Mathematical Handbook for Scientists and Engineers

Mathematical Handbook for Scientists and Engineers
اسم المؤلف
Granino a. Korn and Theresa M. Korn
التاريخ
1 سبتمبر 2020
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Mathematical Handbook for Scientists and Engineers
Definitions, Theorems, and Formulas for Reference and Review
Granino a. Korn and Theresa M. Korn
Contents
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 Plane3.3. 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 a 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 Systems6.3. 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. Tables 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. Introduction9.2. 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 Domains12.4. Models Involving More Than One Class of Mathematical Objects:
Linear Vecto 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 Rotations14.11. 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 Operations
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 Three-dimensional Euclidean Space
17.3. Surfaces in Three-dimensional Euclidean Space
17.4. Curved Spaces
17.5. Related Topics, References, and BibliographyChapter 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 Problems
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 Randomsample 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 Problems20.4. 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 Integration0
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
INDEX
References are to section numbers. References to essential definitions are
printed in boldface numbers to permit the use of this index as a
mathematical dictionary. Numbers preceded by letters (A-2) refer to the
Appendixes.
A priori distribution, 19.9-2, 19.9-4
Abadie, 11.4-3
Abelian group, 12.2-1, 12.2-10
Abel’s integral equation, 15.3-10
Abel’s lemma, 4.8-5
Abel’s test, 4.9-1
Abel’s theorem, 4.10-3
Abscissa, 2.1-2
of absolute convergence, 8.2-2
Absolute bound, 4.3-3
Absolute convergence, abscissa of, 8.2-2
circle of, 8.7-3
of expected values, 18.3-3, 18.4-4, 18.4-8
of infinite products, 4.8-7
of integrals, 4.6-2, 4.6-13, 4.9-3
of Laplace transform, 8.2-2
of series, 4.8-1, 4.8-3, 4.9-1
Absolute derivative, 16.10-8
Absolute differential, 5.5-3, 16.10-1
Absolute differential calculus(seeCovariant differentiation)
Absolute first curvature, 17.4-2
Absolute geodesic curvature, 17.4-2
Absolute moment, 18.3-7
Absolute scalar, 16.2-1
Absolute tensor, 16.2-1
Absolute term, 1.6-3
Absolute value, of complex number, 1.3-2of real number, 1.1-6
of vector, 5.2-5, 16.8-1 (See alsoNorm)
Absorption laws, 12.8-1
Acceleration, 17.2-3
of convergence, 20.2-2d
Accessory conditions(seeConstraint)
Adams-Bashforth predictor, 20.8-3, 20.8-4t
Adams-Moulton corrector, 20.8-4t
Addition formulas, elliptic functions, 21.6-7
hyperbolic functions, 21.2-7
trigonometric functions, 21.2-3
Addition theorem, for binomial coefficients, 21.5-1
for chi-square distribution, 19.5-3
for cylinder functions, 21.8-13
for Legendre polynomials, 21.8-13
for probability distributions, 18.8-9, 19.5-3
for spherical Bessel functions, 21.8-13
Additive group, 12.2-10
Adjoint boundary-value problem, 15.4-3, 15.4-4
Adjoint equations, 11.8-2
Adjoint integral equation, 15.3-7
Adjoint kernel, 15.3-1
Adjoint linear differential equations, 13.6-3
Adjoint matrix, 13.3-1
Adjoint operator, 10.3-6, 10.5-1, 14.4-3, 15.4-3
Adjoint variable, 11.6-8, 11.8-2
Adjoint vector spaces (see Conjugate vector spaces)
Adjugate matrix, 13.3-1
Admissible controls, 11.8-1
Admissible statistical hypothesis, 19.6-1
Admissible transformation, 6.2-1, 16.1-2
Advanced potential, 15.6-10
Affine transformation, 14.10-7
Agnesi, 2.6-1
Aircraft attitude, 14.10-6b
(See alsoRotation)
Aitken-Steffens algorithm, 20.2-2dAitken’s interpolation, 20.5-2c
Algebra, 12.1-2
of classes or sets, 4.3-2, 12.8-4, 12.8-5
Algebraic complement, 1.5-4
Algebraic equation, 1.6-3
Algebraic equations, numerical solution, 20.2-1 to 20.2-8
Algebraic function, 4.2-2
Algebraic multiplicity, 13.4-3, 13.4-5, 13.4-6, 14.8-3, 14.8-4
Algebraic numbers, 1.1-2
Alias-type transformation, 14.1-3, 15.2-7, 16.6-1, 16.1-2
Alibi-type transformation, 14.1-3, 14.5-3, 15.2-7 “Almost everywhere,”
4.6-14
Alternating group, 12.2-8
Alternating matrix (see Skew-hermitian matrix) Alternating product,
16.5-4, 16.10-7
Altitude, of spherical triangle, B-6
of trapezoid, A-l of triangle, B-3, B-4
Amplitude, 4.11-4
of a complex number, 1.3-2
of an elliptic function, 21.6-7
Amplitude-modulated sinusoid, Fourier transform of, 4.11-4
Laplace transform of, 8.3-2
Analysis of variance, 19.6-6
Analytic continuation, 7.8-1, 7.8-2
Analytic function, 4.10-4, 4.10-5, 7.3-1
Anchor ring, A-5
Anger, 21.8-4
Angle, between line elements, 17.3-3, 17.4-2
between line segments, 2.1-4, 3.1-7
of rotation, 14.10-2, 14.10-4, 14.10-7
between straight lines, 2.3-2, 3.4-1
in a unitary vector space, 14.2-7
between vectors, 5.2-6, 14.2-7, 16.8-1
Angular bisector, B-3, B-4, B-6
Angular velocity, 5.3-2, 14.10-5
Anharmonic ratio, 7.9-2
Annular Hankel transform, finite, 8.7-K Antecedents, 16.9-1Antiperiodic function, 4.2-2, 4.11-3
Laplace transform of, 8.3-2
Antisymmetric matrix (see Skew-symmetric matrix) Antisymmetry,
12.6-1
Aperiodic component, 18.10-9
Approximate spectrum, 14.8-3, 15.4-5
Approximation of functions, 20.5-1 to 20.6-7
Approximation functions, 20.9-9, 20.9-10
Arc length, 4.6-9, 6.2-3, 6.4-3, 17.2-1, 17.4-2
Arc length, in vector notation, 5.4-4
Archimedes’ spiral, 2.6-2
Area, 4.6-11, 5.4-6, 6.2-3, 17.3-3
element of, 6.4-3, 17.3-3
of plane figures, A-l to A-3, B-4
of spherical triangle, B-6
of triangle, 2.1-4, 2.1-8, B-4
vector representation of, 3.1-10, 5.4-6
Argand plane, 1.3-2
Argument, of a complex number, 1.3-2
of a function, 4.2-1
principle of the, 7.6-9
Aristotelian logic, 12.8-6
Arithmetic, 1.1-2
Arithmetic mean, 4.6-3
Arithmetic progression, 1.2-6
Arithmetic series, E-4
Artificial variables, in linear programming, HA-2d Associate matrix,
13.3-1
Associate operator, 14.4-3
Associated elliptic integrals, 21.6-6
Associated Laguerre functions, 21.7-5
Associated Laguerre polynomials, 21.7-5, 21.7-7
Associated Legendre functions, 21.8-10
Associated Legendre polynomials, 21.8-10, 21.8-12
Associated metric tensor, 16.7-1
Associated tensors, 16.7-2
differentiation of, 16.10-5Associative law, 1.1-2, 12.2-1, 12.4-1
Astroid, 2.6-1
Asymmetrical impulse (see Impulse functions) Asymmetrical step
function (see Step function) Asymptote, 2.5-2, 17.1-6
Asymptotic cone, 3.5-7
Asymptotic direction, 17.3-6
Asymptotic distribution of eigenvalues, 15.4-8
Asymptotic line, 17.3-6
Asymptotic relations, 4.4-3
Asymptotic series, 4.8-6
for associated Legendre polynomials, 21.8-10
for cylinder functions, 21.8-9
for inverse Laplace transform, 8.4-9
Asymptotic stability, 9.5-4, 13.6-5, 13.6-6
in the large, 1.5-4, 13.6-5
Asymptotically efficient estimate, 19.4-1, 19.4-2, 19.4-4
Asymptotically normal random variables. 18.6-4, 18.6-5
Attitude, aircraft, 14.10-66 (See also Rotation) Augmented matrix, 1.9-4
Autocorrelation function, effect of operations, 18.12-1 to 18.12-5
ensemble, 18.9-3, 18.10-2
examples, 18.11-1 to 18.11-3, 18.11-5, 18.11-6
normalized 18.10-26 t average, 18.10-8 to 18.10-10
Autocovariance function, 19.8-1
Automorphism, 12.1-6, 12.2-9
Autonomous system, 13.6-1
stability, 13.6-6
Auxiliary kernels, 15.3-4, 15.3-9
Average, 4.6-3
of periodic waveforms, D-1t (See also Ensemble average; Sample
average; t average) Averaging time, 19.8-2
Axial symmetry, 10.4-3
Axial vector, 16.8-4
Axis, of curvature, 17.2-5
of revolution, 3.1-15
Backward difference, 20.4-1
Backward-difference operator, 20.4-1
Bairstow’s method, 20.2-4Ball, 12.5-3, 12.5-4
Banach space, 14.2-7, 14.8-3
Banachiewicz, 20.3-lc Banach’s contraction-mapping theorem, 12.5-6,
20.2-1, 20.2-6, 20.3-5
Band-limited functions, 18.11-2a Band-limited random process, 18.11-
26
Bang-bang control, 11.8-36
Base, of a logarithm, 1.2-3
powers of, 1.2-1
of a topology, 12.5-1
Base vectors, abstract, 14.5-1, 14.6-1
cartesian, 5.2-1
in curvilinear coordinates, 6.3-3, 16.8-2
differentiation of, 16.10-1, 16.10-3
in n-dimensional space, 14.2-4
in orthogonal coordinates, 6.4-1, 16.8-2
Bashforth-Adams formula, 20.8-3, 20.8-4t
Basic variables in linear programming, 11.4-2
Basis (see Base vectors) orthonormal (see Complete ortho-normal set)
Bayes estimation, 19.9-2, 19.9-4
Bayes test, 19.9-2, 19.9-3
Bayes theorem, 18.2-6, 18.4-5, 19.9-2, 19.9-4
Bellman, 11.8-6, 11.9-1
Beltrami parameters, 17.3-7
Bending invariant, 17.3-8
Bendixson’s theorems, 9.5-3
Bernoulli, 2.6-1
Bernoulli numbers, 19.2-5, 21.5-2, 21.5-3
Bernoulli polynomials, 21.2-12, 21.5-2, 21.5-3
Bernoulli trials, 18.7-3, 18.8-1, 19.2-1
Bernoulli’s differential equation, 9.2-4
Bernoulli’s theorem, 18.1-1, 18.6-5
Bessel functions, approximations, 20.6-32
modified, 21.8-6
spherical, 10.4-4, 21.8-8, 21.8-13 (See also Cylinder functions)
Bessel’s differential equation, 9.3-3, 21.8-1
Green’s function for, 9.3-3modified, 21.8-6
Bessel’s inequality, 14.7-3, 15.2-3
Bessel’s integral formula, 21.8-2
Bessel’s interpolation formula, 20.5-3, 20.7-1
for two-way interpolation, 20.5-6
Beta distribution, 18.8-5, 19.5-3
Beta function, 19.8-5, 21.4-3
Beta-function ratio, 18.8-5
Bias, 19.4-1
Bilateral Laplace transformation, 8.6-2
Bilinear form, 13.5-1
Bilinear transformation, 7.9-2, 21.6-5
Bimodal distribution, 18.3-3
Binomial coefficient, 1.4-1, 21.5-1, 21.5-3, 21.5-4
tables, C-l to C-3
Binomial distribution, 18.7-3, 18.8-1, 18.8-9, 19.4-2
generalized, 18.7-3
negative, 18.8-1
Binomial series, 21.2-12
Binomial theorem, 1.4-1
Vandermonde’s, 21.5-1
Binormal, 17.2-2 to 17.2-4
Bipolar coordinates, 6.5-1
Bisector, angular, B-3, B-4, B-6
Biunique transformation, 12.1-4
Bivector (see Alternating product) Block relaxation, 20.3-2
Body axes, 14.10-4
Bogolyubov, 9.5-5
Bolza, problem of, 11.6-6
Bolzano-Weierstrass theorem, 12.5-4
Bonnet, 17.3-14
Boolean algebra, 12.6-1, 12.8-1 to 12.8-8
Boolean function, 12.8-2, 12.8-7
Borel set, 4.6-14
Borel’s convolution theorem (see Convolution theorems) Bound, 4.3-3
for eigenvalues, 14.8-9
of a linear operator, 14.4-1of a matrix, 13.2-1
Boundary, 4.3-6
in ordered sets, 12.6-1
of a set, 12.5-1
Boundary collocation, 20.9-9
Boundary conditions, numerical representation, 20.9-6
Boundary maxima and minima, 11.2-1, 11.6-7, 11.8-3 (See also Linear
programming problems; Nonlinear programming) Boundary point,
4.3-6
Boundary-value problem, classification, 10.3-4, 10.4-1
of optimal-control theory, 11.8-2
reduction to initial-value problem, 9.3-4
Bounded operator, 15.3-1
Bounded region, 4.3-6, 7.2-4
Bounded representation, 14.9-1
Bounded set, 4.3-3
Bounded variation, 4.4-8
Boundedly compact space, 12.5-3
Box product (see Scalar triple product) Brachistochrone, 11.6-1
Branch, 7.4-1 to 7.4-3, 7.6-2, 7.8-1
Branch cut, 7.4-2, 7.7-2
Branch point, 7.4-2, 7.6-2
Brianchon’s theorem, 2.4-11
Budan’s theorem, 1.6-6
Burnside’s theorem, 14.9-3
Campbell’s theorem, 18.11-5
Cancellation laws, 1.1-2, 12.2-1, 12.3-1
Canonical equations, 10.2-6, 11.8-2
solution of, 10.2-7
Canonical form, of Boolean function, 12.8-2
of partial differential equation, 10.3-3
of quadratic and hermitian forms, 13.5-4
Canonical maxterm, 12.8-7
Canonical minterm, 12.8-7
Canonical transformation, 10.2-6
Canonically conjugate variables, 10.2-6
Cantor, 4.3-1Cap, 12.8-1
Capacity, 16.2-1, 16.10-10
Cardinal number, 4.3-2
Cardioid, 2.6-1
Cartesian coordinates, local, 17.4-7
n-dimensional, 17.4-6, 17.4-7
plane, 2.1-2
right-handed rectangular, 2.1-3, 3.1-4
in space, 3.1-2
Cartesian decomposition, of complex number, 1.3-1
of linear operators, 14.4-8
of matrices, 13.3-4
Cartesian product, 12.7-1
Casoratian determinant, 20.4-4a Catenary, 2.6-2
Cauchy boundary-value problem, 10.2-2, 10.2-4, 10.3-1, 10.3-5
Cauchy-Goursat integral theorem, 7.5-1
Cauchy principal value, 4.6-2, 7.7-3
Cauchy-Riemann equations, 7.3-2, 15.6-8
Cauchy-Schwarz inequality, 1.3-2
for functions, 4.6-19, 15.2-1
for vectors, 14.2-6
Cauchy sequence, 12.5-4, 15.2-2
Cauchy’s distribution, 18.8-5, 18.8-9
Cauchy’s inequality, 7.5-2
Cauchy’s integral formula, 7.5-1
Cauchy’s integral test, 4.9-1
Cauchy’s mean-value theorem, 4.7-1
Cauchy’s ratio and root tests, 4.9-1
Cauchy’s rule for series, 4.8-3
Cauchy’s test for convergence, 4.9-1 to 4.9-4
Causal distribution, 18.8-1, 18.8-5
Cayley-Hamilton theorem, 13.4-7
Cayley-Klein parameters, 14.10-4
Cayley’s theorem, 12.2-9, 14.9-1
Center, of curvature, 17.1-4, 17.2-2, 17.2-5
of gravity, 18.4-4, 18.8-8, 19.7-2
of a triangle, B-3of a group, 12.2-7
Central of a group, 12.2-7
Central conic, 2.4-6
Central difference, 20.4-1
Central-difference operator, 20.4-2
Central factorial moment, 18.3-7
Central limit theorem, 18.6-5, 19.3-1
Central mean, 20.4-1
Central-mean operator, 20.4-2
Central moment, 18.3-7, 18.3-10, 18.4-3, 18.4-8
Central quadric, 3.5-3, 3.5-5
CEP (circular probable error), 18.8-7
Certain event, 18.2-1
Césaro’s means, 4.8-5, 4.11-7
Cetaev’s theorem, 13.6-6
Chain, 12.6-1
Chapman, 18.11-4
Character, of representation, 14.9-4, 14.9-5, 14.9-6
of rotation group, 14.10-8
Characteristic, of integral domain, 12.3-1
of partial differential equation, 10.2-1, 10.3-1, 10.3-2, 10.3-5 to 10.3-
7
of a surface, 17.3-11
Characteristic directions, 10.2-1
Characteristic equation, of a conic, 2.4-5
of an eigenvalue problem, 14.8-5, 14.8-7
of linear differential equation, 9.4-1
of a matrix, 13.4-5, 13.4-7
in perturbation theory, 15.4-11
of quadric, 3.5-4
Characteristic equations, partial differential equations, 10.2-1, 10.2-4
solution of, 10.2-3, 10.2-4
Characteristic function, 18.3-8
addition theorem, 18.5-7
continuity theorem, 18.6-2
multidimensional, 18.4-10
of probability distribution, 18.3-10of a random process, 18.9-3c of special distributions, 18.8-1, 18.8-2,
18.8-8 (See also Eigenfunction) Characteristic oscillations, 10.4-9
[See also Normal modes) Characteristic quadratic form, 3.5-4
Characteristic strip, 10.2-1
Characteristic value (see Eigenvalue) Characteristic vector (see
Eigenvector) Charlier, 19.3-3
Chebyshev polynomials, 21.7-4, 27.1-17, F-22t
shifted, 20.6-4
use for approximation, 20.6-3 to 20.6-5
Chebyshev quadrature formula, 20.7-3
Chebyshev’s inequality, 18.3-5
Chebyshev’s theorem, 18.6-5
Checking computations, 20.1-2
Chipart, 1.6-6
Chi-square distribution, 19.5-3, 19.7-5
Chi-square test, 19.6-7
Cholesky, 20.3-1
Chord of a circle, A-3
Christoffel, 7.9-4
Christoffel three-index symbols, 16.10-1, 16.10-3, 17.4-5
in cylindrical coordinates, 6.5-1
in spherical coordinates, 6.5-1
on surface, 17.3-7
Circle, of curvature, 17.1-4, 17.2-2
formulas for, A-3
of absolute convergence, 8.7-3
properties of, 2.5-1
Circle theorem, 14.8-9
Cicular frequency, 4.11-4, 10.4-8
Circular probability paper, 18.8-7
Circular probable error, 18.8-7
Circumscribed circle, of regular polygons, A-2
of triangle, B-4
Circumscribed cone, B-6
Cissoid, 2.6-1
Clairaut’s differential equation, 9.2-4, 10.2-3
Class frequency, 19.2-2Class interval, 19.2-2
Clebsch-Gordan equation, 14.10-7
Clipped sinusoid, D-1Z Clippinger, 20.8-4c Closed integration formula,
20.8-3c Closed interval, 4.3-4
Closed set, 4.3-6, 12.5-1
Closure of a set, 12.5-1
Closure property, 1.1-2, 12.2-1
Codazzi, 17.3-8
Coded data, 19.2-5
Cof actor, 1.5-2
Coherence, 18.10-9
Collatz, 20.2-2
Collinear points, 2.3-1, 3.4-3
Collocation, 20.9-9, 20.9-10
Column matrix, 13.2-1
Combinations, tables, C-l to C-3
Combinatorial analysis, 18.7-2
tables, C-l to C-3
Common divisors, 1.7-3
Commutative group (see Abelian group) Commutative law, 1.1-2
Commutator, 14.4-2
Commuting operators, 14.4-9, 14.8-6, 14.9-3
Compact set, 12.5-16
Compact space, 12.5-16
Companion matrix, 20.2-5
Comparison of populations, 19.6-6, 19.6-8
Comparison tests for convergence, 4.9-1 to 4.9-4
Comparison theorems, 14.8-9
for eigenvalue problems, 15.4-10
Compatibility conditions, 10.1-2, 17.3-8
Complement, in Boolean algebra, 12.8-1
of an event, 18.2-1
of a set, 4.3-2
Complementary-argument theorem, 21.5-2
Complementary equation, 9.3-1, 15.4-2, 20.4-4« Complementary error
function, 21.3-2
Complementary function, 9.3-1Complementary modular angle, 21.6-6« Complementary modulus, 21.6-
6
Complete additivity, 12.8-8, 18.2-1
Complete beta function, 21.4-4
Complete elliptic integrals, 21.6-6
Complete hermitian kernel, 15.3-4
Complete integral, of ordinary differential equation, 9.1-2
of partial differential equation, 10.2-3, 10.2-4
Complete orthonormal set, of functions, 10.4-2, 10.4-9, 15.2-4, 15.4-6,
15.4-12, 21.8-12 (See also Eigenfunction) of vectors, 14.7-4
Complete primitive (see Complete integral) Complete set of invariants,
12.2-8, 14.1-4
Complete solution of algebraic equation, 1.6-3
Complete space, 12.5-4, 14.2-7, 14.8-4, 15.2-2
Complete stability, of linear system, 9.4-4, 13.6-7
of a solution (see Asymptotic stability) Completely reducible
operator, 14.8-2
Completely reducible representation, 14.9-2, 14.9-4 to 14.9-6
Completely skew-symmetric tensor, 16.5-1 to 16.5-3
Completely stable system, 20.4-8
Completely symmetric tensor, 16.5-1
Complex, 12.2-4
Complex conjugate, 1.3-1
Complex-conjugate matrix, 13.3-1
Complex number, 1.3-1
Complex potential, 15.6-8
Complex vector space, 14o2-l Components, representation in terms of,
14.1-2, 14.2-4, 16.1-3
Composite character, 14.9-4
Composite statistical hypothesis, 19.6-1, 19.6-3, 19.6-4
Composition factor, 12.2-6
Composition series, 12.2-6
Compound distribution, 18.5-8
Compound experiments, 18.2-4
Compound probabilities, 18.2-2
Concave curve, 17.1-4
Conchoid, 2.6-1Conditional entropy, 18.4-12
Conditional expected value, 18.4-5, 18.4-9, 19.9-4
Conditional frequency function, 18.4-5
Conditional mean (see Conditional expected value) Conditional
probability, 18.2-2, 18.4-5
Conditional probability density (see Conditional frequency function)
Conditional probability distributions, of random process, 18.9-2
Conditional risk, 19.9-2
Conditional variance, 18.4-5
Conditionally compact space, 12.5-3
Cone, 3.1-5
Confidence coefficient, 19.6-5
Confidence level, 19.6-5
Confidence limits, 19.6-5
Confidence region, 19.6-5, 19.7-7
Configuration, C-2
Configuration-counting series, C-2
Configuration inventory, C-2
Confluent hypergeometric function, 9.3-10, 21.7-1
Conformable matrices, 13.2-2
Conformai mapping, 7.9-1 to 7.10-1, 15.6-8
of surfaces, 17.3-10
Congruent matrices, 13.4-1
Congruent modulo r, 12.2-10
Conic (see Conic section) Conic section, 2.4-1 to 2.4-9
central, 2.4-3
classification, 2.4-3
degenerate, 2.4-3
improper, 2.4-3
proper, 2.4-3
Conjugate axis, 2.5-2
Conjugate chords, 2.4-6, 3.5-5
Conjugate diameters, 2.5-2, 3.5-9
Conjugate diametral plane, 3.5-5
Conjugate directions, surface, 17.3-6
Conjugate-gradient method, 20.3-2/ Conjugate group elements, 12.2-5,
14.9-3, 14.9-4Conjugate harmonic functions, 15.6-8
Conjugate matrix, 13.3-1
Conjugate operator, 14.4-3, 14.4-9
Conjugate subgroups, 12.2-5, 12.2-9
Conjugate vector spaces, 14, 4-9, 15.4-3
Conjunct, 15.4-3
Conjunctive matrices, 13.4-1
Connected sets, 12.5-1
Consequents, 16.9-1
Conservation of functional equations, 7.8-1
Consistency property, 12.8-1
Constant of integration, 4.6-4, 9.1-2
Constraint, 110.3-4, 11.6-2, 11.6-3, 11.6-7, 11.7-1, 11.8-le, 14.8-9, 15.4-
7, 15.4-10. 20.2-6d (See also Inequality constraints) Construction, of
ellipses and hyperbolas, 2.5-3
of parabolas, 2.5-4
Constructive definition, 12.1-1
Contact (see Osculation) Contact transformation, 9.2-3, 10.2-5 to 10.2-7,
11.5-6
Contagion, 18.8-1
Content, of a configuration, C-2
of a figure, C-2
Contingency table, 19.7-5
Continued-fraction expansion, 4.8-8, 20.5-7, E-9
Continuity, 4.4-6, 12.5-1
Continuity axiom (see Coordinate axiom) Continuity in the mean, 18.9-
3d Continuity theorem, of characteristic function, 18.6-2
for distribution functions, 18.6-2
for Fourier transforms, 4.11-5
for integrals, 4.6-16
for Laplace transforms, 8.3-12
for series, 4.8-4
for z transform, 8.7-32
Continuous function, 4.4-6
Continuous group, 12.2-11, 12.2-12
Continuous in mean, 15.3-1, 18.9-3
Continuous random process, 18.9-1Continuous random variable, 18.3-2, 18.4-3, 18.4-7
Continuous spectrum, 14.8-3, 15.4-5
Continuous vector function, 5.3-1
Continuously differentiate function, 4.5-1, 4.5-2
Contour, 7.2-3
Contour ellipse, of normal distribution, 18.8-6
Contour integrals, 7.2-5, 7.7-3
in Laplace transforms, 8.4-3
Contraction of tensors, 16.3-5, 16.7-4
Contraction mapping, 12.5-6, 20.2-1, 20.2-2, 20.2-6«, 20.3-5
Contraction rule, 16.10-5
Contragredient transformations, 14.7-6, 16.6-1
Contravariant base vectors, 16.6-1
Contravariant components, 6.3-3, 16.2-1
Contravariant vector, 16.2-1, 16.6-1, 16.7-3
Control, optimal, 11.8-1 to 11.9-2
Control variable, 11.8-1
Convergence, of matrices, 13.2-11
in mean, 12.5-12, 15.2-2
for random variables, 18.6-3
in metric space, 12.5-3
in probability, 18.6-1 (See also Improper integrals; Infinite series;
Power series) Convergence acceleration, 4.8-5, 20.2-2d
Convergence criteria, 4.9-1 to 4.9-4
Convex curve, 17.1-4
Convex set, 11.4-16
Convolution, 4.6-18
Convolution integral, 9t4-3, 10.5-4
Convolution theorems, 4.11-52, 8.3-12, 8.3-3, 8.6-2, 8.7-32, 18.10-8
Coordinate axes, 2.1-1, 3.1-2, 3.1-3
Coordinate axiom, 2.1-2, 4.3-1
Coordinate line, 6.2-2
in curved space, 17.4-2
Coordinate surface, 6.2-2
Coordinate system, 2.1-1, 14.1-2, 14.2-4
cartesian, 2.1-2, 3.1-2
n-dimensional, 17.4-6, 17.4-7rectangular, 2.1-3, 3.1-4
right-handed, 2.1-2, 3.1-3
choice of, 10.4-1
curvilinear, 6.2-1
cylindrical, 3.1-6, 6.5-1
orthogonal, 6.4-1 to 6.5-1, 16.8-2, 16.9-1, 16.9-3, 16.10-3, 17.4-7
polar, 2.1-8
special, formulas for, 6.5-1
spherical, 3.1-6, 6.5-1 (See also Base vectors) Coordinate
transformation (see Transformation) Corner conditions, for
extremals, 11.6-7, 11.8-5
Corrections for grouping, 19.2-5
Corrector, 20.8-3 to 20.8-7
Correlation, test for, 19.7-4, 19.7-6
Correlation coefficient, 18.4-4, 18.4-6, 18.4-8
multiple, 18.4-9
partial, 18.4-9
sample, 19.7-2, 19.7-4
Correlation functions, measurement, 19.8-3c (See also Autocorrelation
function; Crosscorrelation function) Correlation matrix, 18.4-8
Coset, 12.2-4, 12.2-11
Cosine integral, 21.3-1
Cosine law, B-4, B-8, B-9
Cosine series, 4.11-2, 8.7-1
Cosine transform, 4.11-3, 4.11-5, D-3
finite, 8.7-1
Cosinus amplitudinis, 21.6-7
Cost, of error, 19.9-1
Cotes, 20.7-2
Count rate, 18.11-4d, 18.11-5
Countable set, 4.3-2
Courant’s minimax principle, 14.8-8, 15.4-7
Covariance, 18.4-4, 18.4-8, 19.7-2 (See also Sample covariance)
Covariant base vectors, 16.6-1
Covariant components, 6.3-3, 16.2-1
Covariant derivative, 16.10-4
Covariant differentiation, 6.3-4, 16.10-1 to 16.10-11on surface, 17.3-7
Covariant vector, 16.2-1, 16.6-1, 16.7-3
Covering theorem, 12.5-4
CPE (circular probable error), 18.8-7
Cramer’s rule, 1.9-2, 14.5-3
Criterion functional, 11.8-1
Critical point, 7.9-1
in phase plane, 9.5-3
Critical region, 19.6-2
Cross-power spectral density, 18.10-5
Cross product (see Vector product) Cross-quadrature spectral density,
18.10-5
Cross ratio, 7.9-2
Cross-spectral density, 18.10-3
in linear systems, 18.12-2 to 18.12-4
non-ensemble, 18.10-8
Crosscorrelation function, 18.9-3, 18.10-2, 18.10-4, 18.12-1 to 18.12-4
Crout, 20.3-1
Cruciform, 2.6-1
Cube, A-6
Cubic equation, 1.6-3, 1.8-3, 1.8-4
Cumulants (see Semi-invariants) Cumulative distribution function (see
Distribution function) Cumulative frequency, 19.2-2
Cumulative relative frequency, 19.2-2
Cup, 12.8-1
Curl, 5.5-1, 5.5-2, 6.4-2, 16.10-7
Curtosis (see Excess) Curvature, of plane curve, 17.1-4
of space curve, 17.2-2, 17.2-3
Curvature invariant, 17.4-6
Curvature tensor, 16.10-6, 17.4-5
Curvature vector, 17.2-2, 17.4-3
Curve, in complex plane, 7.2-3
in curved space, 17.4-2
in plane, 2.1-9, 17.1-1
in space, 3.1-13, 17.2-1
vector representation, 3.1-13, 17.2-1
Curvilinear coordinates, 6.2-1Cusp, 17.1-3
Cycle index, C-2
Cyclic group, 12.2-3
Cyclic permutation, 12.2-8
Cyclic variables, 10.2-7
Cycloid, 2.6-2
Cylinder functions, 10.4-3, 10.4-9, 15.6-10, 21.8-1 to 21.8-9, 21.8-13
approximations, 20.6-3t
Cylindrical coordinates, 3.1-6
vector relations in, 6.5-1
Cylindrical harmonics, 10.4-3, 21.8-1
Cylindrical waves, 10.4-8
d’Alembert’s solution, 10.3-5
Damped wave, 10.4-8
Damping constant, 9.4-1
Damping ratio, 9o4-l Darboux vector, 17.2-3
D-c process, 18.11-1
Decagon, A-2
Deciles, 18.3-3, 19.2-2
Decision function, 19.9-1
Decomposable operator. 14.8-2, 14.9-2, 14.9-4
Decomposition, of matrices, 13.3-4
of operators, 14.4-8
Dedekind, 4.3-1
Dedekind cut, 1.1-2
Defining postulates, 12.1-1
examples, 12.2-1, 12.3-1, 12.4-1, 12.4-2, 12.5-1, 12.5-2
Definite integral, Lebesgue integral, 4.6-15
Riemann integral, 4.6-1
of vector function, 5.3-3
Degenerate conic, 2.4-3
Degenerate eigenvalue, 14.8-3, 14.8-6, 15.3-3, 15.4-8, 15.4-11
Degenerate kernel (see Separable kernel) Degenerate quadric, 3.5-7
Degree, of degeneracy, 14.8-4, 15.3-3, 15.4-5
of freedom, 19.5-3
of a homogeneous function, 4.5-5
of a polynomial, 1.4-3, 1.6-3of a representation, 14.9-1
of truncation, 19.3-4
Del (see Gradient operator) Delambre’s analogies, B-8
Delayed sequence, z transform, 8.7-3
Delta function, multidimensional, 21.9-7 (See also Impulse functions)
De Moivre—Laplace limit theorem. 18.8-1
De Moivre’s theorem, 1.3-3
de Morgan’s laws, 12.8-1
Dense set, 12.5-1
Density, 16.2-1
Denumerable set (see Countable set) Dependent variable, 4.2-1
Derivative, 4.5-1
of complex variables, 7.3-1
Derived set, 12.5-1
Descartes’s rule, 1.6-6
Descriptive definition (see Defining postulates) Detection, 19.9-1 to
19.9-3
Determinant, 1.5-1, 16.5-3
of a linear operator, 14.6-2
of a matrix, 13.2-7, 13.3-2, 13.4-1, 13.4-3, 13.4-5
numerical evaluation, 20.3-1
d’Huilier’s equation, B-8
Diagonal matrix, 13.2-1
Diagonalization, 13.4-4, 13.5-4, 13.5-5, 14.8-6, 14.8-7
Diameter, of a conic, 2.4-6, 2.4-10
conjugate (see Conjugate diameters) of a quadric surface, 3.5-5
Diametral plane, 3.5-5
conjugate, 3.5-5
Difference coefficient, 20.4-3
Difference-differential equation, 10.4-1
Difference equations, 11.7-3, 18.11-4, 20.4-3 to 20.4-8, 20.8-5, 20.9-4,
20.9-8
Difference operators, 20.4-2, 20.9-3
Differentiable function, 4.5-1, 4.5-2. 7.3-1
Differential, 4.5-3
Differential distribution function (see Probability density) Differential
invariant, 5.5-1 to 5.5-8, 16.10-7, 16.10-11, 17.3-7Differential operator, 5.5-1 to 5.5-8, 15.4-1, 16.10-7 (See also
Differential invariant) Differentiation, 4.5-1, 4.5-4
absolute (see Covariant differentiation) of complex functions, 7.3-1
of elliptic functions, 21.6-7
of integrals, 4.6-1
of matrices, 13.2-11
numerical, 20.6-1
of series, 4.8-4
of vectors, 5.3-2
Diffusion equation, 10.4-7, 10.5-3, 10.5-4, 15.5-3, 20.9-4, 20.9-8, 21.6-8
Dimension, 14.1-2, 14.2-4, 14.7-3
of a representation, 14.9-1
Dimsdale, 20.8-4c Diodes, 2.6-1
Dipole, 15.6-5
Dipole radiation, 10.4-8
Dirac (see Impulse functions) Direct methods, calculus of variations,
11.7-1, 11.7-2
of solving linear equations, 20.3-1
Direct product, of groups, 12.7-2
of matrices, 13.2-10
of representations, 14.9-6, 14.10-7
of vector spaces, 12.7-3
Direct sum, of linear algebras, 12.7-5
of matrices, 13.2-10(See also Step matrix) of operators, 14.8-2
of representations, 14.9-2
of rings, 12.7-5
of vector spaces, 12.7-5
Direction cosines, of coordinate lines, 6.3-2
of intersection, 3.4-5
in plane, 2.1-4
in space, 3.1-8
Direction numbers, 3.1-8
Directional derivative, 5.5-3
in Riemann space, 16.10-8
Directrix, of a conic, 2.4-9
of a surface, 3.1-15
Dirichlet integral in potential theory, 15.6-2Dirichlet problem, 7.10-1, 10.4-9, 15.4-10, 15.5-4, 15.6-2, 15.6-6, 15.6-
8, 15.6-9
Dirichlet region, 15.6-2
Dirichlet series, 8.7-3
Dirichlet’s conditions, 4.4-8, 4.11-4
Dirichlet’s integral, 4.11-6, 21.9-4
Dirichlet’s test for convergence, 4.9-1, 4.9-2
Discontinuity of the first kind, 4.4-7
Discrete random process, 18.9-1
Discrete random variable, 18.3-1, 18.4-3, 18.4-7, 18.7-2, 18.7-3
Discrete set, 4.3-6, 4.4-7
Discrete spectrum, 13.4-2, 14.8-3, 15.4-5
Discrete topology, 12.5-1
Discriminant, of an algebraic equation, 1.6-5
of a conic, 2.4-2
of a quadric, 3.5-2
Disjoint elements, 12.8-1
Disjoint events, 18.2-1
Dispersion; 18.3-3, 18.8-7
Displacement operator (see Shift operator) Distance, in abstract space,
12.5-2
in L2, 15.2-2
between lines, 2.3-2
in normed vector space, 12.5-2, 14.2-7
in a plane, 2.1-4, 2.1-8
Distance, between point and line, 2.3-1, 3.4-2
between point and plane, 3.4-2
in space, 3.1-7 (See also Arc length) Distance element, 4.6-9, 6.2-3
in Riemann space, 17.4-2
on surface, 17.3-3 (See also Arc length) Distance function, 12.5-2
Distribution function, 18.2-9, 18.3-1, 18.3-2, 18.4-3, 18.4-7, 18.5-2,
18.6-2
empirical, 19.2-2
Distributions, theory of, 21.9-2
Distributive law, 1.1-2, 12.4-1, 14.2-6
Divergence, 5.5-1, 5.5-2, 6.4-2, 16.10-7
Divergence theorem, 5.6-1, 16.10-11Divergent series, 4.8-1, 4.8-6
Divided differences, 20.5-2, 20.7-1
Division algebra, 12.4-2, 13.2-5, 14.4-2
Division algorithm, 1.7-2
Divisor, 1.7-1
common, 1.7-3
greatest, 1.7-3
of zero, 12.3-1
Dodecahedron, A-6
Domain of definition, 4.2-1, 12.1-4
Dominant eigenvalue, 20.3-5
Doolittle, 20.3-1
Dot product (see Inner product; Scalar product) Double-dot product,
16.9-2
Double point, 17.1-3
Double-precision arithmetic, 20.8-5
Double series, 4.8-3
Doubly periodic function, 21.6-1
Dual vector spaces (see Conjugate vector spaces) Duality, in Boolean
algebra, 12.8-1
in geometry, 3.4-4
in linear programming, 11.4-lc, 11.4-4
Dualization, 12.8-1
Du Bois-Reymond, lemma, 11.6-ld theorem, 11.6-16
Duffing’s equation, 13.6-7
Duhamel, 9.4-3
Duhamel’s formulas, 10.5-3, 10.5-4
Dummy-index notation, 14.7-7, 16.1-3, 16.6-1, 16.10-1
Dyad, 16.9-1
Dyadic, 14.5-4, 16.9-1 to 16.9-3, 16.10-11
Dynamic programming, 11.8-6, 11.9-1, 11.9-2
Eccentricity, 2.4-9
Edge, 17.3-1
Edge, of regression, 17.3-11
Edgeworth, 19.3-2, 19.3-3
Edgeworth series, 19.3-3
Efficiency, of an estimate, 19.4-1Efficient estimate, 19.4-1, 19.4-2, 19.4-4
Eigenfunction, differential equation, 15.4-5
improper, 15.4-5
integral equation, 15.3-3 (See also Eigenvalue problems)
Eigenfunction expansion, 10.4-1, 10.4-2
differential equation, 15.4-6, 15.4-12
of Green’s function, 15.5-2
integral equations, 15.3-4, 15.3-9
of kernel, 15.3-4, 15.3-5
Eigenvalue, 13.4-2, 13.6-2, 13.6-7, 14.8-3, 15.3-3, 15.4-5 (See also
Eigenvalue problems; Hermitian form; Quadratic form) Eigenvalue
problems, differential equations, 15.4-5 to 15.4-11
dyadics, 16.9-3
estimation of solutions, 14.8-9, 15.4-10
generalized, 14.8-7, 15.4-5 to 15.4-11
and group representations, 14.9-3
hermitian, 14.8-4, 15.3-3, 15.4-6
intergral equations, 15.3-3 to 15.3-6
linear operators, 14.8-3 to 14.8-9
matrices, 13.4-2 to 13.4-6
numerical solution, 20.3-5, 20.9-4, 20.9-10
as stationary-value problems, 14.8-8, 15.3-6, 15.4-7
Sturm-Liouville, 15.4-8 to 15.4-10 (See also Characteristic equation;
Diagonalization; Hermitian form; Principal-axes transformation;
Quadratic form; Spectrum) Eigenvector, 14.8-3 to 14.8-9 (See also
Eigenvalue problems; Principal-axes transformation) Einstein tensor,
17.4-5
Elementary event (see Simple event) Elimination of unknowns, 1.9-1,
20.3-1
Ellipse, 2.4-3
construction of, 2.5-3
properties of, 2.5-2
Ellipsoid, 3.5-7
of concentration, 18.4-8
Ellipsoidal coordinates, 6.5-1
Elliptic cone, 3.5-7
Elliptic cylinder, 3.5-7Elliptic differential equation, 10.3-1, 10.3-3, 10.3-4, 10.3-7
Elliptic functions, 21.6-1 to 21.6-9
Elliptic geometry, 17.3-13
Elliptic integrals, 4.6-7, 21.6-4 to 21.6-6
reduction of, 21.6-5
Elliptic paraboloid, 3.5-7
Elliptic point, 17.3-5
Empirical distribution, 19.2-2
Empty set, 4.3-2
Endomorphism, 12.1-6
Energy-integral solution, 9.5-6
Ensemble, 18.9-1, 19.1-2
Ensemble average, 18.9-3 (See also Expected value) Ensemble
correlation functions, 18.9-3, 18.10-2 to 18.10-5
effect of linear operations, 18.12-2
effect of nonlinear operations, 18.12-5, 18.12-6
Ensemble spectral density (see Spectral density) Entire function (see
Integral function) Entrainment, 9.5-5
Entropy, 9.6-2, 18.4-12
Enumerable set (see Countable set) Enumerating generating function, Cl, C-2
Enumerator, C-l, C-2
Envelope, 10.2-3, 17.1-7, 17.3-11
Epicycloid, 2.6-2
Equality, 1.1-3, 12.1-3
Equiareal mapping, 17.3-10
Equilibrium solution, 13.6-6
stability of, 13.6-6
Equipotential lines, 15.6-8
Equipotential surface (see Level surface) Equivalence relation, 12.1-3,
13.4-1
Equivalent bandwidth, of averaging filter, 19.8-2, 19.8-3
Equivalent configurations, C-2
Equivalent linearization, 9.5-5
Equivalent matrices, 13.4-1 (See also Similarity transformation)
Equivalent representations, 14.9-1
Erdmann-Weierstrass conditions, 11.6-7, 11.8-5Ergodic property, 18.10-76
Ergodic random process, 18.10-76
Ergodic theorem, 18.10-76
Error, 20.1-2
of the first kind, 19.6-2
of the second kind, 19.6-2 (See also Residual) Error estimate, 20.2-2
(See also Remainder) Error function, 18.8-3, 21.3-2
table, F-13
Essential singularity, 7.6-2, 7.6-4
of differential equation, 9.3-6
Estimate, 19.7-7
Estimate variance (see Variance, of estimate) Estimation, 19.1-3, 19.4-1
to 19.4-5, 19.7-3
of random-process parameters, 19.8-1 to 19.9-2, 19.9-4
Euclidean geometry, 2.1-7, 17.3-13
Euclidean norm of a matrix, 13.2-1E Euclidean space, 17.4-6
Euclidean vector space, 14.2-7
Euclidean vectors, 5.1-1
Euler angles, 14.10-4 to 14.10-6
Euler diagram, 12.8-5
Euler-Fourier formulas, 4.11-2
Euler-Lagrange equation, 11.6-1, 11.6-2
Euler-MacLaurin summation formula, 4.8-5
Euler-Mascheroni constant, 21.3-1, 21.4-5, 21.8-1
Euler symmetrical parameters, 14.10-3
relation to angular velocity, 14.10-7
Euler’s definition, gamma function, 21.4-1
Euler’s differential equation, 11.6-1, 11.6-2
Euler’s integral, 21.4-4
Euler’s theorem, on Fourier series, 4.11-2
for surfaces, 17.3-5
Euler’s transformation, 4.8-5
Even function, 4.2-2, 4.11-4
table, D-2
Even permutation, 12.2-8, 16.5-3
Event algebra, 12.8-5, 18.2-1, 18.2-2, 18.2-7
Everett’s interpolation formula, 20.6-3Evolute, 17.2-5
Excess, 18.3-3, 19.2-4, 19.5-3
Excluded middle, 12.8-6
Existence theorems, 4.2-1, 9.1-4, 9.2-1, 9.3-5
Expansion theorem, for integral equations, 15.3-4, 15.3-5, 15.3-9
Expected risk, 19.9-1
Expected value, 18.3-3, 18.3-6, 18.4-4, 18.4-8, 18.5-6, 18.5-7
of derivative, 18.9-3d of integral, 18.9-3d (See also Ensemble
average) Explicit method, 20.9-4, 20.9-8
Exponent, 1.2-1
Exponential function, 21.2-9, F-4£ continued-fraction expansion, E-9
power series, E-7
Exponential generating function, 8.7-2
Exponential integral, 21.3-1
Exponential order, 4.4-3, 8.2-4
Extension, 12.3-3
Exterior measure, 4.6-15
Extremals, 11.6-1
Extreme value (see Maxima and minima) F distribution (see v2
distribution) Factor, 1.2-5
Factor group, 12.2-5, 12.2-10
Factor theorem, lo7-l Factorial, 1.2-4
Factorial moment, 18.3-7, 18.3-10
Factorial polynomial, 21.5-1, 21.5-3
Factoring, 1.7-1
Faithful representation, 14.9-1
False alarm, 19.9 3
Feasible solution, of linear programming problem, 11.4-16
Féjer’s integral, 4.11-6
Féjer’s theorem, 4.11-6
Feuerbach circle, B-3
Fibonacci numbers, 8.7-2
Fiducial limits, 19.6-5
Field, 12.3-1
of matrices, 13.2-5
of real numbers, 1.1-2 (See also Potential; Scalar field; Vector field)
Field line, 5.4-3Figure, C-2
Figure-counting series, C-2
Figure inventory, C-2
Figure store, C-2
Filter, averaging, 19.8-2 (See also Linear system) Final-value theorem, z
transform, 8.7-3
Finite-difference methods for differential equations, 20.9-2, 20.9-4 to
20.9-8
Finite induction, 1.1-2
Finite integral transform, 10.5-1
Finite interval, 4.3-4
Finite matrix, 13.2-1
Finite population, 19.5-5
Finite region (see Bounded region) Finite set, 4.3-2
Finite-time average, 19.8-1
sampled-data, 19.8-1
First curvature vector, 17.4-3
First fundamental form, surface, 17.3-3, 17.3-8, 17.3-9
First probability distribution, 18.9-2
Fischer (see Riesz-Fischer theorem) Fisher’s z distribution (see z
distribution) Fisher’s z test, 19.6-6
Fit, 20.5-1, 20.6-1
Fixed point, of a mapping, 12.5-6
Flat space, 16.10-6, 17.4-6
Focal point, in phase plane, 9.5-3, 9.5-4
on a surface, 17.3-11
Focus of a conic, 2.4-9
Forcing function, 9.3-1
periodic, 9.4-6
Forward difference, 20.4-1
Forward-difference operator, 20.4-1, 20.4-2
Fourier analysis, 4.11-4
Fourier-Bessel transform, 8.6-4 (See also Hankel transform) Fourier
coefficients, formulas, 4.11-2, 4.11-5
table, D-l Fourier cosine series, 4.11-3, 4.11-5
Fourier cosine transform, 4.11-3, 10.5-3, D-3
finite, 8.7-1table, D-3
Fourier integral, 4.11-3
multiple, 4.11-8
Fourier-integral representation, of impulse functions, 21.9-5
of step function, 21.9-1
Fourier series, 4.11-2, 10.4-9
multiple, 4.11-8
operations with, 4.11-5 (See also Orthogonal-function expansion)
Fourier sine series, 4.11-3, 4.11-5
Fourier sine transform, 4.11-3, 10.5-3, D-4
finite 8.7-1
table, D-4
Fourier transform, 4.11-3, 4.11-5, 8.6-1
finite, 8.7-1
generalized, 18.10-10
integrated, 18.10-10
properties of, 4.11-5
table, D-2
Fourier-transform pairs, D-2 to D-4
Fractiles, 18.3-3, 19.2-2 (See also Sample fractiles) Fractional error,
21.4-2
Fraser diagram, 20.5-3
Fredholm alternative, 14.8-10, 15.3-7, 15.4-4
Fredholm-type integral equation, 15.3-2, 15.3-3, 15.3-7 to 15.3-9
numerical solution of, 20.8-5
Fredholm’s formulas, 15.3-8
Free index, 16.1-3
Frequency, 4.11-4, 10.4-8
Frequency distribution, 19.2-2
Frequency function (see Probability density) Frequency-response
function, 9.4-7, 20.8-8
Fresnel integrals, 21.3-2
Fritz John theorem, 11.4-3
Frobenius, 9.3-6
Frobenius norm, of a matrix, 13.2-1
Fubini’s theorem, 4.6-8
Fuchs’s theorem, 9.3-6Full linear group, 14.10-7
Full-wave rectified waveform, Fourier series, table, D-2
Laplace transform, 8.3-2
Function, 4.2-1, 12.1-4
Boolean, 12.8-7
of a linear operator, 14.4-2, 14.8-3
of a matrix, 13.2-12, 13.4-5
Function spaces, 12.5-5, 15.2-1 (See also Banach space; Hubert space)
Functional, 12.1-4
Functional analysis, 15.1-1
Functional dependence, 4.5-6
Functional determinant (see Jacobian) Functional equation, 9.1-2
Functional transformation, 8.1-1, 8.6-1, 15.2-7(See also Integral
transformation) Fundamental, 4.11-4
Fundamental form, of surface, 17.3-3, 17.3-5, 17.3-8, 17.3-9
for unitary vector space, 14.7-1
Fundamental probability set (see Sample space) Fundamental region,
7.9-1
Fundamental-solution matrix, 13.6-3
Fundamental system of solutions, 9.3-2, 20.4-4, 21.8-1
Fundamental tensors, 16.7-1 (See also Metric tensor) Fundamental
theorem, of algebra, 1.6-2, 7.6-1
of integral calculus, 4.6-5
of surface theory, 17.3-9
Galerkin, 20.9-9, 20.9-10
Galois field, 12.3-1
Galois theory, 12.3-3
Game theory, 11.4-4
Gamma distribution, 18.8-5
Gamma function 21.4-1 to 21.4-4
incomplete, 18.8-5, 21.4-5
Gauss-Bonnet theorem, 17.3-14
Gauss-Hermite quadrature, 20.7-3
Gauss-Laguerre quadrature, 20.7-3
Gauss plane (see Argand plane) Gauss quadrature formula, 20.7-3, 20.7-
4
Gauss-Seidel method, 20.3-26Gaussian curvature, 17.3-5, 17.3-8, 17.3-13, 17.3-14
Gaussian distribution (see Normal distribution) Gaussian quadrature,
20.7-3, 20.7-4
multiple, 20.7-5
Gaussian random process, 18.11-3, 18.11-5c effect of linear operations,
18.12-2
effect of nonlinear operations, 18.12-6
measurements on, 19.8-3
series expansion, 18.12-56
Gauss’s analogies, B-8
Gauss’s elimination scheme, 20.3-1
Gauss’s equations, surface, 17.3-8
Gauss’s hypergeometric differential equation, 9.3-9
Gauss’s integral formula, 4.6-12
Gauss’s integral theorem (see Divergence theorem) Gauss’s
multiplication theorem, 21.4-1
Gauss’s recursion formulas, 9.3-9
Gauss’s theorem, 5.6-1, 15.6-5
Gauss’s theorema egregium, 17.3-8
Gegenbauer polynomials, 21.7-8
General integral, 10.1-2, 10.2-3, 10.2-4
General solution of partial differential equation (see General integral)
Generalized binomial distribution, 18.7-3
Generalized eigenvalue problem, 14.8-7, 15.4-5 to 15.4-11
Generalized Fourier analysis (see Integrated power spectrum; Spectral
density) Generalized Fourier transform, 4.11-4, 18.10-10
Generalized Laguerre functions, 21.7-5
Generalized Laguerre polynomials, 21.7-5, 21.7-7
Generalized variance, 18.4-8, 19.7-2
Generating function, 8.7-2
of canonical transformation, 10.2-6, 10.2-7
in combinatorial analysis, C-l, C-2
exponential, 8.7-2
as a functional transform, 8.7-2
of orthogonal polynomials, 21.7-1, 21.7-5
of probability distribution, 18.3-8, 18.5-7, 18.8-1
Generator, of quaternions, 12.4-2, 14.10-6of ruled surface, 3.1-15
Generatrix, 3.1-15
Geodesic, 17.4-3
on a surface, 17.3-12
Geodesic circle, 17.3-13
Geodesie curvature, 17.3-4
Geodesie deviation, 17.4-6
Geodesie normal coordinates, 17o3-13
Geodesic null line, 17.4-4
Geodesic parallels, 17t3-13, 17.4-6
Geodesic polar coordinates, 17.3-13
Geodesic triangle, 17.3-13
Geometric distribution, 18.8-1
Geometric progression, 1.2-7
Geometric series, 1.2-7, 21.2-12, E-4, E-5
Geometric multiplicity (see Degree, of degeneracy) Geometrical object,
16.1-3
Geometry, 2.1-1
on a surface, 17.3-13, 17.3-14
Gerschgorin’s circle theorem, 14.8-9
Gibbs phenomenon, 4.11-7
Gibbs vector, 14.10-3
Gill, 20.8-2t
Givens, 20.3-1
Global asymptotic stability, 9.5-4, 13.6-5
Goldstine, 20.3-5
Gradient, 5.5-1, 5.5-2, 6.4-2, 16.2-2, 16.10-7
theorem of the, 3.6-1
Gradient lines, 15.6-8
Gradient method, 20.2-7, 20.3-2
Gradient operator, 5.5-2, 16.10-4, 16.10-7
Graeffe, 20.2-5
Gram polynomials, 20.6-3
Gram-Charlier series, 19.3-3
Gram-Schmidt orthogonalization, 14.7-4, 15.2-5, 20.3-1, 20.6-3, 21.7-1
Gram-Schmidt orthogonalization process, for polynomials, 21.7-1
Gram’s determinant, 5.2-8, 14.2-6, 15.2-1Graph, 4.2-1
Greatest common divisor, 1.7-3
Greatest lower bound (g.l.b.), 4.3-3
Green’s formula (Green’s theorem), 4.6-12, 5.6-1
generalized, 15.4-3, 15.4-8, 15.4-9, 15.6-5
Green’s function, 9.3-3, 10.3-6, 15.5-1 to 15.5-4, 18.12-2
examples, 9.3-3, 15.6-6, 15.6-9, 15.6-10
modified, 9.3-3, 15.5-1
of the second kind, 15.5-4
Green’s matrix, 9.4-3
Green’s resolvent, 15.5-2
Gregory’s quadrature formula, 20.7-2
Group, 12.2-1
of transformations, 12.2-8, 14.9-1
Group relaxation, 20.3-2
Group representation, 12.2-9, 14.9-1
Grouped data, 19.2-2 to 19.2-5, 19.7-3
Guldin’s formulas, 4.6-11
Hadamard’s inequality, 1.5-1
Half-angle formulas, B-4, B-8
Half-wave rectified waveform, 8.3-2
table, D-2
Half width, 18.3-3
Hamilton-Jacobi equation, 10.2-7, 11.6-8, 11.8-6
Hamilton-Jacobi equation, 11.6-8, 11.8-6
Hamiltonian function, 11.8-2
Hamilton’s principle, 11.6-1, 11.6-9
Hamming’s method, 20.8-4
Hankel functions, modified, 21.8-6 (See also Cylinder functions) Hankel
transform, 8.6-4
finite, 8.7-1t finite annular, 8.7-1t table, D-5
Hankel’s integral representation, 21.4-1
Hansen’s integral formula, 21.8-2
Harmonic, 4.11-4
Harmonic analysis, 4.11-4
numerical, 20.5-8
Harmonic division, 2.4-10Harmonic function, 15.6-4, 15.6-8
Harnack’s convergence theorems, 15.6-4
Hastings, 20.6-4
Haversine, B-9
Heat conduction (see Diffusion equation) Heaviside expansion, 8.4-4
asymptotic series, 8.4-9
Heine-Borel theorem, 12.5-4
Heine’s integral formula, 21.8-11
Helmholtz’s decomposition theorem, 5.7-3
Helmholtz’s equation (see Space form of the wave equation)
Helmholtz’s theorem, 15.6-10
Hermite function, 21.7-6
Hermite polynomial, 18.8-3, 19.3-3, 20.7-3, 21.7-1, 21.7-6, 21.7-7
Hermitian conjugate, of a differential operator, 15.4-3, 15.4-4
kernel, 15.3-1
of a linear operator, 14.4-3, 14.7-5
of a matrix, 13.3-1
Hermitian-conjugate boundary-value problems, 15.4-3
Hermitian-conjugate integral transformations, 15.3-1
Hermitian form, 13.5-3 to 13.5-6, 14.7-1
Hermitian inner product (see Inner product) Hermitian integral form,
15.3-6
Hermitian integral transformation, 15.3-1
Hermitian kernel, 15.3-1, 15.3-3 to 15.3-8
Hermitian matrix, 13.3-2 to 13.3-4, 13.4-2, 13.4-4, 13.5-3, 13.5-6, 14.8-
9
Hermitian operator, 14.4-4, 14.7-5, 14.8-9, 15.4-3
Hermitian part, of linear operator, 14.4-8
of matrix, 13.3-4
Heron’s algorithm, 20.2-26
Heun, 20.8-2t
Hexagon, A-2
Hilbert space, 14.2-7, 15.2-2
Hill climbing, 20.2-6, 20.2-7
H?lder’s inequality, 4.6-19
Holomorphic function, 7.3-3
Homeomorphism, 12.5-1Homogeneous boundary conditions, 15.4-7, 15.5-1
Homogeneous differential equation, 9.1-2, 9.1-5, 9.2-4, 9.3-1, 9.3-6, 9.4-
1, 13.6-2, 15.4-2
partial, 10.1-2, 15.4.2
Homogeneous function, 4.5-5, 9.1-5
Homogeneous integral equation, 15.3-2
Homogeneous linear equations, 1.9-5
Homogeneous polynomial, 1.4-3
Homomorphism, 12.1-6
of groups, 12.2-9
Horner’s method, 20.2-3, 20.2-5
Householder, 20.3-1
Hydrogenlike wave functions, 10.4-6
Hyperbola, 2.4-3
construction of, 2.5-3
properties of, 2.5-2
rectangular, 2.5-2
Hyperbolic cylinder, 3.5-7
Hyperbolic differential equation, 10.3-1 to 10.3-3, 10.3-6, 10.3-7
Hyperbolic functions, 21.2-5 to 21.2-9, F-4t infinite products, E-8
power series, E-7
Hyperbolic geometry, 17=3-13
Hyperbolic paraboloid, 3.5-7
Hyperbolic point, 17.3-5
Hyperboloid, 3.5-7
Hypercomplex numbers, 12.4-2
Hypergeometric differential equation, 9.3-9
Hypergeometric distribution, 18, 8-1
Hypergeometric function, 9.3-9, 21.6-6, 21.7-1
Hypergeometric polynomials, 9.3-9, 21.7-8
Hypergeometric series, 9.3-9, 21.7-1
Hypocycloid, 2.6-2
Hypothesis (see Statistical hypothesis) Icosahedron, A-6
Ideal, 12.3-2
Idempotent element, 12.4-2
Idempotent property, 12.8-1
Identity, additive, 1.1-2of a group, 12.2-1
of a ring, 12.3-1
Identity matrix 13.2-3
Identity relation, 1.1-4
Identity transformation, 14.3-4
Image charge, 15.6-6
Imaginary axis, 1.3-2
Imaginary number, 1.3-1
Imaginary part, 1.3-1
Imbedding, 11.9-2
Implicit functions, 4.5-7
Implicit method, 20.9-4, 20.9-8
Impossible event, 18.2-1
Improper conic, 2, 4-3
Improper eigenfunction, 15.4-5
Improper integrals, 4.6-2
convergence criteria for, 4.9-3, 4.9-4
Improper quadric, 3.5-7
Improper rotation, 14.10-1
Improvement of convergence, 4.8-5
Impulse functions, 21.9-2 to 21.9-7
approximations to, 21.9-4, 21.9-6
asymmetrical, 21.9-6
Laplace transform of, 8.5-1
Impulse noise, 18.1
l-5c Impulse response, 9.4-3, 18.12-2
Impulse train, 20.4-6, D-2t Inclusion relation, 4.3-2, 12.8-3, 18.2-1
Incomplete beta function, 18.8-5, 21.4-5
Incomplete beta-function ratio, 18.8-5, 21.4-5
Incomplete gamma function, 18.8-5, 21.4-5
Indefinite form, 13.5-2 to 13.5-6
Indefinite integral, 4.6-4
Indefinite matrix, 13.5-2 to 13.5-6
Indefinite metric, 14.2-6, 17.4-4
Indefinite operator, 14.4-5
Independent experiments, 18.2-4
Independent trials, 18.2-4Independent variable, 4.2-1
Indeterminate forms, 4.7-2
Index, in phase plane, 9.5-3
of a subgroup, 12.2-2, 12.2-5, 12.2-7
Indicial equation, 9.3-6
Indiscrete topology, 12.5-1
Induced transformation, 16.1-4, 16.2-1
Induction, finite, 1.1-2 INDEX 1112
Inequalities, 1.1-5
examples, 21.2-13
for transcendental functions, 21.2-13
Inequality constraints, on control variables, 11.8-1, 11.8-3
on state variables, 11.8-5 (See also Linear programming problems;
Nonlinear programming) Infinite determinant, 13.2-7
Infinite integral (see Improper integrals) Infinite product, 4.8-7, 7.6-6
examples of, 21.4-5, E-8
Infinite series, convergence of, 4.8-1 to 4.8-6, 4.9-1, 4.9-2
examples, E-5 to E-8
Infinite set, 4.3-2
Infinitesimal displacement, 16.2-2
Infinitesimal dyadic, 14.4-10
Infinitesimal rotation, 14.10-5
Infinitesimal transformation, 14.4-10, 14.10-5
Infinitesimals, 4.5-3
Infinity, 4.3-5, 4.4-1
in complex-number plane, 7.2-2
Inflection, 17.1-5
Initial-state manifold, 11.8-1c Initial-value theorem, Laplace transform,
8.3-1t z transform, 8.7-3
Inner automorphism, 12.2-9
Inner product, of dyadics, 16.9-2
of functions, 15.2-1
of tensors, 16.3-7, 16.7-4
of vectors, 14.2-6, 14.2-7, 14.7-1
of vectors defined on Riemann space, 16.8-1, 16.8-2 (See also Scalar
product) Inscribed circle, of regular polygons, A-2
of a triangle, B-4Inscribed cone, B-6
Instantaneous axis of rotation, 14.10-5
Integers, 1.1-2
Integrability conditions, 10.1-2, 17.3-8
Integral curvature, 17.3-14
Integral domain, 12.3-1
Integral equation, for Karhunen-Loéve expansion, 18.9-5
Integral equations, numerical solution, 20.9-10
types, 15.3-2
Integral function, 4.2-2, 7.6-5
Integral transform, finite, 8.7-1
Integral-transform methods, 9.3-7, 10.5-1
Integral transformation, 15.3-1, 15.5-1 to 15.5-4 (See also Kernel)
Integrated Fourier transform, 18.10-10
Integrated power spectrum 18.10-10
Integrating factor, 9.2-4
for Pfaffian differential equation, 9.6-2
Integration, numerical, 20.7-2 to 20.7-5
by parts, 4.6-1
of vectors, 5.3-3
Integration methods, 4.6-6
Interior measure, 4.6-15
Interpolating function (see Sampling function) Interpolation 18.11-2,
20.5-1 to 20.5-7
Interpolation coefficients, tables, 20.5-3
Interquartile range, 18.3-3
distribution of, 19.5-2
Intersection, in Boolean algebra, 12.8-1
of cone by plane, 2.4-9
of curves, 2.1-9
of events, 18.2-1
of lines, 2.3-2
of planes, 3.3-1, 3.4-3
of sets, 4.3-2
of surfaces, 3.1-16
Interval halving, 20.2-2f
Intrinsic derivative, 5.5-3, 16.10-8Intrinsic differential geometry, 16.7-1, 17.3-9
Intrinsic equation of a curve, 17.2-3
Intrinsic geometry of a surface, 17.3-9
Invariance, 2.1-7, 12.1-5, 14.1-4
of tensor equations, 16.4-1
Invariant manifold, 14.8-2, 14.8-4
Invariant points, 7.9-2
Invariants, 12.2-8, 16.1-3
of a conic, 2.4-2
of elliptic functions, 21.6-2
of a quadric, 3.5-2
Inverse, additive, 11-2
in a group, 12.2-1
multiplicative, 1.1-2
Inverse Fourier transform, 4.11-4
Inverse function, 4.2-2
Inverse hyperbolic functions, 21.2-8, 21.2-10 to 21.2-12
Inverse interpolation, 20.5-4
Inverse Laplace transform, 8.2-5
Inverse operator, 14.3-5
Inverse probability, 19.7-7
Inverse transformation, 12.1-4, 14.3-5
Inverse trigonometric functions, 21.2-4, 21.2-10 to 21.2-12
Inversion, 7.9-2
Inversion theorem, 15.6-3
for Hankel transforms 8.6-4
Kelvin’s, 15.6-3, 15.6-7
for Laplace transforms, 8.2-6
for other integral transforms, 8.6-1, 8.6-2, 8.6-4
Inversion theorem, for z transforms, 8.7-3
Involute, 17.2-5
Irrational numbers, 1.1-2
Irreducible representation, 14.9-2 14.9-3, 14.9-5, 14.9-6
Irrotational vector field, 5.7-1, 5.7-3, 15.6-1
Isoclines, 9.2-2, 9.5-2
Isogonal mapping, 7.9-1
Isogonal trajectories, 17.1-8Isolated point, of a curve, 17.1-3
of a set, 4.3-6
Isolated set, 4.3-6
Isolated singularity, of differential equation, 9.3-6
of a function, 7.6-2
Isometric mapping, 17.3-10
Isometric spaces, 12.5-2
Isometric surface coordinates, 17.3-10
Isomorphism, 12.1-6, 16.1-4
of Boolean algebras, 12.8-5
of fields, 12.6-3
of groups, 12.2-9
of linear algebras, 14.9-7
of vector spaces, 14.2-4
Isoperimetric problem, 11.6-3, 11.7-1, 11.8-le Isothermic surface
coordinates, 17.3-10
Isotropic surface, 17.3-10
Iterated-interpolation method, 12.5-2
Iterated kernels, 15.3-5
Iteration methods, 9.2-5, 15.3-8, 20.2-2, 20.2-4, 20.2-6, 20.2-7, 20.3-2,
20.3-5, 20.8-3, 20.8-7, 20.9-2 to 20.9-4
Jacobi-Anger formula, 21.8-4
Jacobi polynomial, 9.3-9, 21.7-8
Jacobi-Sylvester law of inertia, 13.5-4
Jacobian, 4.5-6, 4.6-13, 6.2-1, 6.2-3, 7.9-1, 16.1-2
Jacobi’s condition, 11.6-10
Jacobi’s elliptic functions, 21.6-1, 21.6-7, 21.6-9
Jacobi’s method, 20.3-5c£ Jacobi’s theta functions, 21.6-8, 21.6-9
Join (see Union) Joint distribution, 18.4-1, 18.4-7
Joint entropy, 18.4-12
Joint estimates, 19.4-1 to 19.4-3, 19.4-4
Jointly ergodic random processes, 18.10-7
Jointly stationary random processes, 18.10-1
Jordan curve, 7.2-3
Jordan separation theorem, 7.2-4
Jordan’s lemma, 7.7-3
Jordan’s test, 4.11-4Jump function, 20.4-6
Jump relations for potentials, 15.6-5
Jury, 20.4-8
Kalman-Bertram theorem, 13.6-6
Kantor, 4.3-1
Kantorovich theorem (see Newton- Raphson method; Quasilinearization) Kapteyn, 19.3-1
Karhunen-Loéve theorem, 18.9-4, 18.11-1
Karnaugh map, 12.8-7
Kelvin’s inversion theorem, 15.6-3, 15.6-7
Kernel, of homomorphism, 12.2-9
of integral transform tion, 15.3-1
Khintchine’s theorem, 18.6-5
Klein-Gordon equation, 10.4-4, 15.6-10
Kolmogorov, 18.11-4
Kotelnikov sampling theorem, 18.11-2a Kronecker delta, 16.5-2
Kronecker product, 14.9-6
Krylov, 9.5-5
Kuhn-Tucker theorem, 11 «4-3
Kummer function, 9.3-10
Kummer’s transformation, 4.8-5
Kutta (see Runge-Kutta methods) Lagrange multiplier, 11.3-4, 11.4-3,
11.6-2, 11.6-3, 11.7-1, 11.8-1, 11.8-2, 11.8-5
Lagrange’s differential equation, 9.2-4
Lagrange’s equations, 11.6-1
Lagrange’s interpolation formula, 20.5-2
Lagrange’s remainder formula, 4.10-4
Laguerre functions, 10.4-6, 21.7-5
Laguerre polynomials, 8.4-8, 10.4-6, 20.7-3, 21.7-1
associated, 21.7-5, 21.7-7
Lamellar vector field, 5.7-1, 5.7-3
Laplace development, 1.5-4
Laplace transform, bilateral, 8.6-2, 18.12-5
of matrix, 13.6-2c of periodic function, 8.3-2
s-multiplied, 8.6-1
Stieltjes-integral form, 8.6-3
Laplace transform pairs, tables, D-6, D-7Laplace-transform solution, of difference equations, 20.4-66
of matrix differential equations, 13.6-2
of ordinary differential equations, 9.3-7, 9.4-5, 13.6-2c Laplacetransform solution, of partial differential equations, 10.5-2, 10.5-3
Laplace transformation, 8.2-1 (See also Laplace transform) Laplace’s
differential equation, 15.6-1 to 15.6-9
numerical solution of, 20.9-4 to 20.9-7
particular solutions, 10.4-3, 10.4-5, 10.4-9
two-dimensional, 10.4-5, 15.6-7 (See also Potential) Laplace’s
distribution, 18.8-5
Laplace’s integral, 21.7-7
Laplacian operator, 5.5-5, 6.4-2, 6.5-1, 16.10-7
La Salle’s theorem, 13.6-6
Latent root (see Eigenvalue) Lattice, 20.6-1
Latus rectum, 2.4-9
Laurent series, 7.5-3
Law of cosines, B-4, B-8
Law of large numbers, 18.1-1, 18.6-5
Law of sines, B-4, B-8
Law of small numbers, 18.8-1
Leaf of Descartes, 2.6-1
Least-squares approximations, 4.11-2, 12.5-4, 15.2-6, 20.6-1 to 20.6-3,
20.9-9, 20.9-10 (See also Estimation; Projection theorem;
Regression) Least upper bound (l.u.b.), 4.3-3
Lebesgue convergence theorem, 4.6-16
Lebesgue integral, 4.6-15, 4.6-16, 15.2-2
Lebesgue measure, 4.6-14
Lebesgue-Stieltjes integral, 4.6-17, 18.3-6
Lebesgue-Stieltjes measure, 4.6-17
Left-continuous function, 4.4-7
Left-hand derivative, 4.5-1
Left-handed coordinate system, 6.2-3, 16.7-1
Legendre functions, 10.4-3, 21.7-2
associated, 21.8-10
Legendre polynomials, 21.7-1, 21.7-2, 21.7-8, 21.8-12
associated, 21.8-10, 21.8-12
Legendre transformation, 9.2-3, 10.2-5Legendre’s condition, 11.6-16
Legendre’s differential equation, 21.7-1, 21.7-3
Green’s function for, 9.3-3
Legendre’s normal elliptic integrals, 21.6-1, 21.6-3, 21.6-5, 21.6-6
Legendre’s relation, 21.6-6
Legendre’s strong condition, 11.6-10
Leibnitz’s rule, 4.6-1
Lemniscate, 2.6-1
Lerch’s theorem, 8.2-8
Level of significance, 19.6-3, 19.6-4
Level surface, 5.4-2
PHôpital’s rule, 4.7-2
Liénard-Chipart test, 1.6-6
Likelihood function, 19.1-2
Likelihood ratio, 19.6-3, 19.9-3
Limaçon, 2.6-1
Limit cycle, 9.5-3
Limit-in-mean, 15.2-2
Limit point, 4.3-6, 12.5-1
Limit theorems, of probability theory, 18.6-5
Limits, 4.4-1
frequently used, 4.7-2
of matrices, 13.2-11
multiple, 4.4-5
operations with, 4.4-3
of vector functions, 5.3-1
Lindeberg conditions, 18.6-5
Lindeberg-Lévy theorem, 18.6-5
Line of curvature, 17.3-6
Line coordinates, 2.3-3
Line-distribution potential, 15.6-7
Line element, 9.2-2
Line integral, 4.6-10, 5.4-5, 6.2-3, 6.4-3
Linear algebra, 12.4-2, 13.2-5, 14.4-2, 14.9-7
Linear dependence (see Linear independence) Linear difference
equation, 20.4-4 to 20.4-8
Linear differential equation, ordinary, 9.3-1partial, 10.2-1
of physics, 10.4-1 (See also Boundary-value problem) Linear
dimension (see Dimension) Linear equations, 1.8-1
homogeneous, 1.9-5
in matrix form, 14.5-3
numerical solutions of, 20.3-1 to 20.3-4
systems of, 1.9-2 to 1.9-5, 14.5-3
Linear fractional transformation (see Bilinear transformation) Linear
function, 4.2-2
Linear independence, of equations, 1.9-3, 2.3-2
of functions, 1.9-3, 9.3-2, 15.2-1
of sets of numbers, 1.9-3
of solutions of differential equations, 9.3-2
of solutions of equations, 1.9-3
of vectors, 5.2-2, 14.2-3, 14.3-5
Linear integral transformation (see Integral transformation; Kernel)
Linear manifold, 14.2-1, 14.2-2
Linear operation, on a random process, 18.12-1 to 18.12-4 (See also
Linear operator) Linear operator, 14.3-1
matrix representation, 14.5-1, 14.7-5
notation, 14.7-7
Linear point set, 4.3-1
Linear-programming problems, 11.4-1 to 11.4-4
canonical form, 11.4-2
dual, 11.4-lc, 11.4-4c standard form, 11.4-16
Linear spiral, 2.6-2
Linear system, with random input, 18.12-2 to 18.12-4
Linear transformation, 14.3-1 (See also Linear operator) Linear vector
function, 14.3-1
Linear vector space, 12.4-1, 14.2-1
Liouville’s theorem, 7t6-5
Liouville’s theorems, on elliptic functions, 21.6-1
Lipschitz condition, 9.2-1
Lituus, 2.6-1
Local base vectors, 16.6-1
differentiation of, 16.10-1, 16.10-3
inner products of, 16.8-2 to 16.8-4 (See also Base vectors) Localcartesian coordinates, 17.4-7
Local dyadic, 16.10-7
Loéve-Karhunen theorem, 18.9-4, 18.11-1
Logarithm, 1.2-3, 21.2-10 to 21.2-12
continued-fraction expansion, E-9
numerical approximation, 20.5-42
power series, E-7
tables, F-2 to F-5
Logarithmic decrement, 9.4-1
Logarithmic integral, 21.3-1
Logarithmic normal distribution, 19.3-2
Logarithmic potential, 15.6-7
Logarithmic property, 1.2-3
Logical addition, 12.8-1
Logical inclusion (see Inclusion relation) Logical multiplication, 12.8-1
Logical product (see Intersection) Logical sum (see Union) Lommel’s
integrals, 21.8-2
Long division, 1.7-2
Lowering of indices, 16.7-2
Lozenge diagrams, 20.5-3
Lyapunov, direct method, 9.5-4, 13.6-5 to 13.6-7
stability theory, for difference equations, 20.4-8
Lyapunov function, 13.6-2 to 13.6-7
MacLaurin, 4.8-5, 20.7-2
MacLaurin’s series, 4.10-4
Magnitude (see Absolute value) Mainardi-Codazzi equations, 17.3-8
Major axis, 2.5-2
Manifold (see Linear manifold) Mapping, isogonal, 7.9-1
of surfaces, 17.3-10(See also Conformai mapping; Transformation)
Marginal distribution, 18.4-2, 18.4-7
Markov chain, 18.11-4
Mascheroni (see Euler-Mascheroni constant) Matched filter, 19.9-3
Mathematical expectation (see Expected value) Mathematical model,
12.1-1
Matrix differential equation, 13.6-1 to 13.6-7
Matrix inversion, 13.2-3
numerical, 20.3-1 to 20.3-4Matrix norms, 13.2-1
Matrix notation, for difference equations, 20.4-7
Matrix operations, 13.2-2 to 13.2-12
Matrix representation, of groups (see Group representation) of integral
transformation, 15.3-1
of linear algebras, 14.9-7
of vectors and linear operators, 14.5-2 to 14.6-2
Maxima and minima, functions of a real variable, 4.3-2, 11.2-1, 11.2-2
functions of n real variables, 11.3-1 to 11.3-5
of integrals, 11.5-2
of multiple integrals, 11.6-9
numerical methods, 20.2-6, 20.2-7, 20.3-2
Maximizing player, 11.4-4
Maximum-likelihood estimates, 19.4-4, 19.9-2
Maximum-modulus theorem, for analytic functions, 7.3-5
for harmonic functions, 15.6-4
Maximum principle, 11.8-2 to 11.8-6
Mayer, 9.6-2
problem of, 11.6-6
Mean, arithmetic, 4.6-3
Mean count rate, 18.11-4d, 18.11-5
Mean curvature, 17.3-5
Mean deviation, 18.3-3
of normal distribution, 18.8-4
Mean radial deviation, 18.8-7
Mean radial error, 18.8-7
Mean square, measurement of, 19.8-36
Mean-square contingency, 19.7-5
Mean-square continuity, 18.9-3d Mean-square error, in Fourier
expansion, 4.11-2
in orthogonal-function expansion, 15.2-6
Mean-square regression, 18.4-6, 18.4-9, 19.7-2
Mean-square value of periodic waveforms, table D-l Mean value, over a
group, 12.2-12, 14.9-5 (See also Expected value) Mean-value
theorem, for derivative, 4.7-1
for harmonic functions, 15.6-4
for integrals, 4.7-1Measureable function, 4.6-15
Measurable set, 4.6-15
Measure, 4.6-15, 12.8-8 (See also Lebesgue measure; Stieltjes measure)
Measure of dispersion, 18.3-3, 19.2-4
Measure of effectiveness, 19.6-9
Measure of location, 18.3-3
Measure algebra, 12.8-8
Measurements (see Estimation) Median, 18.3-3, 19.2-2
of a triangle, B-3, B-4, B-6 (See also Sample median) Meet (see
Intersection) Mellin transform, 8.6-1
Membrane, vibrations of, 10.4-9, 15.4-10
Mercer’s theorem, 15.3-4, 15.5-2, 18.9-4
Meromorphic function, 7.6-7
Metric, 12.5-1
in L2, 15.2-2
in normed vector space, 14.2-7
Metric equality, 12.5-2
Metric invariant, 12.5-2
Metric space, 12.5-2
Metric tensor, 6.2-3, 16.7-1, 16.10-5
on a surface, 17.3-7
Meusnier’s theorem, 17.3-4
Midpoint of line segment, 2.1-4, 3.1-7
Milne’s method, 20.8-4/ Minimal curve, 17.4-4
Minimal polynomial, 12.8-2, 12.8-7
Minimal surface, 17.3-6, 17.3-10
Minimax principle, Courant’s, 14.8-8, 15.4-7
Minimax test, 19.9-2
Minimax theorem, 11.4-46
Minimizing player, 11.4-4
Minimum (see Maxima and minima) Minimum feasible solution, 11.4-
16
Minkowski’s inequality, 4.6-19, 14.2-4 (See also Triangle property)
Minor, 1.5-2
complementary, 1.5-4
principal, 1.5-4
Minor axis, 2.5-2Mittag-Leffler’s theorem, 7.6-8
Mixed-continuous group, 12.2-11
Mixed tensor, 16.2-1
Modal column, 14.8-5
Modal matrix, 14.8-6
Mode, 18.3-3
Model, 12.1-1
Modes of vibration, 10.4-9 (See also Normal modes) Modified Bessel
functions, 21.8-6
Modified Green’s function, 9.3-3, 9.4-3, 15.5-1
Modified Hankel functions, 21.8-6
Modifier formulas, 20.8-3
Modular angle, 21.6-6a Modulation theorem, for Fourier transforms,
4.11-5
for Laplace transforms, 8.3-2
Module, of elliptic integral, 21.6-6 (See also Additive group) “Modulo,”
12.2-10
Modulus, of complex number, 1.3-2
of elliptic integral, 21.6-6
Moebius strip, 3.1-14
Moebius transformation (see Bilinear transformation) Moment, 18.3-7,
18.3-10, 18.4-4, 18.4-8, 18.4-10
Moment-generating function, 18.3-8, 18.3-10, 18.6-2, 18.12-6
Moment matrix, 18.4-8, 18.8-8, 19.7-2
Moment method of estimation, 19.4-3
Monge axis, 10.2-1
Monge cone, 10.2-1
Monogenic analytic function, 7.4-3, 7.8-1
Monomial matrix, 13.2-1
Monotonie function, 4.4-8
Most powerful test, 19.6-3, 19.6-4
Moulton’s corrector, 20.8-4/ Moving trihedron, 17.2-2 to 17.2-4
Muller’s method, 20.2-4
Multimodal distribution, 18.3-3
Multinomial coefficient, tables, C-l to C-3
Multinomial distribution, 18.8-2
Multiple correlation coefficient, 18.4-9, 19.7-2Multiple integrals, 4.6-8
Multiple interpolation, 12.5-6
Multiple Poisson distribution, 18.8-2
Multiple roots, 1.6-7, 20.2-2
Multiple-valued complex functions, 7.4-1 to 7.4-3, 7.8-1
Multiplication of probabilities, 18.2-2
Multiplication theorem for Bernoulli polynomials, 21.5-2
Multipole expansion, 15.6-5, 21.8-12
Multistep method, 20.8-3
Multivariate sample, 19.7-2 to 19.7-7
Mutually exclusive (see under Disjoint) Nabla, 5.5-2
Napier’s analogies, B-8
Napier’s rules, B-7
Natural boundary, 7.8-1
Natural boundary conditions, 11.6-5, 11.8-26
Natural circular frequency, 9.4-1
undamped, 9.4-1
Navel point (see Umbilic point) Negative binomial distribution, 18.8-1
Negative-definite form, 13.5-2, 13.5-3 to 13.5-6
integral form, 15.3-6
matrix, 13.5-2, 13.5-3
operator, 14.4-5
Negative semidefinite form, 13.5-2, 13.5-3
Neighborhood, 4.3-5, 12.5-1
in complex-number plane, 7.2-2
of infinity, 4.3-5, 7.2-2
in a metric space, 12.5-3
in a normed vector space, 14.2-7
Neil’s parabola, 2.6-1
Neumann functions (see Cylinder functions) Neumann problem, 15.4-
10, 15.5-4, 15.6-2, 15.6-8
Neumann series, 15.8-8
Newton-Cotes formulas, 20.7-2o, 20.8-5
Newton-Gregory interpolation formulas, 20.5-3
Newton-Raphson method, 20.2-2, 20.2-8
generalized, 20.9-3
Newton’s formulas, for roots, 1.6-4for symmetric functions, 1.4-3
Newton’s interpolation formula, 20.5-3, 20.7-1
Neyman-Pearson criteria, 19.6-3
Nilpotent operator, 12.4-2
Nine-point circle, B-3
Nodal point, 9.5-3, 9.5-4
Noise, 18.11-3, 18.11-5
effect on detection and measurements, 19.9-1 to 19.9-4
Nonautonomous system, 13.6-6
Nondecreasing function, 4.4-8
Non-Euclidean geometry, 17.3-13, B-6
Nonhomogeneous differential equation, 13.6-2
Nonincreasing function, 4.4-8
Nonlinear operation, on a random process, 18.12-5, 18.12-6
Nonlinear programming, 11.4-3
Nonnegative form. 13.5-2, 13.5-3 to 13.5-6
Nonnegative integral form, 15.3-6
Nonnegative matrix, 13.5-2, 13.5-3
Nonnegative operator, 14.4-5
Nonparametric statistics, 19.6-8, 19.7-5
Nonparametric test, 19.1-3
Nonpositive form, 13.5-2, 13.5-3 to 13.5-6
Nonpositive integral form, 15.3-6
Nonpositive matrix, 13.5-2, 13.5-3
Nonpositive operator, 14.4-5
Nonsingular matrix, 13c2-3, 13.4-2, 14.5-3
Nonsingular operator, 14.3-5, 14.4-5
Norm, of a complex number, 1.3-2
of a function, 15.2-1
of a matrix, 13.2-1
of a vector, 14.2-5, 14.2-7, 14.2-8, 14.7-1 (See also Absolute value)
Normal, of conic section, 2.4-10
of plane, 3.2-1
of plane curve, 17.1-2
of quadric, 3.5-8
of straight line, 2.2-1
of surface, 17.3-2Normal acceleration, 17.2-3
Normal coordinates, 9.4-8, 13.5-4, 13.6-2
Normal curvature, 17.3-4
Normal derivative, 4.6-12, 5.6-1, 10.3-1, 15.4-3, 15.5-4, 15.6-6
of a potential, 15.6-5
Normal deviate (see Standardized normal distribution) Normal
distribution, 18.8-3, 18.8-4 18.8-9
circular, 18.8-7
n-dimensional, 18.8-8
two-dimensional, 18.8-6
Normal divisor (see Normal subgroup) Normal elliptic integral, 21.6-5
Normal error integral (see Error function) Normal form of quadratic or
hermitian forms, 13.5-4
Normal matrix, 13.3-4, 13.4-2, 13.4-4
Normal-mode oscillations, 9.4-8
Normal modes, 8.4-4, 9.4-1, 13.6-2
Normal operator, 14.4-8, 14.8-3, 14.8-6
Normal plane, 17.2-2, 17.2-4
Normal population, 19.4-2g 19.6-4, 19.6-5
Normal random numbers, table, F-20
Normal random process (see Gaussian random process) Normal random
variable, 18.8-3, 18.8-4
Normal response, 9-4-2
Laplace transform of, 9.4-5
Normal samples. 19.5-3, 19.7-3, 19.7-4
Normal section. 17.3-4, 17.3-5
Normal series, 12.2-6
Normal subgroup, 12.2-5, 12.2-6, 12.2-10, 12.2-11
Normal vector of a surface, 17.3-2
Normalizable function, 15.2-1
Normalizable kernel, 15.3-1, 15.3-3, 15.3-4, 15.3-8
Normalization, 15.2-1, 18.3-4
Normalization factor, 18.3-4
Normalizer, 12.2-7
Normed vector space, 14.2-5, 14.2-6
Null curve, 17.4-4
Null direction; 17.4-4Null displacement, 17.4-4
Null geodesic, 17.4-4
Null hypothesis, 19.6-3, 19 9-3
Null matrix, 13-2-3
Null space, 14.3-2
Null tensor, 16.3-2
Null transformation, 14.3-3
Null vector, 12.4-1, 14.2-1
Nullity, 14.3-2
Numerical stability, 20.1-2, 20.3-ld, 20.8-5, 20.9-8
Numerov’s method, 20.8-7d Nyquist criterion, 7.6-9
Nyquist’s sampling theorem, 18.11-2a Object function, z transform, 8.7-
3
Objective function (see Criterion functional) Oblate ellipsoid, 3.5-7
Oblate spheroid, 3.5-7, A-5
Oblate spheroidal coordinates, 6.5-1
Occupancy of cells, table, C-3
Occupation number, 19.2-2
Octagon, A-2
Octahedron, A-6
Octupole, 15.6-5
Odd function, 4.2-2, 4.11-4, D-2
One-sided limits, 4.4-7
One-step method, 20.8-2
One-tailed test, 19.6-4, 19.6-8
One-way classification, 19.6-6
Open ball, 12.5-3
Open integration formula, 20.8-36
Open interval, 4.3-4
Open set, 4.3-6, 12.5-1
Operand, 12.1-1, 12.1-4
Operating characteristic, 19.6-2
Operation, 12.1-1, 12.1-4
Operational calculus, 8.3-1
Operations, on random processes, 18.5-1 to 18.5-8, 18.12-1 to 18.12-6
Operator (see Linear operator) Optimal control, 11.8-1 to 11.9-2
Optimal policy, 11.9-1, 11.9-2Optimal trajectory, 11.8-1
Optimality principle, 11.8-6, 11.9-2
Optimum-interval interpolation, 20.5-5, 20.6-3« Orbit-transfer problem,
11.8-3c Order, of Bernoulli number, 21.5-2
of Bernoulli polynomial, 21.5-2
of a branch point, 7.4-2
of a curve, 2.1-9
of a determinant, 1.5-1
of difference, 20.4-1
of difference equation, 20.4-3
of a differential equation, 9.1-2
of an elliptic function, 21.6-1
of a function, 4.4-3
of a group, 12.2-1
of a group element, 12.2-3
of an inequality constraint, 11.8-6
of an integration formula, 20.8-2
of linear algebra, 12.4-2
of a matrix, 13.2-1
of a moment, 18.3-7, 18.4-4, 18.4-8
of a partial differential equation, 10.1-2
of a pole, 7.6-2
of polynomial interpolation, 20.5-2
of a random process, 18.11-4
of a system of differential equations, 9.1-3
of a zero, 7.6-1
Order-complete set, 12.6-1, 12.6-3
Order statistics, 19.2-6
Ordered field, 12.6-3
Ordinary difference equation, 20.4-3
Ordinary differential equation, 9.1-2
linear, 9.3-1
first-order, 9.2-4
Ordinate, 2.1-2
Origin. 2.1-2. 3.1-2
Orthogonal coordinates, 6.4-1 to 6.5-1, 16.8-2, 16.9-1, 16.9-3, 17.4-7
Christoffel three-index symbols for.» 6.5-1, 16.10-3Orthogonal dimension; 14.7-4
Orthogonal-function expansion, 10.4-2, 10.4-9, 15.2-6, 18.10-6 (See
also Fourier series; Laguerre polynomials; Orthogonal polynomials;
Spherical harmonics) Orthogonal functions, 15.2-3
Orthogonal matrix, 13.3-2 to 13.3-4, 13.4-4, 14.10-1
Orthogonal-polynomial expansion, 20.6-1 to 20.6-5
Orthogonal polynomials, 21.7-1 to 21.7-8
zeros of, 20.7-3, 21.7-2
Orthogonal projection, of a vector space, 14.2-8
Orthogonal representation, 14.9-1
Orthogonal trajectories, 15.6-8, 17.1-8
Orthogonal transformation, 13.5-5, 14.4-6, 14.4-7, 14.10-1
Orthogonal vectors, 14.7-3
Orthogonality, of eigenfunctions, 15.4-6
of eigenvectors, 14.8-4
of group representations, 14.9-5
Orthogonalization, 14.7-4, 15.2-5, 20.3-1, 20.6-3, 21.7-1
Orthonormal basis (see Complete ortho-normal set) Orthonormalfunction expansion, of a random process, 18.9-4, 18.11-1
Orthonormal functions, 10.4-2, 15.2-3, 21.8-12
Osculating circle, 17.2-2
Osculating plane, 17.2-2, 17.2-4
Osculating sphere, 17.2-5
Osculation; 17.1-5, 17, 2-6
Outer product, 12.7-2
of matrices, 13.2-10
of tensors, 16.3-6, 16.9-1
Ovals of Cassini, 2.6-1
Overrelaxation, 20.3-2
Padé table, 20.6-7
Paley-Wiener theorem, 4.11-4e Paperitz notation, 9.3-9
Parabola, 2.4-3
construction of, 2.5-4
properties of, 2.5-4
Parabolic cylinder, 3.5-7
Parabolic differential equation, 10.3-1, 10.3-3, 10.3-4, 10.3-7
Parabolic point, 17.3-5Paraboloid, 3.5-7
Parallel displacement of a vector, 16.10-9, 17.4-6
Parallelism, 16.10-9
Parallelogram, A-l Parallelogram law, 5.2-1
Parameter of a population, 19.1-2, 19.1-3
Parameter-influence coefficient, 13.6-4
Parameter space, 19.6-1
Parametric line (see Coordinate line) Parametric representation, of
quadrics, 3.5-10 (See also Curve; Plane; Straight line; Surface)
Pareto’s distribution, 19.3-4
Parseval’s identity, 14.7-4
Parseval’s theorem, 4.11-4, 4.11-5
Partial correlation coefficient, 18.4-9, 19.7-2
Partial derivative, 4.5-2
Partial difference equation, 20.4-3, 20.9-4, 20.9-8
Partial-fraction expansion, 1.7-4, 7.6-8
of Laplace transform, 8.4-5
Partially ordered set, 12.6-1
Particular integral, of ordinary differential equation, 9.1-2
of partial differential equation, 10.1-2, 10.2-3
Partition, 12.1-3, 12.2-4
of a group into classes, 12.2-5
table, C-l Partition theorem, chi-square distribution, 19.5-3
Partitioning, of matrices, 13.2-8, 20.3-4 (See also Step matrix) Pascal’s
distribution, 18.8-1
Pascal’s theorem, 2.4-11
Patching curve (see Characteristic) Pattern, C-2
Pattern enumerator, C-2
Pauli spin matrices, 14 10-4
Payoff matrix, 11.4-4a Peano’s axioms, 1.1-2
Pearson’s distributions, 19.3-5
Pearson’s measure of skewness, 18.3-3, 18.3-5
Penalty function, 20.2-6d Penny matching, 11.4-46
Pentagon, A-2
Percentiles, 18.3-3, 19.2-2
Period, 4.2-2
of group element, 12.2-3Period parallelogram, 21.6-1
Periodic components, effect on correlation functions and spectra, 18.10-
9
unknown, 20.6-6c Periodic forcing function, 9.4-6
Periodic function, 4.2-2, 4.11-4
Laplace transform of, 8.3-2
Periodic random process, 18.9-4, 18.11-1
Periodic sampling, 18.11-6
Periodicity conditions, 9.3-3, 15.4-8, 15.4-10
Permutation, 12.2-8
table, C-l Permutation group, 12.2-8, 14.9-2
Permutation matrix, 13.2-6 (See also Regular representation)
Permutation symbols, 16.5-3. 16.7-2, 16.8-4, 16.10-7
Perpendicular bisector, B-3, B-6
Perturbation methods, 10.2-7c, 13.6-4, 15.4-11
Perturbation theory, Hamilton-Jacobi equation, 10.2-7
Pfaffian differential equation 9.6-1, 9.6-2
Phase, 411-4 (See also Simple event) Phase plane, 9.5-2
Phase velocity 10.3-5, 10.4-8
Phasor, 9t4-6
rotating. 9.4-6
Physical components 6.3-2, 16.8-3, 16.9-1
Physical readability, 9.4-3
Picard’s method, 9.2-5, 20.7-4
Picard’s theorem, 7.6-4
Piecewise continuous function, 4.4-7
Piecewise continuously differentiate function, 4.5-1, 4.5-2
Pivotal condensation, 20.3-1
Planar element, 10.2-1
Plane, equation of, 3.2-1, 3.2-2
Plane coordinates, 3.4-4
Plane wave, 10.4-8
Pochhammer’s notation, 9.3-11
Poincaré, 9.5-3, 9.5-4
Poincaré’s index, 9.5-3
Point charge, 15.6-5
Point-charge radiation, 10.4-8Point spectrum. 14.8-3
Points of inflection. 17.1-5
loci of, 9.2-2
Poisson brackets, 10.2-6
Poisson distribution, 18.8-1, 18.8-9, 18.11-4d multiple, 18.8-2 (See also
Poisson process) Poisson integral, diffusion equation, 15.5-3
Poisson process, 18.11-4d, 18.11-5
Poisson’s differential equation, 15.6-1, 15.6-5, 15.6-7, 15.6-9
Poisson’s identity^ 10.2-6
Poisson’s integral formula, 15.6-6, 15.6-9
for Bessel functions, 21.8-2
Poisson’s summation formula, 4.8-5
Polar of a conic, 2.4-10
Polar axis, 2, 1-8
Polar coordinates. 2.1-8
Polar curve, 17 2-5
Polar decomposition, of complex numbers, 1.3-2
of linear operators, 14.4-8
of matrices, 13.3-4
Polar developable. 17.2-5
Polar line of space curve, 17.2-5
Polar plane, 3.5-8
Polar surface. 17 2-5
Polar triangle, B-6
Polar vector, 16.8-4
Pole; of a complex function 7.6-2, 7.6-9
of a conic, 2.4-10
of a quadric, 3.5-8
Polya’s counting theorem, C-2
Polya’s distribution, 18.8-1
Polygonal function, 11.6-3
Polynomial approximations, 20.6-1 to 20.6-5
Polynomials, 1.4-3
numerical evaluation, 20.2-3
Pontryagin’s maximum principle, 11.8-2 to 11.8-6
Pooled-sample statistics, 19.6-6
Pooled variance, 19.6-6Population. 19.1-7
Population distribution, 19.1-2
Population moment, 19.2-4
Population parameter 19.1-2, 19.1-3
Position vector, 3t1-5
Positive definite form, 13.5-2 to 13.5-6
Positive definite inner product 14.2-5, 14.2-6
Positive definite integral form, 15.3-6
Positive definite matrix, 13 5-2^ 13.5-3, 20.3-2
Positive definite operator, 14.4-5
Positive direction, 3.3-1
Positive normal, 5.4-6, 15.6-6, 17.1-2. 17.3-2
Positive semidefinite form, 13.5-25 13.5-3
Positive semidefinite matrix, 13.5-2
Potential, 15.6-1, 15.6-5
Power, of a complex number, 1.3-3
of a linear operator, 14.3-6
of a matrix, 13.2-4
measurement of, 19.8-36
of a real number, 1.2-1
spectral decomposition, 18.10-5
of test, 19.6-2
Power function, 19.6-2
Power series, 4.10-2, 7.2-1
economization of, 20.6-5
tables, E-6, E-7
Power spectral density. 18.10-3 to 18.10-10
effect of linear operations, 18.12-1 to 18.12-4
examples, 18.11-1 to 18.11-6
non-ensemble, 18.10-8
of real process, 18.10-4
Precision measure, 18.8-4
Precompact set, 12.5-3
Predictor-corrector methods. 20.8-3 to 20.8-8
Pre-Hilbert space, 14.2-7
Price’s theorem, 18.12-6
Primitive character, 14.9-4Primitive period, 21.6-1
Prncipal-axes transformation, 2.4-8, 3.5-7 13.5-4, 14.8-6
Principal axis, of a conic, 2.4-7
of a quadric, 3.5-6
Principal curvatures, 17.3-5
Principal normal, 17.2-2 to 17.2-4
in curved space, 17.4-3
Principal normal section, 17.3-5, 17.3-6
Principal part, of change. 11.3-2
of function, 7.5-3, 7.6-8
Principal value, of integral (see Cauchy principal value) of inverse
trigonometric functions, 21.2-4
Principle of optimality, 11.8-6, 11-9-2
Prism, A-4
Probability, 18.2-2, 18.3-1
Probability density, 18.3-2, 18.4-3, 18.5-2, 18.5-4
Probability differential (sec Probability element) Probability
distribution, 18.2-7, 18.2-8, 18.9-2
Probability element, 18.3-2, 18.4-3, 18.4-7
Probability function, 18.2-7
Probability integral (see Error function) Probable deviation, 18.8-4
Product, infinite, 4.8-7, 7.6-6, E-8
Product expansion, 7.6-6
of special functions, E-8
Product space, 12.7-3
Product topology, 12.7-3
Projection, 3.1-9
of a curve, 3.1-16
in a vector space, 14.2-8
Projection theorem, for triangles, B-4
for vector spaces, 14.2-8
Prolate ellipsoid, 3.5-7
Prolate spheroid, 3.5-7, A-5
Prolate spheroidal coordinates, 6.5-1
Prony’s method, 20.6-6c Propagated error, 20.8-1 (See also Numerical
stability) Propagation of disturbance, 10.4-1 (See also Wave
equation) Proper conic, 2.4-3Proper function (see Eigenluiietion) Proper quadric, 3.5-3
Proper rotation, 14.10-7
Proper subgroup, 12.2-2
Proper subset, 4.3-2
Proper subspace, 14.2-2
Proper value (see Eigenvalue) Proper vector (see Eigenvector)
Proportions, 1.4-2
Pseudosphere, 17.3-13
Pseudotensor (see Relative tensor) Psi function. 21.4-3
Pure strategy, 11.4-4
Purely random process, 18.11-46
Pyramid, A-4
QD algorithm, 20.2-5a Quadrant, 2.1-2
Quadrantal triangle, B-7
Quadratic equation, 1.6-3, 1.8-2
Quadratic form, 13.5-2, 13.5-4 to 13.5-6
Quadratic-variation theorem, 18.10-10
Quadratically integrable function, 15.2-1
Quadrature formulas (see Integration, numerical) Quadric surfaces, 3.5-1
to 3.5-10, 16.9-3
Quadrupole, 15.6-5
Quartic equation, 1.6-3, 18-5, 1.8-6
Quartiles, 18.3-3, 19.2-2, 19.5-2
of normal distribution, 18.8-4 (See also Fractiles) Quasilinear
differential equation, 10.3-1
Quasilinearization, 20.9-3
Quaternions, 12.4-2, 14.10-4
Quotient-difference algorithm, 20»2-5a Quotient group (see Factor
group) r distribution, 19.5-3, 19.7-4 R test, 19.6-6
Raabe’s test, 4.9-1
Radial deviation, 18.8-7
Radial error, 18.8-7
Radiation, 10.4-8
Radical axis, 2.5-1
Radical center, 2.5-1
Radicand, 1.2-1
Radius, of convergence, 4.10-2, 7.2-1, 7.5-2of curvature, 17.1-4, 17.2-3
of torsion, 17.2-3
Radius vector, 2.1-8
Raising of indices, 16.7-2
Random numbers, generation of, 20.10-4
normal, F-20
tables, F-19, F-20
Random-perturbation optimization, 20.2-6c Random phase, 18.11-1
Random process, 18.9-1 to 18.12-6
Random processes, examples, 18.11-1 to 18.11-6
Random sample, 19.1-2
multivariate, 19.7-2
Random series, 18.9-1
Random sine wave, 18.11-2
Random telegraph wave, 18.11-3
Random variables, 18.2-8
transformation of, 18.5-1 to 18.5-8
Randomized blocks, 19.6-6
Range, of a distribution, 18.3-3
distribution of, 19.2-6, 19.5-4
of function or transformation, 4.2-1, 12.1-4
of a linear operator, 14.3-2
of a sample, 19.2-6
Rank, of distribution, 18.4-8
of a hermitian form, 13.5-4
of a linear operator, 14.3-2
of a matrix, 1.9-3, 1.9-4, 13.2-7, 13.4-1
of a quadratic form, 13.5-4
of a representation, 14.9-1
of a tensor, 16.2-1
Rank correlation, 19.7-6
Rank statistics, 19.2-6
Raphson (see Newton-Raphson method) Rational algebraic function,
inverse Laplace transforms of, 8.4-4
table, D-6
Rational-fraction interpolation, 20.5-7
Rational function, 1.7-4, 4.2-2Rational-function approximations, 20.5-7, 20.6-7
Rational integral function, 1.6-3
Rational numbers, 1.1-2
Rationalizing denominators, 1.2-2
Rayleigh-Ritz method, 11.7-2
Rayleigh’s quotient, 14.8-8, 15 4-7
Real axis, 1.3-1
Real numbers, 1.1-2
Real part, 1.3-1
Real roots of algebraic equations, 1.6-6, 20.2-1, 20.2-3
Real vector space, 14.2-1
Realization of a group, 12.2-9
Reciprocal, 1.1-2
Reciprocal bases, 6.3-3, 14.7-65 16.7-3, 16.8-2
Reciprocal differences, 20.5-7
Reciprocal kernel (see Resolvent kernel) Reciprocal one-to-one
correspondence, 12.1-4
Rectangular distribution (see Uniform distribution) Rectangular
hyperbola, 2.5-2, 21.2-5
Rectangular pulses, D-l Rectifiable curve, 4.6-9
Rectified waveform, 8.3-2
table, D-2
Rectifying plane, 17.2-2, 17.2-4
Recurrence relation (see Recursion formulas) Recursion formulas, for
associated Legendre polynomials, 21.8-10
for cylinder functions, 21.8-1, 21.8-6, 21.8-8
for orthogonal polynomials, 21.7-1
Reduced equation (see Complementary equation) Reducible operator,
14.8-2
Reducible representation, 14.9-2
Reducibility, 14.9-2
Reduction of elliptic integrals, 21.6-5
Reference system (see Coordinate system) Reflected wave, 10.3-5
Reflection, 7.9-2
of extremals, 11.6-7, 11.8-5
principle of, 7.8-2
Reflection-rotation group, 12.2-11Reflexivity, 12.1-3
Refraction, of extremals, 11.6-7, 11.5-2a, 11.8-5
Regression, 18.4-6, 18.4-9, 19.7-2, 19.9-4
Regression coefficient, 18.4-6, 18.4-9, 19.7-2
distribution of, 19.7-4
test for, 19.7-4
Regula falsi, 20.2-2
Regular arc, 3.1-13, 17.2-1, 17.4-2
Regular column, 20.2-56
Regular curve, 3.1-13
Regular function, 7.3-3
Regular operator, 14.3-5
Regular point, of a curve, 3.1-13, 17.1-1
of a differential equation. 9.3-6
of a surface, 3.1-14
Regular polygons, A-2
Regular polyhedra, A-6
Regular representation, of a group, 12.2-9; 14.9-1
of a linear algebra, 14.9-7
Regular singular point, 9.3-6
Regular surface, 3t1-14
Regular surface element, 17.3-1
Rejection region (see Critical region) Relative frequency (see Statistical
relative frequency) Relative scalar, 16.2-1
Relative stability, 20.8-56
Relative tensor, 16.2-1
covariant derivative of, 16.10-2
Relative topology, 12.5-1
Relatively prime polynomials, 1.7-3
Relativity theory, 16.7-1, 17.4-4, 17.4-6
Relaxation methods, 20.2-66, 20.3-5, 20.9-4
Remainder, in interpolation, 20.5-2, 20.5-3
of Laurent series, 7.5-3
of quotient, 1.7-2, 20.2-3
of series, 4.8-1
of Taylor’s series, 4.10-4, 4.10-5, 7.5-2
Remainder theorem, 1.7-2Removable singularity, 7.6-2
Rendezvous problem, 11.8-3c Repeated trials (see Independent trials)
Repetition in combinations, table, C-2
Replacement, C-2
Representation, 12.2-9
of groups (see Group representation) Representation space, 14.9-1
Residual, 20.2-4, 20.3-2
of a regression, 18.4-9
Residual spectrum, 14.8-3, 15.4-5
Residue, 7.7-1
at infinity, 7.7-1
Residue class, 12.2-10
Residue theorem, 7.7-2
Resolvent kernel, 15.3-7, 15.3-8, 15.5-2
Resolvent matrix, 13.4-2
Resolvent operator, 14.8-3
Resubstitution of solutions, 1.6-2, 9.1-2, 20.1-2
Result function, z transform, 8.7-3
Resultant of an algebraic equation, 1.6-5, 1.7-3
Retarded potential, 15.6-10
Reversion of series, 20.5-4
Rhombus rules, 20.2-5a Riccati equation, 9.2-4
Ricci principal directions, 17.4-5
Ricci tensor, 17.4-5
Ricci’s theorem, 16.10-5
Riemann-Christoffel curvature tensor (see Curvature tensor) RiemannGreen function, 10.3-6
Riemann integral, 4.6-1, 4.6-16
Riemann-Lebesgue theorem, 4.11-2
Riemann space, 16.7-1 to 16.10-11, 17.3-7, 17.4-1 to 17.4-7
Riemann-Stieltjes integral, 4.6-17
Riemann surface, 7.4-3
Riemann-Volterra method, 10.3-6
Riemannian coordinates, 17.4-5
Riemann’s differential equation, 9.3-9
Riemann’s elliptic integrals, 21.6-5
Riemann’s mapping theorem, 7.10-1, 15.6-9Riemann’s zeta function, E-5
Riesz-Fischer theorem, 15.2-4
generalized, 15.2-2
Right continuous function, 4.4-7
Right-hand derivative, 4.5-1
Right-handed coordinate system, 3.1-3, 6.2-3, 16.7-1
Right spherical triangle, B-7
Right triangle, B-l, B-2
Ring, 12.3-1
Risk, conditional, 19.9-2
expected, 19.9-1
Rodrigues’s formula, 21.7-1/ Rolle’s theorem, 1.6-6
Root-squaring method, 20.2-56
Roots, of equations, 1.6-2
of numbers, 1.2-1
real, location of, 1.6-6
Rotation, Cayley-Klein parameters, 14.10-4
in complex plane, 7.9-2
of coordinate axes, 2.1-6, 2.1-7, 3.1-12
about coordinate axis, 14.10-6
Euler angles, 14.10-6
with reflection, 12.2-11, 14.10-1, representation by matrix, 14.10-1
in space, 14.10-1 to 14.10-8
Rotation axis, 14.10-2
Rotation group, 12.2-11, 14.10-8
Rotation-reflection group, 12.2-11
Rotational, theorem of the, 5.6-1 (See also Curl) Rouché’s theorem, 7.6-
1
Round-off error, 20.1-2, 20.8-8 (See also Numerical Stability) RouthHurwitz criterion, 1.6-6
Row matrix, 13.2-1
Rule of correspondence, 12.1-1
Ruled surface, 3.1-15
Run, 18.7-3
Runge-Kutta methods, 20.8-2, 20.8-5 to 20.8-7
Saddle point, of a game, 11.4-4
in phase plane, 9.5-3, 9.5-4of a surface, 17.3-5
Saddle-point method, 7.7-3
Saltus, 4.4-7
Sample, 19.1-1
combination, table, C-2
Sample average, 19.2-2, 19.7-2
distribution of, 19.5-3
from grouped data, 19.2-2, 19.2-5
for random process, 19.8-4
Sample central moments, 19.2-4
Sample covariance, 19.7-2
Sample deciles, 19.2-2
Sample dispersion, 19.2-4
Sample distribution, 19.2-2
Sample fractiles, 19.2-2
distribution of, 19.5-2
Sample function, 18.9-1
Sample mean (see Sample average) Sample median, 19.2-2
distribution of, 19.5-2
Sample moments, 19.2-4
functions of, 19.4-2, 19.4-3, 19.5-2
Sample percentiles, 19.2-2
Sample point, 18.2-7, 18.9-1
Sample quantiles, 19.2-2
Sample quartiles, 19.2-2
Sample range, 19.2-6
distribution of, 19.5-4
Sample size, 19.1-2
Sample space, 18.2-7, 18.7-1, 18.9-1, 19.6-1
Sample standard deviation, 19.2-4
Sample values of a random process, 18.9-1
Sample variance, 19.2-4, 19.2-5, 19.7-2
distribution of, 19.5-3
from grouped data, 19.2-5
Sampled-data frequency-response function, 20.8-8
Sampled-data measurements, 19.8-1 to 19.8-3
Sampling function, 18.11-2, 21.3-1, D-2, F-21Sampling property, of impulse function, 21.9-2
of sine function, 18.1
l-2a Sampling ratio, 19.5-5
Sampling theorem, 18.11-2
Scalar, 5.2-1, 12.4-1, 13.2-2, 14.2-1
of a dyadic, 16.9-2 (See also Absolute scalar) Scalar curvature, 17.4-6
Scalar field, 5.4-2
Scalar potential, 5.7-1, 5.7-3
Scalar product, 5.2-1, 16.8-1, 16.8-2
of a dyadic, 16.9-2
in Riemann space, 16.8-1
in terms of curvilinear coordinates, 6.3-4, 6.4-2 (See also Inner
product) Scalar triple product 5.2-8, 6.4-2, 16.8-4
Scatter coefficient, 18.4-8
Scheme of measurements, 5.2-4, 14.1-5, 14.6-2, 16.1-4, 16.2-1, 16.6-2
Schlaefli’s integral, 21.7-7
Schmidt (see Gram-Schmidt orthogonali- zation process) SchmidtHilbert formula, 15.3-8
Schr?dinger equation, 10.4-6
Schur-Cohn test, 20.4-8
Schur’s lemma, 14.9-2
Schwarz, 21.9-2 (See also Cauchy-Schwarz inequality) SchwarzChristoffel transformation, 7.9-4, 7.10-1
Search, for maxima and minima, 20.2-6, 20.2-7
Second fundamental form of a surface, 17.3-5, 17.3-8, 17.3-9
Second probability distribution, 18.9-2
Sector of a circle, A-3
Sectorial spherical harmonics, 10.4-3, 21.8-12
Secular equation (see Characteristic equation) Segment, of a circle, A-3
of a sphere, A-5
Self-adjoint operator (see Hermitian operator) Self-conjugate operator,
14.4-4
Self-conjugate vector space, 14.4-9
Self-osculation, 17.1-3
Semiconvergence, 4.8-6
Semidefinite form, 13.5-2 to 13.5-6
Semidefinite integral form, 15.3-6Semidefinite matrix, 13.5-2 to 13.5-6
Semidefinite operator, 14.4-5
Semi-invariants, 18.3-9, 18.3-10, 18.4-10, 18.5-3
Sensitivity coefficient, 13.6-4
Separable kernel, 15.3-1, 15.3-3
Separable space, 12.5-1, 14.2-7
Separated sets, 12.5-1
Separation, of variables, 9.2-4, 10.1-3, 10.2-3, 10.4-1, 10.4-2
Separation constant, 10.1-3
Sequence, 4.2-1
convergent, 4.4-1
Sequential test, 19.6-9
Series, operations with, table, E-l reversion of, 20.5-4
tables, E-5 to E-7
Serret-Frenet formulas, 17.2-3
Set function, 4.6-15, 18.2-7
Shannon’s sampling theorem, 18.11-2« Sheppard’s corrections, 19.2-5
Shift operator, 20.4-1, 20.4-3
Shift theorem, Fourier transforms, 4.11-5
Laplace transforms, 8.3-1
Sigma function, 21.6-3
Sign test, 19.6-8
Signal detection, 19.9-3
Signal extraction, 19.9-4
Signature, 13.5-4
Similar matrices, 13.4-1, 13.4-2, 14.6-2 (See also Similarity
transformation) Similar representations, 14.9-1
Similarity theorem, for Fourier transforms, 4.11-5
for Laplace transforms, 8.3-1
Similarity transformation, 13.3-3, 13.4-1, 13.4-3, 13.4-4, 14.6-2, 14.9-1
(See also Similar matrices) Simple character, 14.9-4
Simple closed curve, 3.1-13
Simple curve, 3.1-13
Simple event, 18.2-7, 18.7-2
Simple group, 12.2-5
Simple statistical hypothesis, 19.6-1
Simple surface, 3.1-14Simplex method, 11.4-2
Simplex tableau, 11.4-2
Simply ordered set, 12.6-2
Simpson’s rule, 20.7-2
Simultaneous equations, 1.9-1
linear (see Linear equations) sine function, 18.11-2, 21.3-1
table, F-21
Sine integral, 21.3-1
Sine series, 4.11-3, 4.11-5
Sine transform, finite, 8.7-1 (See also Fourier sine transform) Singlevalued function, 4.2-2, 12.1-4
Singular distribution, 18.4-8
Singular integral, of an ordinary differential equation, 9.1-2, 9.2-2
of a partial differential equation, 10.1-2, 10.2-1, 10.2-4
Singular kernel, 15.3-8, 15.3-10
Singular matrix, 13«2-3
Singular operator, 14.3-5
Singular point, of a complex function, 7.6-2
of a curve, 17.1-3
at infinity, 7.6-3
Singular point, in phase plane, 9.5-3
of a surface, 17.3-1
Singular transformation, 14.3-5
Sinus amplitudinis, 21.6-7
Skew field, 12.3-1
Skew-hermitian matrix, 13.3-2
Skew-hermitian operator, 14.4-4, 14.4-7, 14.4-10
Skew-symmetric dyadic, 16.9-2
Skew-symmetric matrix, 14.10-5
Skew-symmetric operator, 14.4-6, 14.4-7, 14.10-5
Skew-symmetric part, of linear operator, 14.4-8
of matrix, 13.3-4
Skew-symmetry of tensors, 16.5-1
Skewness, 18.3-3, 19.2-4, 19, 5-3
Slack variable, 11.4-16
Slope of tangent, 4.5-1, 17.1-1
Smoluchovski, 18.11-4Smoothing, 20.6-1, 20.7-lc Snedecor (see v2 distribution) Solenoidal
vector field, 5.7-2, 5.7-3
Solid angle, 15.6-5
Solution, of game, 11.4-4
spherical, B-7 to B-9
of triangles, B-4
Solvable group, 12.2-6
Sommerfeld’s integral, 21.8-2
Space form of the wave equation, solutions, 10.4-4
Spaces, of sequences and functions, 12.5-5, 15.2-1
Sparse matrix, 20.3-2, 20.9-2, 20.9-4
Spearman, 19.7-6
Special unitary group, 14.10-7
Spectral decomposition, of power, 18.10-5
Spectral density, 18.10-3 to 18.10-10
non-ensemble, 18.10-8
Spectral representation, 14.8-4
Spectrum, of a linear operator, 14.8-3
of a probability distribution, 18.2-1, 18.2-2, 18.4-2, 18.4-7 (See also
Eigenvalue) Sphere, 3.5-9, A-5
geometry on a, 17.3-13
in metric space, 12.5-3
Spherical Bessel function, 10.4-4, 21.8-8, 21.8-13
Spherical coordinates, 3.1-6
vector relations in, 6.5-1
Spherical defect, B-6
Spherical excess, B-6
Spherical harmonies, 10.4-3, 14.10-7, 21.8-12
expansion in series of, 10.4-9
Spherical triangle, B-5 to B-9
Spherical waves, 10.4-8
Spheroid, 3.5-7, A-5
Spin matrices, 14.10-6
Spiral, of Archimedes, 2.6-2
linear, 2.6-2
logarithmic, 2.6-2
parabolic, 2.6-2Spur (see Trace) Square, A-2
Square root, computation of, 20 2-2
Stability, of difference equations, 20.4-8, 20.8-5 & of equilibrium, 9.5-4,
13.6-5, 13.6-6
of finite-difference approximations, 20.8-5, 20.9-5, 20.9-8
of limit cycle, 9.5-3
of linear system, 9.4-4
Lyapunov’s theory, 13.6-5
numerical, 20.1-2. 20.3-ld, 20.8-5, 20.9-8
of solutions, 13.6-5 to 13.6-7
Standard deviation, 18.3-3, 19.2-4
Standard form, of a conic, 2.4-8
of a quadric, 3.5-7
Standardized normal distribution, 18.8-3, 18.8-4, 18.8-8
Standardized random variable, 18.5-3, 18.5-5
Standing waves, 10.4-8. 10.4-9 (See also Space form of the wave
equation) State equations, 11.8-1, 11.9-1, 13.6-1
difference equations, 20.4-7
State-transition matrix, 13.6-2
for difference equations, 20.4-7
State variable, 11.8-1, 13.6-1
State vector, 13.6-1
Stationary random process, 18.10-1 to 18.10-10
Stationary values, and eigenvalue problems, 14.8-8 (See also Maxima
and minima) Statistic, 19.1-1
Statistical dependence, 18.4-12
Statistical hypothesis, 19.1-3. 19.6-1
Statistical independence, 18.2-3 to 18.2-5, 18.4-11, 18.6-2, 18.8-8
test for, 19.7-5, 19.7-6
Statistical relative frequency, 18.2-1
Steady-state solution, 9.4-2, 9.4-6
Steepest descent, 20.2-7, 20.3-2 (See also Saddle-point method)
Steffens-Aitken algorithm, 20.2-2d Steffensen’s interpolation
formula, 20.5-3
Step function, 21.9-1
Step matrix, 13.2-9, 13.4-6, 14.8-6 (See also Direct sum) Step-size
change, in optimization, 20.2-7in solution of ordinary differential equations, 20.8-3c, 20.8-5
in solution of partial differential equations, 20.9-5
Stereographic projection, 7.2-4. 14.10-6
Stieltjes integral, 4.6-17, 14.8-4, 18.3-6 21.9-2
Stieltjes-integral form of Laplace transformation, 8.6-3
Stieltjes measure, 4.6-15
Stirling numbers, 21.5-1
Stirling’s formula, 21.4-2, 21.5-4
Stirling’s interpolation formula, 20.5-3, 20.7-1
Stirling’s series, 21.4-2
Stochastic independence (sec Statistical independence) Stochastic
process (see Random process) Stochastic relation. 19.7-7
Stochastic variables (see Random variables) Stokes’s theorem, 5.6-2
Store enumerator, C-2
Straight line, in plane, equation, 2.2-1, 2.2-2
normal form, 2.2-1
in space, equation, 3.3-1, 3.3-2
Strategy, mixed, 11. 4-46
pure, 11.4-4a Streamlines, 5.4-3, 15.6-8
Strictly triangular matrix, 13.2-1
String, vibrations of, 10.4-9, 15.4-10
String property, of ellipse, 2.5-2
of involute, 17.2-5
Strip condition, 10.2-1, 10.2-4
Strongly monotonie function, 4.4-8
Strophoid, 2.6-1
Student’s ratio, 19.5-3
Student’s t (see t distribution) Sturm-Liouville operator, 15.4-3, 15.4-8
to 15.4-10, 15.4-12
Sturm-Liouville problem, 15.4-8 to 15.4-10, 15.5-2, 21.7-1
Sturm’s method, 1.6-6
Subfield, 12.3-2
Subgroup, 12.2-2
Subharmonic resonance, 9.5-5
Subinterval, 4.3-4
Submatrix, 13.2-8
Subring, 12, 3-2Subset, 4.3-2
Subspace, 12.5-1, 14=2-2
Successive approximations (see Iteration methods; Picard’s method)
Sufficient estimate, 19.4-1, 19.4-2, 19.4-4
Summable function, 4.6-15
Summation, by arithmetic means, 4.8-5, 411-7
of series, 4.8-5, 21.5-3, E-4
toE-7
by use of residues, 7.7-4
Sums, finite, 20.4-3, E-4
Superposition integral, 9.4-3, 18.12-2, 18.12-3
Superposition theorems, 9.3-19 10.4-1, 10.4-2, 13.6-2, 14.3-1, 15.4-2
for linear difference equations, 20.4-4
Surface, 3.1-14
of revolution, 3.1-15
Surface areas, formulas, A-4 to A-6
Surface coordinates, 17.3-1
Surface discontinuity, 5.6-3
Surface-distribution potential, 15.6-5, 15.6-7
Surface divergence, 5.6-3
Surface gradient, 5.6-3
Surface integral, 4.6-12, 5.4-6, 6.2-3, 6.4-3, 17.3-3
Surface normal, 17.3-2
Surface rotational, 5.6-3
Sylvester’s criterion, 13.5-6
Sylvester’s dialytic method, 1.9-1
Sylvester’s theorem, 13.4-7, 13.6-2
Symbolic differential equation, 9.4-3, 15.5-1
Symbolic function, 21.9-2 (See also Impulse functions) Symbolic logic,
12.8-5
Symmetric dyadic, 16.9-2, 16.9-3
Symmetric function, 1.4-3
Symmetric group, 12.2-8
Symmetric integral form, 15.3-6
Symmetric interpolation formulas, 20.5-3
Symmetric kernel, 15, 3-1
Symmetric linear operator, 14.4-6, 14.4-7Symmetric matrix, 13.3-2 to 13.3-4, 13.4-4, 13.5-6, 20.3-1, 20.3-2
Symmetric part, of a linear operator, 14, 4-8
of a matrix, 13.3-4
Symmetric quadratic form, 13.5-2
Symmetrical game, 11.4-4
Symmetrical step function (see Step function) Symmetry, of a relation,
1.1-3, 12.1-3
of tensors, 16.5-1
System determinant, 1.9-2, 9.4-5
System matrix, 1.9-4
Systematic overtaxation, 20.3-2d Systems of differential equations, 9.1-
3, 9.5-4, 10.1-2c, 13.6-1 to 13.6-7
numerical solution of, 20.8-6 t average, 18.10-7 I distribution, 19.5-3,
19.6-6, 19.7-4 t test, 19.6-4, 19.6-6
Tangent, to a conic, 2.4-10 to a plane curve, 17.1-1 to a space curve,
17.2-2 to 17.2-4
Tangent plane, 17.3-2
of a quadric, 3.5-8
Tangent surface, 17.2-5
Tangent vector, 17.2-2
in curved space, 17.4-3
Tangential acceleration, 17.2-3
Tangential curvature {see Geodesic curvature) Tangential developable,
17.2-5
Tauber’s theorem, 4.10-3
Taxicab norm, 13.2-1t, 13.6-5
Taylor’s series expansion, 4.10-4
complex, 7.5-2
multidimensional, 4.10-5
operator notation, 20.4-2
for solution of differential equation, 9.1-5, 9.2-5, 9.3-5, 20.8-1
vector notation, 5.5-4
Telegrapher’s equation, 10.4-8, 10.5-4
Tensor equality, 16.3-1
Term, 1.2-5
Terminal-state manifold, 11.8-lc Tesseral spherical harmonics, 10.4-3,
21.8-21Test, of significance, 19.6-4 to 19.6-6
in statistics, 19.1-3, 19.6-2, 19.7-7
with random parameters, 19.9-1 to 19.9-3
Test statistic, 19.6-4, 19.6-6, 19.9-3
Tetrahedron, A-6
Theta functions, 21.6-8, 21.6-9
Thiele’s interpolation formula, 20.5-7
Three-index symbols (see Christoff el three-index symbols) Time
average (see Finite-time average; t average) Time constant, 9.4-1
Time-invariant system, 13.6-2, 18.12-3
Time-optimal control, 11.8-36, 11.8-3c Time series (see Random
process) Tolerance interval, 19.6-4
Tolerance limits, 19.6-4
of normal deviate, 18.8-4
Topological spaces, 12.5-1
examples, 12.5-5
Topology, 12.5-1, 12.5-3
in a normed vector space, 12.5-2, 14.2-7
Toroidal coordinates, 6.5-1
Torsion; 17.2-3, 17.2-4
Torus, A-5
Total curvature, 17.2-3
Total differential equation, 9.6-1, 9.6-2
Trace, of direct product, 13.2-10
of a linear operator, 14 6-2
of a matrix, 13.2-7, 13.4-1, 13.4-3, 13.4-5, 20.3-3
of a tensor, 16.3-5
Tractrix, 2.6-2, 17.3-13
Transcendental numbers, 1.1-2
Transfer function, 9.4-7
Transfer matrix, 9=4-7
Transfinite number. 4 3-2
Transform, 12.1-4
Transformation, of coordinates, 2.1-5 to 2.1-7, 3.1-12, 6.5-1, 14.6-1
admissible, 6.2-1, 16.1-2
of elliptic functions, 21.6-7
of elliptic integrals, 21.6-6of linear operators, 14.6-2
of quadratic and hermitian forms, 13.5-4
of variables in a differential equation, 9.2-3, 9.3-8
of vector components, 6.3-3, 14.6-1, 16.2-1 (See also Linear operator)
Transformation theory of dynamics (see Hamilton -Jacobi
equation) Transient, 9.4-2
Transitivity of a relation, 1.1-3, 12.1-3
Translation, in complex plane, 7.9-2
of coordinate axes, 2.1-5, 2.1-7, 3.1-12
Transmission-line equation, 10o£-8, 10.5-4 I Transpose, of a linear
operator, ‘J4.4-6
of a matrix, 13.3-1
o Transposed dyadic, 16.10-11 u1 Transposed kernel, 15.3-1 \
Transversality condition, 11.6-8,’11.8-2, 11.8-3
Transverse axis, 2.5-2
Trapezoid, A-l Trapezoidal pulses, table, D-2
Trapezoidal rule, 20, 7-2, 20.8-3
Triangle area, plane, 2.1-4
space, 3.1-10
Triangle computations, B-l to B-4
for spherical triangles, B-5 to B-9
Triangle property, 12.5-2 (See also Minkowski’s inequality) Triangular
matrix, 13.2-1, 13.4-3
Triangular pulses, table, D-2
Trigonometric functions, 21.2-1 to 21.2-13
continued fractions, E-9
infinite products, E-8
power series, E-7
Trigonometric interpolation, 20.6-6
Trigonometric polynomial, 4.7-3, 4.11-2, 20.6-6
Trigonometric series, 4.11-2
Triple scalar product (see Scalar triple product) Trisectrix, 2.6-1
True representation, 14.9-1
Truncated normal distribution, 19.3-4
Truncation error, 20.1-2
in differential-equation solutions, 20.8-1
local, 20.8-1Truth table, 12.8-7
Truth value, 12.8-6
Tcchebycheff (see under Chebyshev) Twelve-ordinate scheme, 20.6-6
Two-person game, 11.4-4
Two-sided Laplace transform, 8.6-2, 18.12-5
Two-tailed test, 19.6-4, 19.6-8
Two-valued logic, 12.8-6
Two-way classification, 19.6-6
Type form (see Standard form) Ultraspherical polynomials, 21.7-8
Umbilic point, 17.3-5
Umbral index (see Dummy-index notation) Unbiased estimate, 19.1-3,
19.4-1, 19.8-1
Unbiased test, 19.6-3, 19.6-4
Unconditional convergence, 4.8-3, 4.8-7
Uncorrelated functions, 18.10-9
Uncorrelated variables, 18-4-11, 18.5-5, 18.8-8
test for, 19.7-4
Undamped natural circular frequency, 9.4-1
Undetermined coefficients, 9.4-1, 20.4-5
Uniform bounds, 4.3-3
Uniform continuity, 4.4-6
Uniform convergence, 4.4-4., 12.5-52
of infinite product, 4.8-7
of an integral, 4.6-2
of series, 4.8-2
Uniform distribution, 18.8-5, 19.5-4
Uniformly most powerful test, 19.6-3, 19.6-4
Unilateral continuity, 4.4-7
Unilateral limits, 4o4-7
Unimodal distribution, 18.3-3, 18.3-5
Unimodular group. 14.10-7
Unimodular matrix, 14.10-6
Union, in Boolean algebra, 12.8-1
of events, 18.2-1
of sets, 4.3-2
Uniqueness theorem, for Fourier series, 4.11-2, 411-5
for Fourier transforms, 4oll-5for harmonic functions, 15.6-2
for Laplace transforms, 8.2-8
for orthogonal vector components, 14, 7-4
for power series, 4.10-2
Unit impulse (see Impulse functions) Unit-step function (see Step
function) Unit-step response, 9.4-3
Unit vector, 5.2-4, 14.2-5, 14.7-3, 16 8-1
Unitary matrix, 13.3-2, 13.3-3, 13.3-4, 13.4-4
Unitary operator, 14.4-5, 14.4-7
Unitary representation, 14.9-1
Unitary transformation, 13.5-5, 14.4-5 (See also Unitary operator)
Unitary vector space, 14.2-6, 14.7-1
Unity, 12.3-1
Universe (see Population) Unknown periodic components, 12.6-6
Unstable solution, 13.6-5 v2 distribution, 19.5-3, 19.6-6
table, F-18 v2 test, 19.6-6
Value, of a game, 11.4-4
Vandermonde’s determinant, 1.6-5, 13.4-7
binomial theorem, 21.5-1
Van der Pol’s differential equation, 9.5-4, 9.5-5
Van der Pol’s method of solution, 9.5-5
Variance, 18.3-3, 18, 4-4, 18.4-8, 18.5-6; 18.5-7
of estimate, 19.2-1 to 19.2-4, 19.7-3, 19.8-1 to 19.8-4 (See also
Sample variance) Variance law, 18.5-6
Variance ratio (see v2 distribution) Variation, 11.4-1
of constants, 9.3-3, 13.6-3
for difference equation 20.4-4
total, 4.4-8
Vector of a dyadic 16.9-2
Vector field, 5.4-3
Vector potential, 5.7-2, 5.7-3
Vector product, 5 2-7, 6.3-4, 6.4-2, 16.8-4
dyadic, 16.9-2
Vector space (see Linear vector space) Venn diagram, 12.8-5
Versed cosine, B-9
Versed sine, B-9
Vertex of a conic, 2.4-9Vibrating membrane, 15.4-10
Vibrating string, 10.4-9, 11.5-7, 15.4-10
Volterra-type integral equation, 15.3-2, 15.3-10
Volume, 3.1-11, 4.6-11, 5.4-6 6.2-3, 17.3-3
formulas, A-4 to A-6
Volume-distribution potential, 15.6-5
Volume element, 6.2-3, 6.4-3
in Riemann space, I60IO-IO Volume integral, 4.6-12, 5.4-6, 5.4-7,
6.2-3, 6.4-3
Von Neumann-Goldstine rotation method, 20.3-5
Vortex point, 9.5-3, 9.5-4
Wave equation, 10.3-5, 10.4-8, 10.4-9, 11.5-7, 15.6-10, 20.9-8
two-dimensional, 10.4-6, 10.4-8 (See also Space form of the wave
equation) Wave number, 10.4-8
Wavelength, 10.4-8
Weakly monotonie function, 4 4-8
Weber, 21.8-1
Weddle’s rule 20.7-2« Weierstrass E function, 11.6-10
Weierstrass-Erdmann conditions, 11.6-7, 11.8-5
Weierstrass’s approximation theorems, 4.7-3
Weierstrass’s elliptic functions, 21.6-1, 21.6-2, 21.6-3, 21.6-9
Weierstrass’s necessary condition, 11.6-10, 11.8-1
Weierstrass’s normal elliptic integrals, 21.6-1 to 21.6-3, 21.6-5
of the first kind. 21.6-2
of the second kind, 21.6-3
Weierstrass’s test for convergence, 4.9-2
Weierstrass’s theorem, on essential singularities, 7-6-4
on product expansion, 7.6-6
Weight, of a configuration, C-2
of a figure, C-2
of a relative tensor, 16.2-1
Weighting function, 9.3-3, 9.4-3, 9.4-7, 18.12-2, 19.8-2
of an inner product, 15.2-1 (See also Green’s function) Weightingfunction method, Galerkin’s, 20.9-9, 20.9-10
Weingarten equations, 17.3-8
Well-defined transformation, 12.1-4
Well-ordered set, 1.1-2. 12.6-2Wiener-Khinchine relations, 18.10-3
for integrated spectra, 18.10-105
for non-ensemble spectral densities, 18.10-8
for real processes, 18.10-4
Wiener-Lee relations, 18.12-2, 18.12-3
Wiener-Paley theorem, 4.11-4e Wiener’s quadratic-variation theorem,
18.10-10
Witch of Agnesi, 2.6-1
Wronskian, 9.3-2
for cylinder functions, 21.8-1 z distribution, 19.5-3, 19.6-6 z
transform, 8.7-3, 20.4-6
Zermelo’s navigation problem, 11.8-3a Zero (number), 1.1-2
of complex function, 7t 6-1, 7.6-9
at infinity, 7.6-3
Zero-argument values, of theta functions, 21.6-8
Zero divisor, 13.2-5
Zero-sum game, 11 4-4
Zeros, of Bessel functions, 21.8-3
of cylinder functions, 21.8-3
of orthogonal polynomials, 21.7-2
of solutions, 9.3-8
Zeta function, Riemann’s, E-5
Weierstrass’s, 21.6-3
Zonal spherical harmonics, 10.4-4, 21.8-12
Zone, sphere, A-5
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