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, 1.3-2
of 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, 20.2-2d
Aitken'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, 16.9-1
Antiperiodic 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
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, 16.10-5
Associative 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
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
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
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, 20.2-4
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 theorem, 18.2-6, 18.4-5, 19.9-2, 19.9-4
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, 9.3-3
modified, 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
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, 14.4-1
of 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 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
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, 4.3-1
Cap, 12.8-1
Cardinal number, 4.3-2
Cardioid, 2.6-1
Cartesian coordinates, local, 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 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-3
of 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
CEP (circular probable error), 18.8-7
Certain event, 18.2-1
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
in perturbation theory, 15.4-11
of quadric, 3.5-4
Characteristic equations, partial differential equations, 10.2-1, 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, 18.3-10
of 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, 19.2-2
Class 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
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
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, 9.3-1
Complementary 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, 2.6-1
Conditional 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, 14.9-4
Conjugate 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 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, 18.9-1
Continuous 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
choice of, 10.4-1
curvilinear, 6.2-1
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
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
Correlation functions, measurement, 19.8-3c (See also Autocorrelation function; Crosscorrelation function) Correlation matrix, 18.4-8
Cosine integral, 21.3-1
Cosine law, B-4, B-8, B-9
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
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 16.10-11
on 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
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
vector representation, 3.1-13, 17.2-1
Curvilinear coordinates, 6.2-1
Cusp, 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
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 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
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, 17.3-7
Differential 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, 15.6-2
Dirichlet 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 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
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
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, 16.10-11
Divergent 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 series, 19.3-3
Efficiency, of an estimate, 19.4-1
Efficient 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
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
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, 3.5-7
Elliptic 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 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, C-l, 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, 11.8-5
Ergodic 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, 13.2-1Ê 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
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, 20.6-3
Evolute, 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 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, 5.4-3
Figure, 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
Fixed point, of a mapping, 12.5-6
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, 8.7-1
table, 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 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, 9.3-6
Full linear group, 14.10-7
Full-wave rectified waveform, Fourier series, table, D-2
Laplace transform, 8.3-2
Boolean, 12.8-7
of a linear operator, 14.4-2, 14.8-3
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
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
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, 20.3-26
Gaussian 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, 14.10-6
of 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, 15.2-1
Graph, 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, 2.4-10
Harmonic 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
Hölder’s inequality, 4.6-19
Holomorphic function, 7.3-3
Homeomorphism, 12.5-1
Homogeneous 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, 1.1-2
of 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, 18.2-4
Independent 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
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 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-4
Inscribed 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 sets, 4.3-2
of surfaces, 3.1-16
Interval halving, 20.2-2f
Intrinsic derivative, 5.5-3, 16.10-8
Intrinsic 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
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
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, 17.1-8
Isolated 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
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, 4.11-4
Jump 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; Quasilineariza- tion) 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
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-7
Laplace-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 Laplace-transform 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
Legendre transformation, 9.2-3, 10.2-5
Legendre’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 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, 9.3-1
partial, 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) Local cartesian 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’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
Matrix 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, 4.7-1
Measureable 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, 2.5-2
Mittag-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, 19.7-2
Multiple 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
operator, 14.4-5
Negative semidefinite form, 13.5-2, 13.5-3
in complex-number plane, 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, 1.6-4
for 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
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, 17.3-2
Normal 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 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 factor, 18.3-4
Normalizer, 12.2-7
Normed vector space, 14.2-5, 14.2-6
Null curve, 17.4-4
Null direction; 17.4-4
Null 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
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
Operating characteristic, 19.6-2
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, 11.9-2
Optimal 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
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, 16.10-3
Orthogonal 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
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) Orthonormal-function 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
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, 17.3-5
Paraboloid, 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
Period, 4.2-2
of group element, 12.2-3
Period 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
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, 10.4-8
Point 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, 19.6-6
Population. 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
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
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
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, 14.9-4
Primitive 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
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 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, 2.4-3
Proper 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
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, 7.5-2
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, 4.2-2
Rational-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
principle of, 7.8-2
Reflection-rotation group, 12.2-11
Reflexivity, 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 series, 4.8-1
of Taylor’s series, 4.10-4, 4.10-5, 7.5-2
Remainder theorem, 1.7-2
Removable 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
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) Riemann-Green 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, 15.6-9
Riemann’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
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) Routh-Hurwitz 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, 9.5-4
of 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 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-21
Sampling 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) Schmidt-Hilbert formula, 15.3-8
Schrödinger 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) Schwarz-Christoffel 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, 15.3-6
Semidefinite 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
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 group, 12.2-5
Simple statistical hypothesis, 19.6-1
Simple surface, 3.1-14
Simplex 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 transform, finite, 8.7-1 (See also Fourier sine transform) Single-valued 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, 18.11-4
Smoothing, 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, 2.6-2
Spur (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
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, 20.2-7
in 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, 3-2
Subset, 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
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, 14.4-7
Symmetric 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
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, 21.8-21
Test, 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
in a normed vector space, 12.5-2, 14.2-7
Toroidal coordinates, 6.5-1
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
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
of elliptic functions, 21.6-7
of elliptic integrals, 21.6-6
of 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, 20.8-1
Truth 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
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-5
for 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, 2.4-9
Vibrating 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
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) Weighting-function 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. 12.6-2
Wiener-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