《MATHEMATICAL HANDBOOK FOR SCIENTISTS AND ENGINEERS KORN and KORN》求取 ⇩

Chapter 1．Real and Complex Numbers．Elementary Algebra1

1．1．Introduction．The Real-number System2

1．2．Powers,Roots,Logarithms,and Factorials．Sum and Product Notation4

1．3．Complex Numbers7

1．4．Miscellaneous Formulas10

1．5．Determinants12

1．6．Algebraic Equations:General Theorems15

1．7．Factoring of Polynomials and Quotients of Polynomials．Partial Frac-tions19

1．8．Linear,Quadratic,Cubic,and Quartic Equations22

1．9．Systems of Simultaneous Equations24

1．10．Related Topics,References,and Bibliography27

Chapter 2．Plane Analytic Geometry28

2．1．Introduction and Basic Concepts29

2．2．The Straight Line35

2．3．Relations Involving Points and Straight Lines37

2．4．Second-order Curves(Conic Sections)39

2．5．Properties of Circles,Ellipses,Hyperbolas and Parabolas46

2．6．Higher Plane Curves51

2．7．Related Topics,References,and Bibliography52

Chapter 3．Solid Analytic Geometry54

3．1．Introduction and Basic Concepts55

3．2．The Plane64

3．3．The Straight Line66

3．4．Relations Involving Points,Planes,and Straight Lines67

3．5．Quadric Surfaces71

3．6．Related Topics,References,and Bibliography79

Chapter 4．Funetions and Limits．Differential and Integral Caleulus80

4．1．Introduction82

4．2．Functions82

4．3．Point Sets,Intervals,and Regions84

4．4．Limits,Continuous Functions,and Related Topics87

4．5．Differential Caiculus92

4．6．Integrals and Integration99

4．7．Mean-value Theorems．Values of indeterminate Forms．Weierstrass＇s Approximation Theorems115

4．8．Infinite Series,Infnite Products,and Continued Fractions118

4．9．Tests for the Convergence and Uniform Convergence of Infinite Series and Improper Integrals124

4．10．Representation of Functions by Infinite Series and Integrals．Power Series and Taylor＇s Expansion128

4．11．Fourier Series and Fourier Integrals131

4．12．Related Topics,References,and Bibliography140

Chapter 5．Vector Analysis141

5．1．Introduction142

5．2．Vector Algebra143

5．3．Vector Calculus:Functions of a Scalar Parameter147

5．4．Scalar and Vector Fields149

5．5．Differential Operators153

5．6．Integral Theorems158

5．7．Specification of a Vector Field in Terms of Its Curl and Divergence160

5．8．Related Topics,References,and Bibliography162

Chapter 6．Curvilinear Coordinate Systems164

6．1．Introduction164

6．2．Curvilinear Coordinate Systems165

6．3．Representation of Vectors in Terms of Components167

6．4．Orthogonal Coordinate Systems．Vector Relations in Terms of Orthog-onal Components169

6．5．Formulas Relating to Special Orthogonal Coordinate Systems181

6．6．Related Topics,References,and Bibliography182

Chapter 7．Functions of a Complex Variable183

7．1．Introduction184

7．2．Functions of a Complex Variable．Regions of the Complex-number Plane184

7．3．Analytic(Regular,Holomorphic)Functions188

7．4．Treatment of Multiple-valued Functions189

7．5．Integral Theorems and Series Expansions191

7．6．Zeros and Isolated Singularities193

7．7．Residues and Contour Integration197

7．8．Analytic Continuation199

7．9．Conformal Mapping200

7．10．Functions Mapping Specified Regions onto the Unit Circle213

7．11．Related Topics,References,and Bibliography213

Chapter 8．The Laplace Transformation and Other Integral Transformations215

8．1．Introduction216

8．2．The Laplace Transformation216

8．3．Correspondence between Operations on Object and Result Functions219

8．4．Tables of Laplace-transform Pairs and Computation of Inverse Laplace Transforms222

8．5．＂Formal＂Laplace Transformation of Impulse-function Terms227

8．6．Some Other Functional Transformations227

8．7．Related Topics,References,and Bibliography231

Chapter 9．Ordinary Differential Equations233

9．1．Introduction234

9．2．First-order Equations237

9．3．Linear Differential Equations242

9．4．Linear Differential Equations with Constant Coefficients253

9．5．Nonlinear Second-order Equations264

9．6．Pfaffian Differential Equations270

9．7．Related Topics,References,and Bibliography272

Chapter 10．Partial Differential Equations273

10．1．Introduction and Survey274

10．2．Partial Differential Equations of the First Order276

10．3．Hyperbolic,Parabolic,and Elliptic Partial Differential Equations．Characteristics288

10．4．Linear Partial Differential Equations of Physics．Particular Solutions297

10．5．Integral-transform Methods310

10．6．Related Topics,References,and Bibliography314

Chapter 11．Maxima and Minima315

11．1．Introduction316

11．2．Maxima and Minima(Extreme Values)of Functions of One Real Variable316

11．3．Maxima and Minima(Extreme Values)of Functions of Two or More Real Variables317

11．4．Calculus of Variations．Maxima and Minima of Definite Integrals320

11．5．Solution of Variation Problems in Terms of Differential Equations322

11．6．Solution of Variation Problems by Dircct Methods328

11．7．Related Topics,Rcferences,and Bibliography329

Chapter 12．Definition of Mathematical Models:Modern(Abstract)Alsebra and Abstract Spaces331

12．1．Introduction332

12．2．Algebra of Models with a Single Defining Operation:Groups336

12．3．Algebra of Models with Two Defining Operations:Rings,Fields,and Inte-gral Domains340

12．4．Models Involving More Than One Class of Mathematical Objects:Linear Vector Spaces and Linear Algebras342

12．5．Models Permitting the Definition of Limiting Processes:Topological Spaces344

12．6．Order347

12．7．Combination of Models:Direct Products,Product Spaces,and Direct Sums348

12．8．Boolean Algebras350

12．9．Related Topics,References,and Bibliography353

Chapter 13．Matrices．Quadratic and Hermitian Forms355

13．1．Introduction356

13．2．Matrix Algebra and Matrix Calculus356

13．3．Matrices with Special Symmetry Properties362

13．4．Equivalent Matrices．Eigenvalues,Diagonalization,and Related Topics364

13．5．Quadratic and Hermitian Forms368

13．6．Related Topics,References,and Bibliography372

Chapter 14．Linear Vector Spaces and Linear Transformations(Linear Operators)．Representation of Mathematical Models in Terms of Matrices374

14．1．Introduction．Reference Systems and Coordinate Transformations376

14．2．Linear Vector Spaces378

14．3．Linear Transformations(Linear Operators)382

14．4．Linear Transformations of a Normed or Unitary Vector Space into Itself．Hermitian and Unitary Transformations(Operators)384

14．5．Matrix Representation of Vectors and Linear Transformations(Opera-tors)388

14．6．Change of Reference System390

14．7．Representation of Inner Products．Orthonormal Bases392

14．8．Eigenveetors and Eigenvalues of Linear Operators396

14．9．Group Representations and Related Topics406

14．10．Mathematical Description of Rotations411

14．11．Reiated Topics,References,and Bibliography418

Chapter 15．Linear Integral Equations,Boundary-value Problems,and Eigenvalue Problems420

15．1．Introdnetion422

15．2．Functions as Vectors．Expansions in Terms of Orthogonal Functions423

15．3．Linear Integral Transformations and Linear Integral Equations428

15．4．Linear Boundary-value Problems and Eigenvalue Problems Involving Differential Equations438

15．5．Green＇s Functions．Relation of Boundary-value Problems and Eigen-value Problems to Integral Equations451

15．6．Potential Theory456

15．7．Related Topics,References,and Bibliography467

Chapter 16．Representation of Mathematical Models:Tensor Algebra and Analysis469

16．1．Introduction470

16．2．Absolute and Relative Tensors473

16．3．Tensor Algebra:Definition of Basic Operations476

16．4．Tensor Algebra:Invariance of Tensor Equations479

16．5．Symmetric and Skew-symmetric Tensors479

16．6．Local Systems of Base Vectors481

16．7．Tensors Defined on Riemann Spaces．Associated Tensors482

16．8．Scalar Products and Related Topics485

16．9．Tensors of Rank Two(Dyadics)Defined on Riemann Spaces487

16．10．The Absolute Differential Calculus．Covariant Differentiation488

16．11．Related Topics,References,and Bibliography496

Chapter 17．Differential Geometry497

17．1．Curves in the Euclidean Piane498

17．2．Curves in Three-dimensional Euclidean Space501

17．3．Surfaces in Three-dimensional Euclidean Space505

17．4．Curved Spaces515

17．5．Related Topics,References,and Bibliography520

Chapter 18．Probability Theory and Random Processes521

18．1．Introduction523

18．2．Definition and Representation of Probability Models524

18．3．One-dimensional Probability Distributions529

18．4．Multidimensional Probability Distributions538

18．5．Functions of Random Variables．Change of Variables550

18．6．Convergence in Probability and Limit Theorems554

18．7．Special Techniques for Solving Probability Problems557

18．8．Special Probability Distributions559

18．9．Description of Random Processes571

18．10．Generalized Fourier Analysis for Stationary Random Processes．Corre-lation Functions and Spectra577

18．11．Examples of Random Processes583

18．12．Related Topics,References,and Bibliography585

Chapter 19．Mathematical Statistics587

19．1．Introduction to Statistical Methods588

19．2．Statistical Description．Definition and Computation of Random-sample Statistics591

19．3．General-purpose Probability Distributions597

19．4．Estimation of Parameters599

19．5．Sampling Distributions603

19．6．Tests of Statistical Hypotheses609

19．7．Some Statistics,Sampling Distributions,and Tests for Multivariate Dis-tributions619

19．8．Related Topics,References,and Bibliography626

Chapter 20．Numerical Calculations and Finite Differences627

20．1．Introduction629

20．2．Numerical Solution of Equations630

20．3．Linear Simultaneous Equations and Matrix Inversion．Eigenvalues and Eigenvectors of Matrices635

20．4．Finite Differences and Difference Equations641

20．5．Polynomial Interpolation,Numerical Harmonic Analysis,and Other Approximation Methods650

20．6．Numerical Differentiation and Integration662

20．7．Numerical Solution of Ordinary Differential Equations666

20．8．Numerical Solution of Partial Differential Equations,Boundary-value Problems,and Integral Equations671

20．9．Related Topics,References,and Bibliography680

Chapter 21．Special Functions682

21．1．Introduction684

21．2．The Elementary Transcendental Functions684

21．3．Some Functions Defined by Transcendental Integrals695

21．4．The Gamma Function and Related Functions697

21．5．Binomial Coefficients and Factorial Polynomials．Bernoulli Polynomials and Bernoulli Numbers700

21．6．Elliptic Functions,Elliptic Integrals,and Related Functions703

21．7．Orthogonal Polynomials722

21．8．Cylinder Functions,Associated Legendre Functions,and Spherical Har-monics727

21．9．Step Functions and Symbolic Impulse Functions740

21．10．References and Bibliography746

Appendix A．Formulas Describing Plane Figures and Solids747

Appendix B．Plane and Spherical Trigonometry751

Appendix C．Permutations,Combinations,and Related Topics760

Appendix D．Tables of Fourier Expansions and Laplace-transform Pairs763

Appendix E．Tables of indefinite and Definite Integrals787

Appendix F．Numerical Tables827

Squares828

Logarithms831

Trigonometric Functions848

Exponential and Hyperbolic Functions856

Natural Logarithms863

Sine Integral865

Cosine Integral866

Exponential and Related Integrals867

Complete Elliptie Integrals871

Factorials and Their Reciprocals872

Binomial Coefficients872

Gamma and Factorial Functions873

Bessel Functions875

Legendre Polynomials898

Error Function899

Normal-distribution Areas900

Normal-curve Ordinates901

Distribution of t902

Distribution of x2903

Distribution of F904

Glossary of Symbols and Notations909

Index915