MATHEMATICAL HANDBOOK FOR SCIENTISTS AND ENGINEERS

Definitions, Theorems, and Formulas

for Reference and Review

SECOND, ENLARGED AND REVISED EDITION

G R A N I N O A. K O R N , P h . D .

Professor of Electrical Engineering The University of Arizona

THERESA M. KORN, M.S.

M c G R A W - H I L L B O O K C O M P A N Y

New York San Francisco Toronto London Sydney

CONTENTS

Preface vü

C h a p t e r 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 Nürnberg 7 1.4. Miscellaneous Formulas 10 1.5. Determinants 12 1.6. Algebraic Equat ions: General Theorems 15 1.7. Factoring of Polynomials and Quotients of Polynomials. Partial Frac-

tions 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 Ana ly t i c G e o m e t r y 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 G e o m e t r y 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. F u n c t i o n s a n d L imi t s . 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

sl

CONTENTS xii

4.6. Integrals and Integration 102 4.7. Mean-value Theorems. Valuesof IndeterminateForms. Weierstrass's

Approximation Theorems 118 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 146 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 I t s Curl and Divergence. . 164 5.8. Related Topics, References, and Bibliography 166

Chapter 6. Curvilinear Coordinate S y s t e m s 168

6.1. Introduction . . 168 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 Orthog­

onal Components 174 6.5. Formulas Relating to Special Orthogonal Coordinate Systems . . . 177 6.6. Related Topics, References, and Bibliography 177

Chapter 7. Funct ions of a Complex Variable 187

7.1. Introduction 188 7.2. Functions of a Complex Variable. Regions of the Complex-number

Plane 188 7.3. Analytic (Regulär, Holomorphic) Functions 192 7.4. Trea tment 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 Transformat ion and Other Funct iona l 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

xiii CONTENTS

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, lieferences, and Bibliography 240

Chapter 9. Ordinary Differential E q u a t i o n s 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 265 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 Equat ions 287

10.1. Introduction and Survey 288 10.2. Part ial Differential Equations of the First Order 290 10.3. Hyperbolic, Parabolic, and Elliptic Partial Differential Equations.

Characteristics 302 10.4. Linear Part ial 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 M i n i m a and Opt imizat ion 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. Extremais as Solutions of Differential Equat ions: 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 Mathemat ica l Models : Modern ( A b s t r a c t ) 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 *^\2.5. Models Permit t ing 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

C O N T E N T S xiv

C h a p t e r 13. M a t r i c e s . Q u a d r a t i c a n d H e r m i t i a n F o r m a . . . . 402

13.1. Introduction 403 13.2. Matr ix 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 Equa-

tions). Perturbations and Lyapunov Stability Theory 420 13.7. Related Topics, References, and Bibliography 430

C h a p t e r 14. L i n e a r Vec to r Spaces a n d L i n e a r T r a n s f o r m a t i o n s ( L i n e a r O p e r a t o r s ) . R e p r e s e n t a t i o n of M a t h e m a t i c a l Models i n T e r m s of M a t r i c e s 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 (Opera­

tors) 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

C h a p t e r 15. L i n e a r I n t e g r a l E q u a t i o n s , B o u n d a r y - v a l u e P r o b l e m s , a n d E igenva lue P r o b l e m s 484

15.1. Introduct ion. Functional Analysis 486 15.2. Functions as Vectors. Expansions in Terms of Orthogonal Funct ions . 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 Eigen­

value Problems to Integral Equations 515 15.6. Potential Theory 520 15.7. Related Topics, References, and Bibliography 531

C h a p t e r 16. R e p r e s e n t a t i o n of M a t h e m a t i c a l M o d e l s : T e n s o r Algebra a n d Analys is " 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. Symmetrie 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

xv CONTENTS

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. Probabil ity Theory and R a n d o m Processes 585

18.1. Introduotion 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. Mathemat ica l Stat is t ics 664

19.1. Introduotion 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 Est imation 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 Calculat ions and Fini te Differences . . . . 715

20.1. Introduotion 718 20.2. Numerical Solution of Equations 719 20.3. Linear Simultaneous Equations, Matr ix Inversion, and Matrix Eigen-

value 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

CONTENTS xvi

20.8. Numerieal Solution of Ordinary Differential Equations 777 20.9. Numerical Solution of Boundary-value Problems, Part ial Differential

Equations, and Integral Equations 785 20.10. Monte-Carlo Techniques 797 20.11. Related Topics, References, and Bibliography 800

Chapter 21. Special Func t ions 804

21.1. Introduction 806 21.2. The Elementary Transcendental Functions 806 21.3. Some Functions Defined by Transeendental Integrals 818 21.4. The Gamma Function and Related Functions 822 21.5. Binomial Coefficients and Factorial Polynomials. Bernoulli Poly-

nomials 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 Figuren and S o l i d s . . . . 881

Appendix B . P lane a n d Spherical Tr igonometry 885

Appendix C. Permuta t ions , Combinat ions , and Related Topics . 894

Appendix D. Tables of Fourier Expansions and Laplace-transform Pairs 900

Appendix E. Integrals , S u m s , Infinite Series a n d Products , a n d Cont inued Fract ions 925

Appendix F. Numerica l Tables 989

Squares . 990 Logarithms 993 Trigonometrie Functions 1010 Exponential and Hyperbolic Functions 1018 Natura l 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 x

2 1065

xvii CONTENTS

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