eigenvalues of Linear Operators 457
14.9. Group Representations and Related Topics 467
14.10. Mathematical Description of Rotations 471
14.11. Related Topics, References, and Bibliography 482
Chapter 15. Linear Integral Equations, Boundary-value Problems, and Eigenvalue Problems 484
15.1. Introduction. Functional Analysis 486
15.2. Functions as Vectors. Expansions in Terms of Orthogonal Functions. 487
15.3. Linear Integral Transformations and Linear Integral Equations . 492
15.4. Linear Boundary-value Problems and Eigenvalue Problems Involving Differential Equations 502
15.5. Green's Functions. Relation of Boundary-value Problems and Eigenvalue Problems to Integral Equations 515
15.6. Potential Theory ... 520
15.7. Related Topics, References, and Bibliography 531
Chapter 16. Representation of Mathematical Models: Tensor Algebra and Analysis 533
16.1. Introduction 534
16.2. Absolute and Relative Tensors 537
16.3. Tensor Algebra: Definition of Basic Operations 540
16.4. Tensor Algebra: Invariance of Tensor Equations 543
16.5. Symmetric and Skew-symmetric Tensors 543
16.6. Local Systems of Base Vectors 545
16.7. Tensors Defined on Riemann Spaces. Associated Tensors .... 546
16.8. Scalar Products and Related Topics 549
16.9. Tensors of Rank Two (Dyadics) Defined on Riemann Spaces. . . . 551
16.10. The Absolute Differential Calculus. Covariant Differentiation. . . 552
16.11. Related Topics, References, and Bibliography 560
Chapter 17. Differential Geometry 561
17.1. Curves in the Euclidean Plane 562
17.2. Curves in Three-dimensional Euclidean Space 565
17.3. Surfaces in Three-dimensional Euclidean Space 569
17.4. Curved Spaces 579
17.5. Related Topics, References, and Bibliography 584
Chapter 18. Probability Theory and Random Processes 585
18.1. Introduction 587
18.2. Definition and Representation of Probability Models 588
18.3. One-dimensional Probability Distributions 593
18.4. Multidimensional Probability Distributions 602
18.5. Functions of Random Variables. Change of Variables 614
18.6. Convergence in Probability and Limit Theorems 620
18.7. Special Techniques for Solving Probability Problems 623
18.8. Special Probability Distributions 625
18.9. Mathematical Description of Random Processes 637
18.10. Stationary Random Processes. Correlation Functions and Spectral Densities 641
18.11. Special Classes of Random Processes. Examples 650
18.12. Operations on Random Processes 659
18.13. Related Topics, References, and Bibliography 662
Chapter 19. Mathematical Statistics 664
19.1. Introduction to Statistical Methods 665
19.2. Statistical Description. Definition and Computation of Random- sample Statistics 668
19.3. General-purpose Probability Distributions . 674
19.4. Classical Parameter Estimation 676
19.5. Sampling Distributions 680
19.6. Classical Statistical Tests 686
19.7. Some Statistics, Sampling Distributions, and Tests for Multivariate Distributions 697
19.8. Random-process Statistics and Measurements. ....... 703
19.9. Testing and Estimation with Random Parameters 708
19.10. Related Topics, References, and Bibliography 713
Chapter 20. Numerical Calculations and Finite Differences .... 715
20.1. Introduction 718
20.2. Numerical Solution of Equations 719
20.3. Linear Simultaneous Equations, Matrix Inversion, and Matrix Eigenvalue Problems 729
20.4. Finite Differences and Difference Equations 737
20.5. Approximation of Functions by Interpolation 746
20.6. Approximation by Orthogonal Polynomials, Truncated Fourier Series, and Other Methods 755
20.7. Numerical Differentiation and Integration 770
20.8. Numerical Solution of Ordinary Differential Equations 777
20.9. Numerical Solution of Boundary-value Problems, Partial Differential Equations, and Integral Equations 785
20.10. Monte-Carlo Techniques 797
20.11. Related Topics, References, and Bibliography 800
Chapter 21. Special Functions ... 804
21.1. Introduction 806
21.2. The Elementary Transcendental Functions 806
21.3. Some Functions Defined by Transcendental Integrals 818
21.4. The Gamma Function and Related Functions 822
21.5. Binomial Coefficients and Factorial Polynomials. Bernoulli Polynomials and Bernoulli Numbers 824
21.6. Elliptic Functions, Elliptic Integrals, and Related Functions. . . . 827
21.7. Orthogonal Polynomials 848
21.8. Cylinder Functions, Associated Legendre Functions, and Spherical Harmonics 857
21.9. Step Functions and Symbolic Impulse Functions 874
21.10. References and Bibliography 880
Appendix A. Formulas Describing Plane Figures and Solids. . 881
Appendix B. Plane and Spherical Trigonometry 885
Appendix C. Permutations, Combinations, and Related Topics. . 894
Appendix D. Tables of Fourier Expansions and Laplace-transform Pairs 900
Appendix E. Integrals, Sums, Infinite Series and Products, and Continued Fractions 925
Appendix F. Numerical Tables 989
Squares 990
Logarithms 993
Trigonometric Functions 1010
Exponential and Hyperbolic Functions 1018
Natural Logarithms 1025
Sine Integral . 1027
Cosine Integral 1028
Exponential and Related Integrals 1029
Complete Elliptic Integrals 1033
Factorials and Their Reciprocals . 1034
Binomial Coefficients 1034
Gamma and Factorial Functions 1035
Bessel Functions 1037
Legendre Polynomials 1060
Error Function 1061
Normal-distribution Areas 1062
Normal-curve Ordinates 1063
Distribution of t 1064
Distribution of x2 1065
Distribution of F 1066
Random Numbers ... 1070
Normal Random Numbers 1075
sin x/x 1080
Chebyshev Polynomials 1089
Glossary of Symbols and Notations 1090
Index 1097
