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General Index

abbreviation, 9Abelian

group, 114semigroup, 114

Adams-Bashforthmethod, 406, 415

Adams-Bashforth-Moultonmethod, 406

Adams-Moultonmethod, 416

algebraabstract, 113fundamental theorem of, 344

algebraicnumbers, 81simplification, 89structures, 69

algorithmcomplexity of, 343

analyticalderivation, 349, 389, 394, 412

antiderivative, 219approximant, 374approximation

discrete, 370, 372least squares, 357, 373–377of derivatives, 379rational, 368, 369trigonometric, 368

arc length, 224, 225Archimedes spiral, 195arithmetic

arbitrary precision, 73

modular, 171associative laws, 76asymptotic solution, 304, 306averaging transformations, 299Axiom, 174axis of rotation, 201

backward difference, 380Banach

fixed point theorem, 344space, 156

basis ofcolumn space, 155nullspace, 155row space, 155

Bessel functions, 265, 276Beta function, 262binary

hardware system, 8number system, 8relation, 85

binomial coefficient, 71bounded region

graphs of, 59volume of, 242

Bourbaki, N., 114

cardioid, 195Cauchy

integral formula, 254integral theorem, 254principal value, 317problem, 408

General Index 447

Cauchy–Riemann equations, 249Cayley

table, 119theorem, 119

Cayley, A., 113Cayley-Hamilton theorem, 162centered difference, 380, 381characteristic

matrix, 162polynomial, 162

Chebyshevapproximation, 368interpolation, 362, 366nodes, 362, 366polynomials, 263series expansion, 368

Chebyshev-Padeapproximation, 368

Christoffel symbol, 186, 187Church, A., 86circular membrane, 283Clairaut, A., 364commutative laws, 76commutator relation, 183complex

algebra, 245argument, 245conjugate, 245functions

graphs of, 253integration, 253numbers, 81, 139, 245

absolute value of, 245evaluation of, 245geometric interpretation, 251imaginary part of, 245limit of, 251polar representation of, 245real part of, 245sets of, 252sign of, 245

plane, 252, 256

vectors in, 139computation

domain, 89time, 18, 35, 37, 39

computational cost, 343conformal mapping, 259constants, 35

mathematical, 11, 35predefined, 11

continued fraction, 368, 369continuity equation, 305, 308convergence

analysis, 402, 404, 409, 411rate, 346, 348, 349, 351, 352

convolution theorem, 322, 338Cooley, J.W., 372coordinate

basis, 187coordinates

cylindrical, 204polar, 65, 252spherical, 205

coprimes, 195Cramer rule, 396Crank-Nicolson method, 437curve fitting, 356, 373curves

animations of, 64area between, 196, 224implicitly defined, 194in polar coordinates, 194

cyclic group, 124

d’Alembert solution, 276data

approximation of, 356object, 134, 141structure, 20type, 10, 33, 87

De Moivre theorem, 247decimal number system, 8, 29degenerate kernel, 331, 332, 338

448 General Index

degree of freedom, 17, 41density plot, 61, 62derivative

covariant, 186directional, 186partial, 235, 236

differentialcalculus, 212–217operator, 306

Diracdelta function, 268notation, 182

discreteFourier transform, 278, 281, 372transform, 278–281

discretizationerror, 380, 381step, 409, 411

dispersion relation, 306distribution functions, 268, 272distributive laws, 78divergence to infinity, 256division algorithm, 13, 37double

factorial, 37, 42pendulum, 17, 41quotes, 34, 35

dynamical system, 256, 299

Efros theorem, 338eigenspace, 162eigenvalue, 162eigenvalues, 166eigenvector, 162Einstein theory, 186elimination methods, 164ellipsoid, 199environment variable, 8, 228equation

algebraic, 285–290integral, 316–340nonlinear, 344

ordinary differential, 265, 291parametric, 195partial differential, 300–316real roots of, 15, 39trascendental, 264, 345

equidistant nodes, 360equivalence relations, 85error

absolute, 79, 342, 409approximation, 232computational, 342discretization, 342, 388, 389estimators, 342function, 262mean square, 370, 374, 375real, 409relative, 79, 342, 386, 387roundoff, 79, 342truncation, 342upper bound of, 231, 232

Euclideandomain, 174space, 154, 189, 316

Eulerconstant, 5, 26equations, 304method, 406, 409, 411, 412

Euler, L., 379Euler-Maclaurin expansion, 379Eulerian variables, 304exponent, 73exponential integral, 276expression, 9, 30, 33

boolean, 10, 34, 87converting, 89denominator of, 89head of, 30numerator of, 89trigonometric, 92type of, 13

external flow, 244

General Index 449

factorial, 13, 37, 42double, 13, 19

Fast Fouriertransform, 278, 279, 372

Fermat spiral, 195FFT, 372Fibonacci

numbers, 13, 18, 37, 39field, 126

algebraic extension of, 130characteristic of, 126finite, 126, 130

finite differences, 380, 381, 424, 426fitting

exponential, 373linear, 373logarithmic, 373nonlinear, 373polynomial, 373power, 373

fixed string, 439floating-point number system, 73fluid motion, 299form

bilinear, 168linear, 168quadratic, 168

forward difference, 380Fourier

coefficient, 232, 233, 281polynomial, 280series, 232–234, 281transform, 269, 274, 276, 281

fractals, 256fraction-free elimination, 164Fredholm

determinant method, 326, 327integral equation, 324–332

freeboundary problem, 313surface, 304vibration, 283

frequencyamplitude dependence, 304, 315correction, 299spectrum, 278

Fresnel integral, 262Frullani integral, 227function

analytic, 249animations of, 22approximation of, 356bijective, 107composition, 14, 15, 38, 39, 105continuity of, 210definitions of, 14derivative, 209, 212, 213digamma, 262elliptic, 263evaluation of, 14, 38exponential, 36graph of, 208harmonic, 249, 250holomorphic, 254hypergeometric, 267inverse, 104iteration, 352limit of, 209multilinear, 178multivalued, 250multivariable, 14, 228, 235, 236periodic, 233, 234, 364, 370, 372periodic extension of, 16, 39piecewise, 15, 39, 61, 100, 268polygamma, 262prime number, 13real, 207recursive, 39, 101rose, 195scalar, 97sign, 13, 37square integrable, 323vector, 14, 15, 37, 38, 97

functional, 168

450 General Index

functionsabbreviations of, 96absolute value of, 13, 37active, 12arithmetic, 30, 31classes of, 96computable, 96elementary trascendental, 12hyperbolic, 12, 36inert, 12library of, 12predefined, 12, 36procedural, 96pure, 36, 291random, 37special, 12, 36specialized, 28trascendental, 36trigonometric, 12, 36user-defined, 12, 14, 36, 37

Galerkin method, 430Galerkin, B.G., 430Galois field, 126Galois, E., 113Gamma function, 262GAP, 115Gauss, K.F., 372, 374, 379, 396Gauss-Chebyshev quadrature, 391Gauss-Jordan elimination, 164Gauss-Jordan method, 397Gauss-Kronrod quadrature, 394Gauss-Legendre quadrature, 391Gauss-Lobatto quadrature, 392Gauss-Radau quadrature, 392Gauss-Seidel method, 400–402Gaussian

elimination, 164, 397quadrature, 379, 391

Gear extrapolation method, 406Gegenbauer polynomials, 263general relativity theory, 178, 187

generalizedcoordinates, 17forces, 17functions, 268

geometricinterpretation, 350shape, 205, 206

globalminimum/maximum, 217

gradient field, 203Gram-Schmidt

orthonormalization, 156graphs

bar, 62intersection of, 190, 289logarithmic, 60multiple, 54–56of bounded region, 59of function, 51of piecewise function, 61options of, 52, 54special, 58–62structures of, 55, 56text strings in, 57with color gradient, 62with legend, 62

greatestcommon divisor, 13, 37, 112

Green function, 432grid lines, 58group, 113, 114

conjugate subgroup, 122cyclic, 124factor, 120, 128generator of, 124normal subgroup, 122permutation, 114, 117quotient, 120set of cosets, 120, 121, 128subgroups of, 121, 122symmetric, 117tables, 119, 128

General Index 451

Hankelfunctions, 265matrix, 149transform, 281

hardware precision, 278hardware-floats, 73harmonic conjugate, 250Heaviside step function, 268Helmholtz equation, 281Hermite

decomposition, 166interpolation, 359polynomials, 263

Hermitianskew-symmetric matrix, 183transpose matrix, 149transpose vector, 139

Heun method, 406, 412Hilbert

matrix, 149space, 156transform, 269

Horner form, 358, 368, 369hyperbolic spiral, 195hyperboloid

of one sheet, 199of two sheets, 199

indeterminate forms, 79inequality, 10, 60inner product, 156integers

congruent modulo, 171nearest, 13

integralabsolute convergence of, 226approximation, 218around poles, 255by parts, 218, 223by substitution, 218, 223convergence of, 225, 226definite, 218, 223

double, 240elliptic, 263exponential, 262Fresnel, 262Frullani, 227improper, 223, 225–227indefinite, 218iterative, 235logarithmic, 262multiple, 235, 236, 385polylogarithmic, 262trigonometric, 262

integral calculus, 218–228integral domain, 125, 129integral equation, 316–340

eigenfunction of, 332eigenvalue of, 332exact solution, 317kernel of, 316linear, 316nonlinear, 316, 333–336of convolution type, 318, 335of the first kind, 316of the second kind, 316of the third kind, 316singular, 317, 337, 338with constant

limits of integration, 317with variable

integration limit, 317integral transform, 269–278interface parametrization, 313interpolation, 356

composite linear, 365cubic spline, 367errors, 361, 362picewise polynomial, 367piecewise linear, 365polynomial, 357, 358rational, 357, 369spline, 357, 365, 367trigonometric, 357, 364, 372

452 General Index

inversefunction, 286hyperbolic functions, 36trigonometric functions, 36

iterationsfixed point, 352number of, 346, 347

Jacobiiterations, 396method, 400, 401, 404polynomials, 263

Jacobi, C.G.J., 396Jacobian, 305, 354Jordan decomposition, 166Julia set, 256

kinetic energy, 17Kleene, S.C., 86Klein four-group, 124knots, 197Kronecker product, 182, 183Kronecker, L., 113

Lagrangeequations, 17, 41, 305interpolation, 358, 360, 361, 364,

365, 367, 375multipliers, 238, 239

Lagrange, J.L., 364Lagrangian, 17

formulation, 304mechanics, 17variables, 304

Laguerre polynomials, 263lambda calculus, 86Lambert function, 264Laplace equation, 281, 434Laplace transform, 269, 294, 318,

322, 335, 338, 339Laurent series, 258least

common multiple, 13, 37

Legendrefunctions, 266polynomials, 36, 263

Leibniz, G.W., 405lemniscate, 195level

curves, 200surfaces, 200

Levi-Civitasymbol, 183tensor, 183

Lie product, 183linear

algebra, 134boundary value problem, 424equation, 158operator, 160space, 134, 154system, 158, 396, 400transformation, 160

Lipschitz condition, 423Lissajous curves, 65, 195list, 43, 45

empty, 43nested, 43–45of coordinates, 84simple, 43, 44

Livermore method, 406logarithm

general, 36natural, 36

logarithmic spiral, 195LU factorization, 397

machinecode, 29epsilon, 79number, 74

Maclaurinpolynomial, 230–232series, 228, 230, 231

Maclaurin, C., 379

General Index 453

Magma, 115, 174magma, 114Mandelbrot set, 256mantissa, 73map of North America, 28mappings, 103mathematical object, 134, 141Mathieu

equation, 57functions, 266

matricescommutative, 152congruence relations of, 164equivalence relations of, 164similarity relations of, 164

matrix, 46adjoint, 150algebra, 145augmented, 158band, 149canonical form of, 164coefficient, 158cofactor, 150column operations, 147components, 142constant, 149corank of, 158data type of, 143decomposition, 164, 396determinant, 7, 150diagonal, 149diagonalization of, 162diagonally dominant, 402, 404dimension, 155eigenvalues of, 144factorization, 164function of, 151identity, 149inverse, 150involutive, 152kernel of, 155, 158minors of, 150

norm, 404normal form of, 164positive definite, 402random, 7, 149rank of, 158row operations, 147sparse, 149special, 149spectral radius of, 402, 404submatrix of, 146symmetric, 144, 402trace of, 162transposition, 149vector product, 146visualization, 148zero, 149

maximum, 40, 238, 239amplitude, 315wave profile, 315

Mellin transform, 269memory feature, 39method

bisection, 346direct, 396, 397elimination, 396iterative, 344, 396, 400least squares, 370, 373linear shooting, 420multi-step, 415, 416nonlinear shooting, 423predictor-corrector, 406secant, 351separation of variables, 433single-step, 409, 411, 412variational, 429

minimaldistance, 216polynomial, 162volume, 216, 217

minimax approximation, 368, 369minimum, 238, 239module, 16, 18, 40, 41

454 General Index

moment of inertia, 240monoid, 114Mozart, W.A., 71multivariate calculus, 234–244musical

chromatic scale, 129composition, 71inversion, 129transposition, 129

Newtoninterpolation, 358iteration formula, 349, 354method, 288, 345, 347, 350, 354method modified, 348

Newton, I., 379, 405Newton-Cotes

quadrature, 379, 385, 389nonlinear

boundary value problem, 426equations, 354frequency, 306

normal equations, 374normed space, 342notation

disjoint cycle, 117engineering, 29mathematical, 180scientific, 29tensor, 180

number ring, 109numbers

fractional part of, 13maximum/minimum of, 13

numericalanalysis, 341approximation, 8, 29, 281, 288differentiation, 379, 382integration, 379, 383solution, 287, 288, 345

ODE, 271, 291–300

analytical solution, 291animations, 406boundary value problem

for, 296, 420, 423, 429, 430exact solution, 291, 293explicit form of solution, 292general solution, 291, 292graphical solution, 293, 295, 406high order, 293, 420homogeneous, 292implicit form of solution, 292initial value problem

for, 271, 273, 274, 291, 295,297, 298, 321, 408

integral transform method, 291interactive

numeric/symbolic solving, 291Laplace transform, 293linear, 291, 292, 319numerical solution, 293, 295, 406phase portrait, 295, 299, 300, 406separation of variables, 292series solution, 291solution classification, 292solution suggestion, 292system, 274, 291, 296, 418vector field, 295, 296, 406with polynomial coefficients, 291

operatorarithmetic, 9, 30, 31assignment, 9, 32composition, 14conditional, 39logical, 9, 10, 31, 34, 87relation, 9, 10, 31, 34, 87replacement, 38unassignment, 9, 33

orthogonalkernels, 329matrices, 156polynomials, 263, 368, 391vectors, 156

General Index 455

orthogonality condition, 309orthonormal vectors, 156

Pade approximation, 368, 369parametric curve, 192, 193, 305Pascal matrix, 153pattern, 31, 37, 42, 44, 208

matching, 31, 89PDE, 276, 283, 300–316

analytical solution, 301boundary value problem

for, 303, 434finite differences for, 437, 439Fourier series expansion

for, 433, 434general solution, 301graphical solution, 302, 303, 432initial value problem

for, 276, 283, 301initial/boundary value problem

for, 271, 432linear system, 307method of characteristics, 301numerical solution, 301–303, 432separation of variables, 301system, 301

perturbation theory, 290, 306phase space, 256Picard-Lindelof theorem, 408pivot element, 399pivoting strategy, 399point

along a curve, 68critical, 215, 237inflection, 215

polynomialcoefficients, 111, 112definition, 109, 110degree, 111, 112distribution, 89division, 111, 112equations, 112

factorization, 89multivariate, 109normal form of, 89quotient, 13remainder, 13roots, 111, 112, 286simplification, 89sorting, 111type, 111, 112univariate, 109, 130

potential energy, 17power series, 290, 306precision

arbitrary, 35machine, 35user-specified, 8, 29with accuracy, 35

previous results, 11, 35primality test, 13prime number, 37principle

minors, 399of induction, 85, 86of well-ordering, 86

problemill-posed, 323well-posed, 313

procedure, 16recursive, 18, 39

projection method, 430

quadraturecomposite, 387equidistant, 379, 383, 385, 388midpoint, 386trapezoidal, 386

random numbers and variables, 13rational approximation, 369recurrence relation, 327reduced row-echelon form, 164reflexivity, 85

456 General Index

regression analysis, 356, 373regularization method, 323relative

extremum, 235, 236minimum/maximum, 217

Remez algorithm, 368removable discontinuity, 251repetition structure, 41resolvent kernel, 320–322, 326–329resonance parameter, 299Richardson extrapolation, 382, 390Richardson, L.F., 382Riemann

curvature tensor, 186, 188integrable function, 384sphere, 256, 259sum, 218, 219, 383, 384surfaces, 260Zeta function, 267

ring, 125commutative, 128factor, 126ideal of, 125quotient, 126

Ritz method, 429Ritz, W., 429Romberg integration, 379, 390Romberg, W., 390rose function, 195Ruffini, P., 113Runge

function, 361, 362phenomenon, 362, 365

Runge, C.D.T., 361Runge-Kutta

method, 406, 412, 418Runge-Kutta Fehlberg

method, 406Runge-Kutta Rosenbrock

method, 406

scalar product, 182

Schoenberg, I.J., 365Schur decomposition, 166Schwarzschild

metric, 188vacuum, 186

Schwarzschild, K., 186scientific

computing, 341notation, 73, 83

secant line, 197semigroup, 114sequence

animations of, 66, 67graph of, 22, 48, 84limit of, 210of arguments, 16of characters, 10, 34

series, 228–234coefficient, 228, 229convergence, 229, 230, 255expansion, 228, 229functional, 228infinite, 267inverse, 91, 229multiplication, 228numerical, 228polynomial

representation of, 228, 229radius of convergence, 256sum, 228, 229

setof random primes, 75operations on, 84subsets of, 70

sign condition, 313signal recuperation, 278, 279significant digits, 8, 29, 73Simpson quadrature, 223, 224, 387singular value decomposition, 166singularity, 159, 160, 166, 167, 258,

317, 337, 338small parameter, 290, 299, 305

General Index 457

Smith decomposition, 166software-floats, 73solution

approximate, 158analytical, 290, 304

exact, 158, 285, 286interactive, 406periodic, 304

special functions, 261–268special options, 16spectral decomposition, 166spectrum, 162stability diagram, 57statement, 9, 16

separators, 10string vibration, 271strings, 10, 34structure

animations of, 68applying a function to, 22control, 19, 41dimension of, 44element of, 21empty, 20inserting, 21length of, 44number of elements, 21properties, 22rank of, 44repetition, 19replacing, 21selection, 19, 41substructure of, 21, 44

structuresconcatenation of, 21difference of, 22division of, 22intersection of, 22multiplication of, 22removing, 22sum of, 22union of, 22

subharmonic resonance, 299successive approximations

method, 323, 324, 333, 334surface

implicitly defined, 198of revolution, 201, 202

volume of, 225parametrically defined, 198tension, 313

Sylvester criterion, 402symbol, 30symmetry, 85system

holonomic, 17, 41of equations, 285–287of integral equations, 339of linear equations, 396of nonlinear equations, 354quantum state of, 182

table, 47tangent

line, 197, 214–216, 350plane, 238

Taylorpolynomial, 228, 232series, 228, 230, 231series method, 406

tensor, 47algebra, 182, 183analysis, 178antisymmetric, 183calculus, 178components, 179contravariant metric, 179, 180covariant metric, 179, 180curvature, 187differentiation, 178, 186index character, 179product, 182, 183rank, 179type, 179

458 General Index

tensorsdefinition of, 179linear combination of, 182, 183operations on, 178, 179

theoremfixed point, 352intermediate value, 346

Tikhonovregularization method, 324

tolerance, 342tonal music, 71, 108, 129total

CPU time, 13derivative, 236, 237differential, 236, 237

transformation rules, 30, 286transitivity, 85transverse vibration, 271trapezoidal

quadrature, 223, 224, 390trascendental equation, 285–287tree structure, 10, 33trigonometric

polynomials, 370, 372triple product, 169tubes, 197Tukey, J., 372type

of brackets, 10, 35of numbers, 11, 35of quotes, 11, 35range, 9

unevaluated form, 8, 29unit

normal, 243, 313roundoff, 79

units of angle, 11upper bound, 360

variableglobal, 16, 40

local, 16, 40with assigned value, 9with no assigned value, 9

vectoralgebra, 137components, 135, 136constant, 139, 140direction of, 134field, 203, 244, 406matrix product, 146norm, 156random, 139, 140space, 134, 154–156sparse, 139, 140special, 140unit, 139, 140visualization, 138zero, 139, 140

vector spacebasis of, 154inner product, 156normed, 156

vectorsangle between, 156representation of, 135, 136special, 139transposition of, 139

Volterraintegral equation, 317–335

wave equation, 65, 271, 301, 303,432, 433, 439

wave number, 306waveform, 234waves

standing, 304traveling, 65

Maple Index

( ), 10, 14->, 14, 22, 235., 137, 146, 156, 235, 240, 243, 332.., 9, 16, 20:, 10:-, 18, 132, 302, 303, 405::, 18, 83, 282, 313:=, 9, 21;, 10<, 9, 135, 137, 142< >, 20, 191<=, 9, 16<>, 9=, 9>, 5, 9>=, 9, 16?, 6@, 14, 15, 57, 105, 212, 223[ ], 10, 15, 20–22, 54, 84, 135, 142,

216, 224, 229–231, 277#, 11$, 237, 238, 250, 263, 301%, 11&*, 22&x, 235, 9

', 61, 109, 233, 238, 251, 257'', 9, 11`, 237, 238``, 11, 17{ }, 10, 20, 54, 287, 294, 303Abs, 227

AddColumn, 396AddFlavor, 75AddRow, 396Add, 137, 145Adjoint, 150ApproximateInt, 218, 383method, 223, 383, 385partition, 223

ArrayInterpolation, 357Array, 20, 135, 293, 297Asymptotes, 210, 211BackwardSubstitute, 165, 396BandMatrix, 149BarChart, 62Basis, 155BesselI, 265BesselJZeros, 265BesselJ, 265, 282BesselK, 265BesselYZeros, 265BesselY, 265, 276Beta, 262BilinearForm, 168conjugate, 168

Bra, 182CanonicalTensors, 179Catalan, 11CharacteristicMatrix, 153, 162CharacteristicPolynomial, 162Christoffel1, 186Christoffel2, 186Ci, 262

460 Maple Index

ColumnDimension, 142, 155ColumnGraph, 62, 63ColumnOperation, 147ColumnSpace, 155CommutatorL, 184CommutatorR, 184Commutator, 182Compile, 83Complex, 11, 81ConstantMatrix, 149ConstantVector, 140ConvertIn, 174ConvertOut, 177Copy, 172CramerRule, 396Create, 172CriticalPoints, 212, 215CrossProduct, 137, 235Curl, 235, 243DBL EPSILON, 79DBL MAX, 77DBL MIN, 77DEFAULT, 211DEplot3d, 295, 405DEplot, 295, 297, 405DGsetup, 179, 181DGtensor, 179Define, 179, 181DeleteColumn, 146DeleteRow, 146DensityPlot, 63Determinant, 7, 145, 150, 158, 172,

174, 327, 331, 332, 396DiagonalMatrix, 149, 404Diagonal, 404Diff, 12, 212, 235, 254, 268Digits, 8, 73, 77, 79, 82, 144, 278,

280, 287, 316, 344Dimension, 135, 142, 155Dirac, 268, 269, 282Distance, 212, 216Divergence, 235, 242–244

DotProduct, 137, 156, 168, 235Doubleint, 240D, 212, 236, 272, 277, 294, 297, 299Eigenvalues, 144, 162Eigenvectors, 162, 166, 297Ei, 262, 276EllipticE, 263EllipticF, 263EllipticK, 263EllipticPi, 263Eq Lagrange, 17Equal, 157, 166Equate, 297Expand mod, 127ExponentialFit, 373ExtremePoints, 212, 215FAIL, 11FFT, 278, 279Factor mod, 111, 132Factors, 127Fib, 18Fit, 368, 373Float, 11, 75ForwardSubstitute, 165, 396FourierTransform, 278, 280, 368FresnelC, 262FresnelS, 262FunctionAdvisor, 12, 261GAMMA, 262GF, 127, 174GaussJordanEliminationTutor, 398

GaussianElimination, 165, 396Tutor, 396, 398

GenerateEquations, 158, 404GenerateMatrix, 158, 331, 404augmented, 158

Generate, 13, 75, 109complex, 246

Gradient, 235, 242, 243, 429GramSchmidt normalized, 156HFloat, 75, 77, 79

Maple Index 461

HankelH1, 265HankelH2, 265HankelMatrix, 149Heaviside, 9, 100, 211, 234, 268–

270, 272, 273, 282HermiteForm, 166HermiteH, 263HermitianTranspose, 137, 145HilbertMatrix, 149Histogram, 62Homotopy, 287, 344IdentityMatrix, 22, 149Im, 81, 246, 280Infinitesimals, 301InflectionPoints, 212, 215IntersectionBasis, 155Int, 12, 218, 224, 242, 254, 268,

327, 383AllSolutions, 325

InverseFourierTransform, 278Inverse, 172Invfunc, 104Irreduc mod, 127Irreduc, 132IsOrthogonal, 156IsUnitary, 156, 168Isolate, 287, 344I, 11, 245JACOBIAN, 355, 427Jacobian, 235JordanBlockMatrix, 166JordanForm, 166JuliaSet2D, 257JuliaSet3D, 257Ket, 182KroneckerDelta, 181bas, 181

KroneckerProduct, 182LUApply, 172LUDecomposition, 165, 172, 396method, 165output, 165

LagrangeMultipliers, 235, 238LambertW, 153Laplace, 301Laplacian, 235, 242, 243, 303, 435LargeExpressions, 134LeastSquares, 373free, 158optimize, 158

LegendreP, 266LegendreQ, 266LeviCivita, 184LieProduct, 184Limit, 12, 209, 227LineChart, 62LinearFit, 373LinearSolve, 158, 174, 331, 396inplace, 172method free, 158

Li, 262LogarithmicFit, 373MAX DIGITS, 74, 77MAX FLOAT, 77MIN FLOAT, 77MandelbrotSet, 257Maple floats, 74, 77Map, 144MathieuA, 57, 266MathieuB, 57, 266MathieuCEPrime, 266MathieuCE, 266MathieuC, 266MathieuExponent, 266MathieuFloquet, 266MathieuSE, 266MathieuS, 266MatrixAdd, 145MatrixBuilder, 142MatrixExponential, 152MatrixFunction, 152MatrixInverse, 145, 150, 162, 166MatrixMatrixMultiply, 145MatrixNorm, 156

462 Maple Index

MatrixOptionsdatatype, 144shape, 144

MatrixPower, 152MatrixScalarMultiply, 145MatrixVectorMultiply, 145, 174Matrix, 20, 22, 142, 297, 327->, 142datatype, 144fill, 142shape, 149storage, 149symbol, 22, 142, 146, 153

MaxMin, 18Maximize, 212, 217, 238, 363Maximum, 16MinimalPolynomial, 162Minimize, 212, 217, 238Minor, 150Mod, 172MultiInt, 235, 240Multiply, 145, 172NGSol, 298NULL, 18, 20, 22, 68, 240, 280, 295,

298, 321, 332Nabla, 235NextAfter, 79NextZero, 287, 344NonlinearFit, 373Normalize, 156Euclidean, 157

Normal, 13Norm, 137, 145, 156NullSpace, 155, 158ON, 272, 277, 284, 301Order, 228OuterProductMatrix, 137PieChart, 62interactive, 62

Pivot, 396Pi, 11PolynomialFit, 373

PolynomialInterpolation, 357form, 357

PowerFit, 373Psi, 262QRDecomposition, 165, 396RETURN, 17RREF, 174RandElement, 115RandomMatrix, 7, 10, 149RandomVariable, 13Normal, 63

RandomVector[column], 140RandomVector, 140Range, 282Rank, 150, 158, 172RationalInterpolation, 357ReducedForm, 301ReducedRowEchelonForm, 10, 165,

174, 396RegPertPoly, 290Rem mod, 127Re, 81, 246, 278, 280RiemannSum, 218, 383method, 219, 383partition, 219

Riemann, 186RombergInt, 390RootOf, 11, 81, 285index, 81

Roots, 127Rounding, 79RowDimension, 142, 155RowOperation, 147, 396RowReduce, 172RowSpace, 155SFloat, 75, 77, 79SampleNormal, 63

ScalarMatrix, 149ScalarVector, 140ScatterPlot, 189, 191SchurForm, 166

Maple Index 463

SetCoordinates, 235, 243, 244Setup, 182dimension, 184signature, 184

Si, 262SmithForm, 166Spline, 357SubMatrix, 142, 146SumBasis, 155Sum, 210SurfaceInt, 235SurfaceOfRevolution, 201, 202axis, 201, 202output, 201, 202surfaceoptions, 201, 202

SwapRow, 396Tangent, 197ThieleInterpolation, 357Trace, 145, 162Transpose, 137, 145, 172Tripleint, 241UnitVector, 140UseHardwareFloats, 8VConj, 250VectorAdd, 137VectorAngle, 156VectorField, 235, 243, 244VectorNorm, 156Vector, 20, 22, 135, 278–280, 295column, 135fill, 135row, 135shape, 140symbol, 135, 146

VolumeOfRevolution, 201axis, 202output, 202

ZeroMatrix, 22, 149ZeroVector, 140Zeta, 267C1, 292, 297seed, 75

about, 246abs, 12, 81, 226, 230, 246, 253, 257,

275, 342addtable, 282, 283add, 17, 109, 210, 228, 233, 280,

290, 321, 327, 331algsubs, 86, 88, 94alias, 9, 81, 96, 264allvalues, 81, 237, 238, 247, 285,

287, 336and, 9, 16, 241angle, 11animate3d, 64, 302animatecurve, 64coords, 65frames, 65thickness, 65

animate, 64, 300, 302, 303, 316background, 68frames, 65, 197view, 197

applyop, 22applyrule, 88apply, 22, 103, 207arccos, arccosh, 12arccot, arccoth, 12arccsc, arccsch, 12arcsec, arcsech, 12arcsin, arcsinh, 12arctan, arctanh, 12argument, 81, 246, 252array, 65, 109, 179, 195, 237, 238,

278, 280, 294plot, 55sparse, 179, 186, 188storage, 140symmetric, 179, 186, 188

arrow, 138, 251, 252assign, 9, 220, 274, 294, 297, 354assume, 85, 212, 225, 246, 247, 269,

282, 337assuming, 223, 225–227, 240, 325

464 Maple Index

asympt, 265bernoulli, 109bernstein, 109binomial, 71, 153break, 9, 19, 20by, 9cartprod, 71casesplit, 301cat, 17, 21, 109ceil, 12center, 115changevar, 218, 221, 223, 241chebpade, 368chebyshev, 363, 368choose, 70circle, 206coeffs, 111coeff, 111, 220, 233, 290coeftayl, 228collect, 111, 233, 290combine, 89, 222, 223, 240, 243, 273trig, 15, 92, 221, 233

completesquare, 221compoly, 111cone, 206conformal3dgrid, 259orientation, 259

conformal, 58, 251, 265grid, 58, 259numxy, 259

confracform, 368conjugate, 246, 249constants, 11content, 111contourplot3d, 200contourplot, 292coloring, 200contours, 200, 201filled, 200grid, 201, 203

contract, 182

convert, 10, 11, 89+, 81, 313Array, 135Bessel, 265Ei, 262Hankel, 282Heaviside, 268, 275Matrix, 135StandardFunctions, 255, 267Vector, 135`+`, 109cos, 221diff, 236disjcyc, 115exp, 92, 277factorial, 262horner, 368hypergeom, 266int, 277, 282, 284list, 17, 185, 237, 290, 313, 331parfrac, 94, 102, 220

sqrfree, 210permlist, 115piecewise, 100, 268, 272, 273polar, 246polynom, 91, 111, 228, 230, 231radical, 81ratpoly, 357, 368set, 233sincos, 94string, 195tan, 221trigh, 92trig, 92, 247

coordplot, 58cartesian, 58grid, 58view, 58

core, 115cos, cosh, 12cosets, 115cost, 343

Maple Index 465

cot, coth, 12cov diff, 186create, 179, 186, 188csc, csch, 12csgn, 246cuboid, 206curry, 227cyclotomic, 109cylinderplotgrid, 204

cylinder, 206d1metric, 186d2metric, 186dchange, 301declare, 212, 236, 272, 277, 284,

301defform, 237degrees, 11degree, 111denom, 74, 89, 254densityplot, 61, 251, 253, 257axes, 62colorstyle, 62numpoints, 62style, 62

dfieldplot, 295, 405diagonal, 22diff table, 301diff, 12, 17, 212, 231, 235, 236,

248, 249, 264, 268dilog, 262directional diff, 186discont, 210, 211discrim, 111display3d, 55, 238, 242insequence, 64, 67, 68orientation, 57, 68scaling, 68

displayGR, 188display allGR, 188display, 55, 57, 59, 60, 65, 190,

195, 238, 278, 280, 294

axes, 58insequence, 22, 64, 66, 102, 195scaling, 58, 62title, 55xtickmarks, 58ytickmarks, 58

divide, 111doublefactorial, 12do, 9dsolve, 249, 273, 277, 297, 320Lie, 291coeffs, 291formal series, 291implicit, 292interactive, 291, 295, 405method, 274, 291numeric, 295, 405method, 405output, 405

output, 295, 297range, 298series, 291, 292type, 298

d, 237elements, 115eliminate, 259, 260end, 9entermetric, 179environment, 8erfc, 262erf, 262, 269euler, 109evala, 81, 89, 111, 127, 212, 231,

286, 353evalb, 84, 85, 249, 270, 286evalc, 12, 81, 246, 247, 249, 250,

252, 253, 255, 260, 278evalf, 5, 8, 12, 14, 82, 207, 229,

231, 233, 234, 237, 238, 287,288

evalhf, 8, 73, 77, 79, 82evalm, 157

466 Maple Index

evaln, 9, 21eval, 14, 104, 190, 231, 273, 287,

301, 379example, 6exp(1), 11expand, 5, 12, 89, 111exp, 12extrema, 235, 238factorial, 12, 327factor, 89, 111, 145, 243false, 11fdiff, 379fdiscont, 210, 211fibonacci, 12, 109fieldplot3daxes, 203scaling, 203

fieldplotarrows, 203, 295grid, 203

floor, 12fnormal, 79, 167, 231, 393forget, 18for, 9, 15, 17–20, 22, 84, 109, 190,

195, 257, 290, 298, 321, 327while, 19, 20, 79, 257

fouriercos, 274, 276, 337fouriersin, 274, 276fourier, 274, 275, 277fracdiff, 212frac, 12, 16frontend, 89, 92fsolve, 59, 111–113, 190, 224, 287,

289, 344, 395, 423.., 287, 288, 346, 348, 349[ ], 392'', 345avoid, 288complex, 287, 288, 345

function, 9galois, 115gamma, 5, 11

gcd, 12, 74, 111genhomosol, 292get char, 179get compts, 179, 188get rank, 179gradplot3daxes, 203grid, 203

gradplotarrows, 203grid, 203

grad, 242grelgroup, 115groupmember, 115grouporder, 115hankel, 282, 284has, 312help, 6histogram, 62hornerform, 368hypergeom, 267iFFT, 278, 279identity, 22ifactor, 74, 127, 132if, 9, 15, 16, 18–20, 61, 100, 216,

233, 251, 257ilog2, 12implicitplot3d, 198implicitplot, 194, 251, 253, 289coloring, 252filledregions, 252

implies, 9indets, 92, 170index[function], 261index[package], 7, 261inequal, 59, 60optionsexcluded, 60optionsfeasible, 60optionsopen, 60

infinity, 11, 337inifcn, 261inifncs, 9

Maple Index 467

ininames, 9, 11interface, 11displayprecision, 82rtablesize, 128showassumed, 85, 212, 225, 247,

252, 270, 272, 275, 282intersect, 22, 84inter, 115intparts, 218, 222introduction, 5int, 12, 218, 233–235, 250, 268, 310AllSolutions, 223, 253CauchyPrincipalValue, 253continuous, 253

inverse, 228invert, 179, 186invfourier, 274, 276, 277invfunc, 104invlaplace, 270, 272, 335, 338invperm, 115is, real, 85isPrimitiveElement, 132isabelian, 115iscont, 210closed, 210, 211open, 210, 211

isnormal, 115isolate, 217, 254, 285, 286, 318,

320, 331, 336isprime, 74, 75, 172issubgroup, 115ithprime, 12, 74, 195kernelopts, 74laplace, 227, 269, 270, 272–274,

294, 318, 335, 339laurent, 258lcm, 12, 111lcoeff, 111ldegree, 111leftbox, 218leftsum, 218lhs, 9, 10, 292

liesol, 291limit, 12, 210, 212, 225, 229, 230,

251, 255, 256, 272, 275complex, 209, 258left, 209real, 209right, 209

lin com, 182line, 206listdensityplot, 148listplot3d, 189, 191listplot, 189, 191, 278, 279ln, 12local, 16log10, 12log[b], 12loglogplot, 60scaling, 60

logplot, 52, 60color, 60style, 60

lprint, 351macro, 88makeproc, 417map2, 103map, 22, 89, 103, 207, 233, 237, 243,

247, 260, 288, 337match, 89matrixDE, 294, 297matrixplot, 148, 189maxdigits, 74maximize, 212location, 217

maximum, 18max, 12, 231minimax, 368minimizelocation, 212, 217

minimum, 18minus, 22, 84min, 12modp, 19, 127, 172

468 Maple Index

mods, 127module, 18export, 18

mod, 16, 127, 172monodromy, 115msolve, 127mtaylor, 228, 351mulperms, 115multiply, 228names, 9nextprime, 74, 75next, 19nops, 10, 17, 21, 87, 190, 222, 237,

238, 289, 290, 298, 313, 331normalizer, 115normal, 17, 89, 111not, 9numbperm, 115numer, 74, 89, 216, 254odeadvisor, 292odeplot, 52, 293–295, 298, 405odetest, 274op, 10, 21, 87, 216, 222, 231, 237,

238, 242, 272, 277, 280, 286orbit, 115orthopoly[G], 263orthopoly[H], 263orthopoly[L], 263orthopoly[P], 263orthopoly[T], 263orthopoly[U], 263or, 9package, 7pade, 368partial diff, 186pdetest, 301pdsolveHINT, 301INTEGRATE, 301build, 301numeric, 302, 303, 405singsol, 301

permgroup, 115permute, 115phaseportrait, 295, 299, 300, 405piecewise, 15, 16, 61, 100, 210–

212, 269, 275, 325, 332plot3d[options], 52plot3d, 5, 51, 52, 193, 198, 260,

302, 303grid, 52, 57, 225, 236orientation, 199, 236shading, 236, 238, 243style, 52, 68

plot[options], 52, 59plot, 5, 16, 22, 51, 54, 302color, 55coords, 194discont, 52, 58, 208, 210, 233filled, 59gridlines, 58linestyle, 55, 214, 264numpoints, 15scaling, 52style, 22, 52, 66, 84, 229symbolsize, 22symbol, 22, 66, 84, 229thickness, 58tickmarks, 52title, 52

pointplot3d, 138, 148, 189, 191axes, 190symbolsize, 190symbol, 190

pointplot, 68, 138, 189, 191, 211,251, 252, 257, 278

symbol, 190, 215poisson, 228polarplot, 194, 251, 252filled, 244numpoints, 195

polar, 246, 252polynomials, 109prevprime, 74

Maple Index 469

primpart, 111primroot, 127, 132printf, 10, 19, 20, 70, 232, 342, 346print, 11, 12, 15, 135, 233, 250, 329procedure, 16proc, 9, 14–19, 61, 100, 215, 231,

233, 257, 310, 313args, 16global, 16local, 16operator, 207option remember, 18, 411options, 16

product, 286prod, 182proot, 111psqrt, 111quo, 12, 111radians, 11random, 60normald, 63

randperm, 115randpoly, 109rand, 13, 75realroot, 111removehas, 22, 84, 113, 170, 216, 310

rem, 12, 111reserved, 9residue, 255restart, 9, 10, 18resultant, 111return, 9reversion, 228rhs, 9, 249, 254, 277, 284, 293, 294,

297rightbox, 218rightsum, 218roots, 111root, 344rotate, 206round, 12

save, 9scale, 206sec, sech, 12selectSol, 112select, 75has, 21, 170, 216, 310, 339type, 231

semilogplot, 60separability, 301separablesol, 292seq, 20, 22, 60, 62, 67, 84, 191, 229,

237, 238, 278series, 12, 91, 228, 230, 231, 258,

264, 266setcolors, 208setoptions3d, 54, 238, 242setoptions, 54, 65, 211, 225, 244,

293shortcut, 6signum, 246sign, 12simplify, 12, 14, 89, 207HermiteH, 263LegendreP, 266log, 220trig, 92

simpson, 218sin, sinh, 12singular, 258solve, 5, 15, 91, 112, 113, 231, 237,

242, 247, 259, 264, 271, 273,285–287

sort, 5, 111, 290spacecurve, 192, 193, 238shading, 67style, 67

sparsematrixplot, 148sparse, 22sphereplotgrid, 205orientation, 205

sphere, 206

470 Maple Index

sprintf, 62sqrfree, 111statplots, 62subgrel, 115subsets, 70subsop, 21, 277, 331, 336subs, 14, 21, 215, 216, 231, 254,

271, 277, 284, 292sum, 71, 109, 210, 228, 229, 234,

255, 256, 267, 320, 329symmetric, 22table, 20tan, tanh, 12tanLine, 215taylor, 228, 368tcoeff, 111tensorsGR, 188testeq, 92, 95textplot3d, 57color, 57font, 57

textplot, 57, 62color, 57font, 57

time, 10, 12, 18, 83, 343translate, 206trapezoid, 218true, 11trunc, 12, 16, 208tubeplotnumpoints, 197radius, 197tubepoints, 197

type, 10, 12, 87, 179+, 310Vector, 135function, 222polynom, 111trigh, 92

unapply, 14, 15, 179, 185, 207, 230,249, 254, 270, 275, 280, 286,301, 320, 327

unassign, 9, 246, 297undefined, 212union, 22, 84, 233, 237, 238, 274,

277, 284, 312, 313, 331, 336units, 11unprotect, 104usage, 6use in, 132, 287, 344useHardwareFloats, 77, 79value, 218, 221, 240, 241, 254, 302,

303, 320, 327vectorProj, 157verboseproc, 11whattype, 10, 87, 135while not, 9, 70, 71with, 7worksheet, 6xor, 9

functionsspecialized, 7standard library of, 7

incorrect response, 10

letterlower/upper-case, 9

Maplebasic concepts, 5basic features, 4comments, 11data structure, 8help system, 6interface, 5kernel, 5language, 8large expression in, 89library, 5memory, 10palettes, 6source code, 11worksheet, 5, 6, 10

Maple Index 471

Maple Application Center, 4Maple Classic Worksheet, 5Maple Community, 4Maple object, 8, 20

array, 20list, 20, 69matrix, 20sequence, 20, 69set, 20, 69table, 20tensor, 179vector, 20

Maple Student Help Center, 4Maple Teacher Resource Center, 4Maplet

application, 6user interface, 6

menuFormat, 6FullTextSearch, 6Help, 6Insert, 6View, 6stop button, 10

module, 302

packageCurveFitting, 357, 373DETools, 405DEtools, 291, 292, 294, 295, 297,

299DifferentialGeometry, 179DiscreteTransforms, 278, 368Groebner, 109LinearAlgebra[Generic], 174LinearAlgebra[Modular], 172LinearAlgebra, 7, 10, 22, 128,

134–184, 235, 278, 297, 396Optimization, 212, 217, 363PDEtools, 272, 277, 284, 301–303Physics, 179–188PolynomialIdeals, 109

PolynomialTools, 109RandomTools, 13, 75, 109, 246RealDomain, 84, 92, 287, 344RegularChains, 109RootFinding, 287, 344SDMPolynom, 109ScientificConstants, 11Statistics, 13, 62, 368, 373Student, 7, 212, 218Calculus1, 197, 201, 210, 212,215, 218, 223, 383, 385LinearAlgebra, 134, 142, 212,396, 398MultivariateCalculus, 235–240, 383Precalculus, 212, 216

Units, 11VectorCalculus, 212, 235–244,

303, 435algcurves, 115codegen, 343, 355, 427combinat, 70, 71, 115–120difforms, 237geom3d, 206geometry, 206group, 115–129inttrans, 269–284, 294, 318–339linalg, 134, 242numapprox, 258, 363, 368numtheory, 127orthopoly, 109, 263, 363, 368plots, 22, 57, 68, 189–206, 215,

251, 257, 278, 295, 297, 299,301, 303, 316

plottools, 138, 206, 251, 252powerseries, 228stats, 60, 62student, 212, 218, 221–223, 240tensor, 179–188subpackage, 7

typesetting Standard Math, 6

Mathematica Index

( ), 35(* *), 35*, 26, 30**, 116, 125*=, 31++, 31, 42, 43+=, 31--, 31, 42-=, 31->, 32, 41, 180, 217, 231, 264, 271,

272, 278, 286, 294., 46, 110, 137, 156, 236, 243/., 31, 32, 38, 190, 213, 215–217,

231, 232, 264, 271, 278, 286,318

//, 29, 35, 83, 95, 112//., 32/;, 32, 39, 61, 211, 318, 337/=, 31/@, 30, 44, 47, 87, 91:=, 32:>, 32, 120, 124, 125, 198, 278;, 30, 52<, 31<<, 28<=, 31=, 32, 33=., 33==, 31, 33, 42===, 33, 34, 47, 87, 95, 249, 267,

270>, 31>=, 31

?, 26, 27, 33??, 27@, 38, 39, 105@@, 47, 91[ ], 35[[ ]], 33, 41, 44, 48, 136, 180, 215,

286, 292, 302#&, 35, 36, 39, 41, 44, 47, 180, 232,

271, 272, 274, 278, 301%, 35&&, 31, 43, 112, 230, 232, 241, 257,

258, 275, 286, 338^:=, 32^=, 32, 29, 31, 32, 37, 39, 208, 217?, 44', 213, 268``, 35{ }, 35, 43, 287, 299Abs, 31, 37, 39, 43, 81, 226, 230,

246, 253, 257, 258, 276, 281AccuracyGoal, 74Accuracy, 74, 79AddOns, 29AddTo, 31AlternatingGroup, 116And, 31Animate, 48, 65, 66, 68, 102, 303AnimationRate, 66, 68, 197

Animator, 48Apart, 91, 102, 210, 221, 222Append, 45, 66–68, 75, 146, 280, 300Apply, 47, 91, 104, 164, 207

Mathematica Index 473

Arg, 81, 246, 253ArrayDepth, 44, 46, 142, 180, 183ArrayFlatten, 149, 168ArrayQ, 142Array, 44, 46, 48, 136, 142, 180,

321, 327, 332, 333, 337Assuming, 85, 226, 227, 240, 241,

268, 270, 273, 275, 276, 301,326, 338

AtomQ, 44BarChart3D, 63AxesLabel, 64

BarChart, 63BarEdges, 64BarOrientation, 64BarStyle, 64ChartElementFunction, 64ChartLegends, 64

BesselI, 265BesselJZero, 265BesselJ, 265, 282–284BesselK, 265BesselYZero, 265BesselY, 265, 276Beta, 262Binomial, 71, 154Block, 40, 41Break, 42, 347BubbleChart, 64Cancel, 91, 332, 333Trig, 93

CartesianProduct, 72Cartesian, 243Cases, 44Catalan, 35Ceiling, 37CharacterRange, 44, 45CharacteristicPolynomial, 163Characteristic, 128, 132, 175Characters, 44ChebyshevT, 264ChebyshevU, 264

CholeskyDecomposition, 397Chop, 168, 279, 280Christoffel1, 187Christoffel2, 187Circle, 206ClearAll, 33Clear, 33CoefficientArrays, 159, 333, 405CoefficientList, 110, 112, 164Coefficient, 110, 112, 221, 233,

290, 322Collect, 112, 233, 278, 290Colordata, 52ColumnForm, 46, 136Column, 136CommutatorL, 184CommutatorRel, 184CommutatorR, 184Compile, 29, 83Complement, 84, 124CompleteSquare, 221ComplexExpand, 81, 93, 246, 247,

249, 251, 253, 256, 260, 279TargetFunctions, 246, 247, 253

ComplexInfinity, 246Complex, 29, 81, 87, 248Composition, 38, 104, 207ConjugateTranspose, 145, 168Conjugate, 36, 96, 137, 169, 246,

249, 259ConstantArray, 136, 140, 142, 146,

149, 168ContinuedFraction, 368Continue, 42ContourPlot3D, 200Contours, 199, 200

ContourPlot, 194, 200, 204, 252,253, 289

ContourLines, 201ContourShading, 201ContourStyle, 201Contours, 201

474 Mathematica Index

CosIntegral, 262CountRoots, 224, 286, 287CrossProduct, 137Cross, 137Cuboid, 206Curl, 236, 243CyclicGroup, 116Cyclotomic, 110Cylinder, 206DSolve, 249, 267, 271–274, 293, 294GeneratedParameters, 291, 292

DTFourierTransform, 281Decompose, 112Decrement, 31Degree, 35DeleteCases, 44, 156, 174, 217Delete, 44Denominator, 74, 91, 221, 255DensityPlot, 62, 252, 253ColorFunction, 62

Depth, 44, 45, 124Derivative, 213, 236–238Det, 145, 150, 159, 163, 180, 327,

333, 397DiagonalMatrix, 149, 168, 405DigitQ, 33, 87DihedralGroup, 116Dimensions, 44, 46, 47, 136, 142,

155, 156, 180DiracDelta, 268, 270, 282Discriminant, 112Disk, 68, 211, 244DivideBy, 31Divide, 30Div, 236, 242–244DotProduct, 137, 236Dot, 137, 145Do, 41–43, 48, 67, 68, 195, 216, 217,

296, 298, 300Drop, 44Dt, 213, 217, 221, 222, 236, 237Constants, 237

DynamicModule, 48Dynamic, 48D, 32, 41, 213, 236, 248, 262, 264,

267, 278, 301, 303, 318EconomizedRationalApproximation, 368

Eigensystem, 163, 166, 168Eigenvalues, 144, 163Eigenvectors, 163ElementData, 35ElementToPolynomial, 178Element, 85, 212, 230, 232, 268, 301Complexes, 246, 289Reals, 282–284, 286

Eliminate, 112, 239, 259EllipticE, 263EllipticF, 263EllipticK, 263EllipticPi, 263EngineeringForm, 29Equal, 31, 260Erfc, 263Erf, 263EuclideanDistance, 213EulerGamma, 26, 35EulerPhi, 128, 132Evaluate, 62, 275, 276, 299ExampleData, 206ExpIntegralEi, 276ExpIntegralE, 262ExpToTrig, 93, 247, 322, 323, 430ExpandAll, 91, 93, 216, 222, 230,

243, 278, 332Expand, 26, 32, 91, 112Modulus, 128Trig, 93

Exponent, 110, 112, 132Exp, 36E, 35FactorInteger, 74, 128, 132FactorList, 112FactorTerms, 91

Mathematica Index 475

Factorial2, 37, 42Factorial, 37Factor, 41, 251Extension, 91, 112, 128Modulus, 128

False, 33, 87Fibonacci, 37Fib, 39FieldIrreducible, 178FileNames, 29FindFit, 371, 373FindInstance, 287, 345FindMaximum, 364MaxIterations, 215WorkingPrecision, 215

FindMinimum, 213, 218MaxIterations, 215WorkingPrecision, 215

FindRoot, 224, 287, 289, 345DampingFactor, 289MaxIterations, 289WorkingPrecision, 289

First, 44, 132Fit, 373Flatten, 45, 159, 164, 190, 191,

217, 232, 259, 300Floor, 37FoldList, 124FortranForm, 76For, 41, 42, 233, 234, 280, 290FourierCosTransform, 276, 338FourierSinTransform, 276FourierTransform, 275, 278FourierParameters, 275, 276

FourierTrigSeries, 232Fourier, 278–280, 372FractionalPart, 37FreeQ, 85, 113, 174, 318, 337FresnelC, 263FresnelS, 263FromContinuedFraction, 368FromCycles, 116

FromElementCode, 178FullForm, 30, 79, 87NumberMarks, 76

FullSimplify, 282, 301, 302, 318FunctionExpand, 91, 93, 227, 262,

266–268, 283, 284FunctionInterpolation, 357Function, 37, 38, 68, 104, 207, 251GCD, 37, 74GF, 128, 132, 175Gamma, 262GegenbauerC, 264GeneralizedBarChart, 63GeometricTransformation, 206GoldenRatio, 35Goto, 42GradientFieldPlot3D, 203, 204GradientFieldPlot, 203BaseStyle, 204

Grad, 236, 242, 243Graphics3D, 191, 206Cuboid, 57Text, 57

GraphicsColumn, 55, 56, 61, 193GraphicsGrid, 55, 56, 64, 293GraphicsRow, 39, 55, 56, 68, 196,

201, 209, 225, 265, 268, 270,279, 280, 298

Background, 56Graphics, 62, 68, 206, 244Arrowheads, 138, 252, 253Arrow, 138, 252, 253Text, 57

GreaterEqual, 31Greater, 31GroebnerBasis, 112HamiltonianFieldPlot, 203HankelH1, 265HankelH2, 265HankelMatrix, 149HankelTransform, 282–284Head, 30, 87

476 Mathematica Index

HeavisideTheta, 268, 270, 282HermiteDecomposition, 166, 168HermiteH, 264HermitianMatrixQ, 169HilbertMatrix, 149Histogram3D, 63Histogram, 63HoldForm, 82, 210, 226, 241, 268,

392, 403, 405, 437Hold, 91HornerForm, 368Hypergeometric0F1, 267Hypergeometric1F1, 267Hypergeometric2F1, 267HypergeometricPFQ, 267HypergeometricU, 267IdentityMatrix, 149, 164, 398IdentityPermutation, 122If, 40–42, 217, 221, 257, 322Implies, 31Im, 81, 246, 251, 280Increment, 31Infinity, 35, 39, 338Inner, 137, 145, 156, 157, 183InputForm, 25, 30, 229, 247, 284NumberMarks, 76

Input, 142Insert, 45, 165IntegerPart, 37IntegerQ, 32, 33, 37, 87Integer, 29, 87, 146Integrate, 91, 196, 218, 236, 249,

250, 254, 255, 262, 267, 268,318, 327, 383, 387, 389, 392,395, 431, 433, 434, 437

Assumptions, 223, 282PrincipalValue, 223, 253

InterpolateRoot, 287, 345InterpolatingPolynomial, 357Interpolation, 357Intersection, 46, 84, 113InverseDTFourierTransform, 281

InverseFourierTransform, 275InverseFourier, 278–280InverseFunction, 105InverseLaplaceTransform, 269–271, 336, 340

InversePermutation, 122InverseSeries, 91, 229Inverse, 150, 152, 163, 180IrreduciblePolynomial, 128, 132IsolatingInterval, 286I, 29, 35, 245JacobiP, 264JacobianMatrix, 236, 355, 428Join, 42, 45, 146, 165, 183, 257JordanDecomposition, 166, 168KroneckerProduct, 183LCM, 37LUDecomposition, 165, 397Label, 42LaguerreL, 264LaplaceTransform, 269–274, 323Assumptions, 270GenerateConditions, 270

Laplacian, 236, 242, 243Last, 44, 165LeafCount, 44LeastSquares, 159, 373LegendreP, 36, 96, 264, 267, 394LegendreQ, 267Length, 33, 41, 44, 46, 47, 70, 136,

155, 180, 190, 217, 237, 248,299, 332, 333, 337

LessEqual, 31Less, 31Level, 45LeviCivitaT, 184LieProduct, 184Limit, 91, 210, 226, 230, 251, 255,

256, 259, 262Analytic, 209, 349Assumptions, 209, 212, 272, 276Direction, 209, 212

Mathematica Index 477

LinearSolve, 159, 165, 397, 398Modulus, 175

Line, 198ListAnimate, 66–68, 300AnimationRate, 68, 195

ListContourPlot3D, 200Contours, 201DataRange, 201

ListContourPlot, 200DataRange, 201

ListDensityPlot, 62, 258ColorFunction, 257Mesh, 257PlotRange, 62

ListInterpolation, 357ListLinePlot, 56, 138, 189, 191ListLogLinearPlot, 60ListLogLogPlot, 60ListLogPlot, 60, 61ListPlot3D, 148, 191MeshShading, 189

ListPlot, 48, 56, 138, 251–253, 281FillingStyle, 189, 191Filling, 191ImageSize, 279Joined, 189, 191, 279, 280MaxPlotPoints, 189PlotMarkers, 66, 67, 189PlotRange, 230PlotStyle, 190, 229, 230

ListPointPlot3D, 68, 138, 189PointSize, 68

ListPolarPlot, 66, 252, 253ListVectorFieldPlot3D, 203ListVectorFieldPlot, 203, 204List, 43, 69LogIntegral, 262LogLinearPlot, 60LogLogPlot, 60, 61LogPlot, 60LogicalExpand, 31, 34, 87Log, 31, 36

MachineNumberQ, 79Manipulate, 48Manipulator, 48MapAll, 104, 207MapThread, 156Map, 41, 47, 91, 104, 207, 210, 267,

271, 272, 274, 278, 340MathieuCPrime, 266MathieuCharacteristicA, 57, 266MathieuCharacteristicB, 57, 266MathieuCharacteristicExponent, 266

MathieuC, 266MathieuS, 266MatrixExp, 152MatrixForm, 46, 136, 142, 180, 327MatrixPlot, 148MatrixPower, 145, 152, 164MatrixQ, 142MatrixRank, 150, 156, 159Maximize, 213, 286Max, 37, 232MemberQ, 122, 124, 132MiniMaxApproximation, 368Minimize, 213, 286Minors, 150, 164Minus, 30Min, 37Modular, 172Module, 40–42, 207, 216, 232, 258,

282, 284, 338, 359, 362, 365,367, 376, 385, 391, 424

Modulus, 172, 174Mod, 36, 42, 128, 132, 172, 175MoebiusStrip, 206MultiplicationTable, 116, 120MultiplicativeOrder, 128, 132NDSolve, 293, 298, 300, 303, 406,

409, 424, 428, 433MaxSteps, 298Method, 295, 406

NDTFourierTransform, 281

478 Mathematica Index

ND, 379Method, 379Terms, 379

NFourierTransform, 281NIntegrate, 218, 236, 253Method, 383, 386WorkingPrecision, 223

NInverseDTFourierTransform, 281

NInverseFourierTransform, 281NMaximize, 213Method, 218

NMinimize, 213, 218NRoots, 112, 287, 345NSolve, 60, 190, 287, 345Names, 28, 29, 174Needs, 29NestList, 38, 39, 104, 207Nest, 38, 39, 104, 183, 207, 224NextPrime, 74NonCommutativeMultiply, 116NormalDistribution, 61Normalize, 156Normal, 91, 159, 229, 231, 262, 333Norm, 145, 156Not, 31NullSpace, 151, 155, 156, 159NumberForm, 29, 62, 83NumberQ, 211, 212Numerator, 74, 91, 216, 255NumericQ, 37N, 26, 29, 38, 42, 43, 46, 47, 62, 196,

218, 234, 248, 257, 258, 287Off, 33, 66, 293Options[FindRoot], 288, 345Options, 27, 52, 174, 200OrbitRepresentatives, 116Orbits, 116Orthogonalize, 156Or, 31Outer, 137, 145, 183PDFNormalDistribution, 64

PaddedForm, 47, 232, 237, 239, 303,343, 347, 417, 426, 440

PadeApproximant, 368Pane, 48ParametricPlot3D, 68, 193, 260BoxRatios, 192ColorFunction, 68PlotPoints, 192Tube, 198

ParametricPlot, 192, 263, 294AspectRatio, 195, 260ColorFunction, 251FrameTicks, 195PlotStyle, 274

Partition, 45, 178, 187, 257Part, 33, 44, 286PercentileBarChart, 63BarLabels, 64

PermutationGroupQ, 116PermutationWithCycle, 116Permutations, 116Permute, 116, 120PieChart, 63ChartStyle, 64

PiecewiseExpand, 268, 272, 326Piecewise, 39, 61, 101, 211, 251,

270, 276, 326, 333Pi, 29, 35Plot3D, 25, 68, 193, 201, 243, 303Axes, 53BaseStyle, 57BoxRatios, 53Boxed, 51, 53FaceForm, 53FaceGrids, 53MeshShading, 53MeshStyle, 53Mesh, 53PlotPoints, 53, 236PlotRange, 53PlotStyle, 53ViewPoint, 53

Mathematica Index 479

Plot, 26, 48, 51, 252AspectRatio, 53AxesLabel, 53AxesOrigin, 53Axes, 53BaseStyle, 57Epilog, 211Evaluate, 57Exclusions, 58, 209, 211, 268Filling, 59, 60, 224FrameTicks, 53Frame, 53GridLines, 53, 58ImageSize, 39PlotLabel, 53PlotLegend, 53PlotRange, 53, 197PlotStyle, 53, 197Ticks, 53

Plus, 30, 47, 137Point, 191PolarPlot, 66, 194, 252, 253FrameTicks, 196ImageSize, 196MaxRecursion, 66PlotPoints, 196PlotStyle, 196, 244

PolyGamma, 262PolyLog, 262PolynomialGCD, 112, 164PolynomialLCM, 112PolynomialMod, 112PolynomialQuotient, 112Modulus, 128Remainder, 112

PolynomialQ, 37, 112PolynomialRemainder, 112Position, 362PowerExpand, 91, 93PowerMod, 128, 132Power, 30, 32PreDecrement, 31

PreIncrement, 31Precision, 74, 79Prepend, 45PrimeQ, 74, 75, 174Prime, 37, 66, 74, 195Print, 37, 42, 43, 195, 233, 251, 337ProductLog, 264Product, 286Projection, 156Protect, 32PseudoInverse, 159QRDecomposition, 165, 397Quotient, 37RandomComplex, 37, 246RandomInteger, 37, 110RandomMatrix, 149RandomPrime, 74, 75RandomReal, 37NormalDistribution, 64

RandomSample, 140Random, 37, 45, 61, 62, 64, 140, 146,

191, 201Range, 44, 75, 110, 136RationalInterpolation, 357, 368Rationalize, 368Rational, 74, 87Reals, 260Real, 29, 87Rectangle, 64Reduce, 85, 112, 230, 260, 264, 286,

289, 350, 352Refine, 91ReflectionMatrix, 206RegionPlot3D, 59RegionPlot, 59, 60, 252, 253BoundaryStyle, 60, 244

ReleaseHold, 86Remove, 33ReplaceAll, 32, 88ReplacePart, 45ReplaceRepeated, 32Replace, 88

480 Mathematica Index

Residue, 253, 255Resolve, 112Rest, 44Resultant, 112Return, 40Reverse, 45RevolutionPlot3D, 202, 204BoxRatios, 202ViewVertical, 202

Re, 81, 246, 251, 266, 280RiemannSumLeft, 385RiemannSumLower, 385RiemannSumL, 219RiemannSumRight, 385RiemannSumUpper, 385RombergInt, 391RootIntervals, 224, 286, 287, 345Complexes, 286

RootReduce, 112Roots, 112, 286Modulus, 128

Root, 35, 81, 82, 112, 286RotateLeft, 45RotateRight, 45Rotate, 206Round, 37, 79RowReduce, 156, 159, 165, 397Modulus, 175

Row, 48, 110SameQ, 33, 87, 124SameTest, 124Scale, 206ScalingMatrix, 206Scan, 104, 174SchurDecomposition, 166, 168ScientificForm, 29, 35, 76, 83SectorChart, 64SeedRandom, 75Select, 44, 75, 113, 174, 232, 385Sequence, 69SeriesCoefficient, 229SeriesData, 229

Series, 91, 229, 232, 258, 262, 264,350, 352, 381, 382, 389, 390

Set2DJulia, 257Set3DJulia, 257Set3DMandelbrot, 258SetAccuracy, 74SetCoordinates, 236, 242–244SetDelayed, 32SetDirectory, 29SetFieldFormat, 175SetOptions, 39, 54, 65, 174, 190–

197, 200, 201, 206, 209, 265,266, 268, 279, 293, 298

Modulus, 172SetPrecision, 74, 79Set, 32Short, 91, 134Show, 56, 57, 61, 62, 68, 189–191,

244, 296–300SphericalRegion, 68ViewCenter, 68ViewPoint, 68

Signature, 180, 184Sign, 37, 246Simplify, 38, 112SinIntegral, 262SingularValueDecomposition, 166, 168

SlideView, 48Slider, 48Solve, 26, 32, 39, 112, 217, 237,

242, 249, 255, 258, 271, 273,274, 286, 289, 318

Modulus, 128VerifySolutions, 286

Sort, 45, 110, 112SparseArray, 140, 149, 165, 180Band, 149

Sphere, 206SphericalPlot3D, 205SplineFit, 357, 366Sqrt, 29

Mathematica Index 481

StackedBarChart, 63BarStyle, 64

StringDrop, 34StringForm, 35StringInsert, 34StringJoin, 29, 34, 45StringLength, 34, 44StringPosition, 34StringQ, 44StringReplace, 34StringReverse, 34StringTake, 34, 174String, 87StyleForm, 62Subscript, 110, 181, 403, 405, 428Subsets, 70Subset, 46Subsuperscript, 181SubtractFrom, 31Subtract, 30Sum, 41, 42, 47, 71, 110, 210, 219,

224, 229, 230, 234, 256, 267,280, 327

Superscript, 181Switch, 41, 42Symbol, 87, 174SymmetricGroup, 116, 120TabView, 48TableForm, 46, 47, 237, 239, 293,

298, 303, 409, 433, 440TableAlignments, 47TableHeadings, 47, 125, 232TableSpacing, 47, 187

Table, 29, 39, 44, 45, 47, 48, 57, 61,62, 64, 68, 83, 110, 136, 140,142, 146, 200, 224, 229, 230,232, 233, 237, 248, 251

Take, 44, 66–68, 136, 142, 146, 159,165, 280

TensorQ, 180TensorRank, 44, 47Text, 62

Thread, 47, 91, 156, 159, 260, 350,352, 353, 392, 405, 414, 428,430, 431

Through, 251, 260TimesBy, 31Times, 30, 137Timing, 29, 37, 39, 83, 281, 343ToCycles, 116ToExpression, 174ToFileName, 29ToRadicals, 112Together, 91, 221, 437Trig, 93

Torus, 206TraditionalForm, 229, 254, 270Translate, 206Transpose, 145, 156, 163, 183TreeForm, 30, 45, 47, 87, 180TrigExpand, 93, 95, 221TrigFactor, 93TrigReduce, 32, 93, 221, 233, 243TrigToExp, 93, 212, 223TrueQ, 221True, 32, 33, 87Tr, 145, 163, 183Tube, 198Unequal, 31Union, 46, 84, 113, 124, 156, 184,

233, 237, 278, 337UnitStep, 39, 61, 101, 211, 234UnitVector, 140Unprotect, 32Variables, 112, 428VectorFieldPlot3D, 203VectorHeads, 204

VectorFieldPlot, 203, 295BaseStyle, 204ColorFunction, 296, 299Frame, 300PlotRange, 297

VectorPlot3D, 295PlotPoints, 204

482 Mathematica Index

VectorPlot, 204, 295, 406VectorColorFunction, 204

VectorQ, 37, 46, 136Which, 41, 42, 101, 211, 212While, 41–43, 79, 257, 258With, 40, 41WorkingPrecision, 74WorldPlot, 28Xor, 31Zeta, 267$ContextPath, 28$Context, 28$HistoryLength, 375, 411$InstallationDirectory, 29$MachineEpsilon, 79$MachinePrecision, 74$MaxMachineNumber, 74, 78$MaxNumber, 74, 78$MinMachineNumber, 74, 78$MinNumber, 78$Packages, 28$RecursionLimit, 39, 411adjointM, 151angle, 157blockJordan, 168centerGroupQ, 124coRank, 151cofactor, 151columnSpace, 156conjSubgr, 124constantVector, 140contract, 183convertIn, 178convertOut, 178doubleFactorial, 42ecLagrange, 41elimGauss, 165, 397ffMat, 175ffVec, 175functionsModp, 174genSubgrs, 124intsolve, 319

irreduciblePolynomialQ, 132mIJ, 151minimalPolynomial, 164mult, 124normalQ, 124pMaclaurin, 232primitiveRootQ, 132primitiveRoot, 132rank, 151regPertPoly, 290rowSpace, 156scalarVector, 140secL, 197subGroupQ, 124tanLine, 216tanL, 197unionSubgrs, 124unitVector, 140unitaryQ, 169vConj, 250zeroVector, 140

Mathematicaanimation frame, 48atomic objects, 87basic concepts, 26basic features, 24basic principles, 30cells, 27chemical elements in, 35comments, 35data structure, 30design, 25Front End, 25, 27help system, 27history, 24initialization, 33interface, 25kernel, 25language, 30Notebook, 25, 27obsolescence messages, 66

Mathematica Index 483

Mathematica objects, 30, 43, 48dynamic, 48list, 69

mathematicalconstants, 35typesetting system, 24

MathLink protocol, 24, 25menuEvaluation, 35Format, 27BackgroundColor, 27Style, 27

Help, 27Insert, 46, 52Palettes, 46, 52BasicMathInput, 27ColorSchemes, 52

package, 28Combinatorica, 46, 84, 116–130FiniteFields, 29, 128, 132, 175FourierSeries, 232–234, 281FunctionApproximations, 287, 368

NumericalCalculus, 379

PhysicalConstants, 35PlotLegends, 53Splines, 357Units, 35VectorAnalysis, 137, 236–244WorldPlot, 28

packagesExtra Packages, 29Legacy Packages, 29, 66contexts in, 28, 35loaded, 28names of, 29names of functions in, 28, 29short names in, 28

transformation rules, 32

Wolfram Research, 24, 66, 206Demonstrations Project, 24, 27Documentation Center, 27Information Center, 24Learning Center, 24Mathematica Virtual Book, 27

Wolfram, Stephen, 24

CD-ROM Contents

The present CD-ROM (packaged with this edition) contains all Mapleand Mathematica solutions (i.e. segments of code) of the problems andexamples as a set of the following PDF files:

ChapterN.pdf — all solutions and examples described in the individualChapterN of the book, where N = 1, 2, ..., 12.

Chapter1-12.pdf — a single file with the whole set of solutions andexamples described in the book.