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 |  موضوع: كتاب Mathematical Handbook for Scientists and Engineers الثلاثاء 01 سبتمبر 2020, 10:43 pm | |
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أخوانى فى الله
أحضرت لكم كتاب
Mathematical Handbook for Scientists and Engineers
Definitions, Theorems, and Formulas for Reference and Review
Granino a. Korn and Theresa M. Korn
و المحتوى كما يلي :
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
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