• References

o Some Books about Mathematica

[AB92/1] Abell, M. and J. Braselton: Mathematica by Example. Academic Press, London, 1992.

[AB92/2] Abell, M. and 1. Braselton: The Mathematica Handbook. Academic Press, London, 1992.

[AB93] Abell, M. and 1. Braselton: Differential Equations with Mathematica. Academic Press, New York, 1993.

[Bla91] Blachman, N.: Mathematica: A Practical Approach. Prentice-Hall, Englewood Cliffs, 1991.

[Bla92] Blachman, N.: Mathematica Quick Reference, Version 2. Addison­Wesley, Redwood City, 1992.

[BKM92] Boyland, P., J. Keiper, E. Martin et al.: Guide to Standard Mathe­matica Packages. Mathematica Technical Report, Wolfram Research, Champaign, 1992.

[BPU91] Brown, D., H. Porta and J. Uhl: Calculus and Mathematica. Addison­Wesley, Redwood City, 1991.

[BD92] Burbulla, D. C. M. and C. T. J. Dodson: Self-Tutor for Computer Cal­culus Using Mathematica. Prentice-Hall, Englewood Cliffs, 1992.

[Cam90] Cameron, S.: The Mathematica Graphics Guidebook. Addison-Wes­ley, Redwood City, 1990.

[Cra91] Crandall, R. E.: Mathematicafor the Sciences. Addison-Wesley, Red­wood City, 1991.

[CR91] Crooke, P. and J. Ratcliffe: Guidebook to Calculus with Mathematica. Wadsworth, London, 1991.

[EL91] Ellis, W. Jr. and E. Lodi: A Tutorial Introduction to Mathematica. Brooks/Cole, Pacific Grove, 1991.

[FL92] Finch, J. and M. Lehman: Exploring Calculus with Mathematica. Addison-Wesley, Redwood City, 1992.

[Fre94] Freeman, J. A.: Simulating Neural Networks with Mathematica. Add­ison-Wesley, Reading, 1994.

396

[GG91] Gray, T. W. and J. Glynn: Exploring Mathematics with Mathematica. Addison-Wesley, Redwood City, 1991.

[GG92] Gray, T. and 1. Glynn: The Beginners Guide to Mathematica 2. Addi­son-Wesley, Redwood City, 1992.

[GKW93] Gaylord, R. 1., S. N. Kamin and P. R. Wellin: Introduction to Pro­gramming with Mathematica. Telos/Springer, New York, 1993.

[Mae91] Maeder, R. E.: Programming in Mathematica, Second Edition. Addi­son-Wesley, Redwood City, 1991.

[Mae94] Maeder, R. E.: The Mathematica Programmer. AP Professional, Bos­ton, 1994.

[ST94] Shaw, W. T. and J. Tigg: Applied Mathematica: Getting Started, Get­ting it Done. Addison-Wesley, Reading, 1994.

[Ske93] Skeel, R. and J. Keiper: Elementary Numerical Computing with Mathematica. MacGraw-Hill, New York, 1993.

[Ski90] Skiena, S.: Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Addison-Wesley, Redwood City, 1990.

[Var91] Vardi, I.: Computational Recreations in Mathematica. Addison-Wes­ley, Redwood City, 1991.

[Vari93] Varian, H. et al.: Economic and Financial Modeling with Mathe­matica. Springer, New York, 1993.

[Vve92] Vvedensky, D.: Partial Differential Equations with Mathematica. Addison-Wesley, Redwood City, 1992.

[Wag91] Wagon, S.: Mathematica in Action. Freeman, New York, 1991.

[WoI88] Wolfram, S.: Mathematica,A System for Doing Mathematics by Com­puter. Addison-Wesley, Redwood City, 1988.

[WoI91] Wolfram, S.: Mathematica, A System for Doing Mathematics by Com­puter, Second Edition. Addison-Wesley, Redwood City, 1991.

[WoI92] Wolfram, S.: Mathematica Reference Guide. Addison-Wesley, Red­wood City, 1992.

o Further References

[AS65] Abramowitz, M. and I. A. Stegun: Handbook of Mathematical Func­tions. Dover, New York, 1965.

References 397

[Akr89] Akritas, A. G.: Elements of Computer Algebra with Applications. Wiley, New York, 1989.

[BB79] Blakley, G. R. and 1. Borosh: Rivest-Shamir-Adleman public key cryptosystems do not always conceal messages. Computers and Mathematics with Applications 5,169-178,1979.

[Bri74] Brigham, E. 0.: The Fast Fourier Transform. Prentice-Hall, Engle­wood Cliffs, 1974.

[DST88] Davenport, J. H., Y. Siret and E. Tournier: Computer Algebra, Sys­tems and Algorithms for Algebraic Computation. Academic Press, London, 1988.

[EK94] Estrada, R. and R. P. Kanwal: Asymptotic Analysis: A Distributional Approach. Birkhauser, Boston, 1994.

[Flii62] Fliigge, W.: Handbook of Engineering Mechanics. McGraw-Hill, New York, 1962.

[Her78] Herlestam, T.: Critical remarks on some public-key cryptosystems. BIT 18,493-496, 1978.

[KS78] Kirchgraber, D. und E. Stiefel: Methoden der analytischen Storungs­rechnung und ihre Anwendungen. Teubner, Stuttgart, 1978.

[Mau89] Maurer, D.: Fast generation of secure RSA-moduli with almost maxi­mal diversity. Advances in Cryptology, EUROCRYPT '89, Springer, 1989, pp. 636-647.

[Nay73] Nayfeh, A.: Perturbation Methods. Wiley, New York, 1973.

[Nay81] Nayfeh, A. H.: Introduction to Perturbation Techniques. Wiley, New York, 1981.

[RA87] Rand, R. H. and D. Armbruster: Perturbation Methods, Bifurcation Theory and Computer Algebra. Springer, New York, 1987.

[Ris68] Risch, R.: On the Integration of Elementary Functions which are built up using Algebraic Operations. Report SP-2801l002100, Sys. Dev. Corp., Santa Monica, 1968.

[RSA 78] Rivest, R. L., A. Shamir and L. Adleman: A method for obtaining dig­ital signatures and public key cryptosystems. Communications of the ACM 21, 120-126, 1978.

[WB79] Williams, H. C. and B. Schmid: Some remarks concerning the M.LT. public-key cryptosystem. BIT 19,525-538, 1979.

• Index

Built-in Mathematica objects are printed bold (e.g., Abs). Functions, which are self-programmed or read from packages, are printed plain (e.g., S1). Italic page numbers point to the documentations of the built-in functions in the summary tables.

Symbols

!,24 !!,32 ! =,60,64 ?,16 ??,16 @,322 @, 15-16,23,322 @@, 146-147 #, 144,147 %,19,23 %%,19,23 %n,23 ",18,24 "=,262 &, 144,147 &&,64 -,23 ",322 ",309-310 ',298 ., 160,163 · .,246,248 · .. ,246,248 .m,26 · ma, 26 :,243,247 : :,327 : : usage, 302,307 :=,68,78,255,261,274,296 : >,253-254,274 ;,64,183 -,17,23 --,288 -=,288

->,46,50,52,253-254 +,17,23 ++,288 +=,288 =,42,78,255,261,274 = .,52 =! =,240-241 ==,54,60,64 ===,240-241 *,322 *,15-17,24,163,322 *=,288 /, 17,24,240,287 /@, 144,147 / .,50,52,251,254 /:, 196,198,262 /;,155,276-277 /=,288 / /,23 / /@, 147 / / . , 252, 254 Itmp,118 <,64 >,64 <=,64 >=,64 «,11,26,31,57,97,306,392 <>,120,323 »,57,390,392 >>>,392 ( ... ),19,23 (* ... *),305,307,309-310 [ ... ], 19,23 [ [ ... ] ], 50,52, 149,156 { ... }, 23,50,52, 138

400

1,244,247 I I, 60, 64 _,233,236 _.,245,247 __ ,245,247 ___ ,245,248 _ Complex, 375 _Integer, 375 _Real, 375 \ 322 \,321-322 \@,322 \ "A, 322 \",322 \ .nn, 322 \:nnnn,322 \ *,322 \ \,322 \b,322 \n,322 \nnn,322 \r,322 \t,322 00,21,25

Global Variables

$Context,297-299,303,307 $ContextPath, 299, 303,307 $DefaultFont, 83, Ill-1l2 $DisplayFunction, 83,104-105,111-

ll2,203 $ModuleNumber, 293 $Packages, 303, 307 $Path,31,221 $Recurs ionLimi t, 334--335 $SessionID,293

Numerics

3D ViewPoint Selector, 110

A

abort,12 Abort Calculation, 12 Abs,24,195,197,213,348,354

AbsArg' , 354-355 AbsArg. m, 354 Absolute

context names, 298 errors, 377-378 time, 224 values, 24, 197,353

AbsoluteDashing,177 AbsolutePointSize,177 AbsoluteThickness,177 AbsoluteTime,224 Accounting notation, 314 AccountingForm,313-314 Accuracy, 377-378

in built-in functions, 378 in numerical functions, 381

Accuracy, 378 AccuracyGoal,381 Action, 12, 14, 110 Adding

attributes, 265 elements to lists, ISO, 156, 288 numbers, 17, 23

AiryAi, 194,213 AiryAiPrime,213 AiryBi,213 AiryBiPrime,213 Algebra'

ReIrn' , 196, 198,353 SymbolicSurn' , 199,201

Algebraic equations, 54, 61 rules, corresponding to equations, 65

AlgebraicRules,65 Algebra, linear, 158,216--217 Algorithms

iterative, 334 Newton's, see also: Newtons's method, 282 numerical, precision and accuracy, 379 Risch, 70, 250

Aliasing, in plots, 84 Alignment, in tables, 142 Alternatives, in patterns, 244, 247 AmbientLight, llO Analysis, statistical, 205-206 Analytic, 79 And, 64

Index

Angle, bending, 84, 88 Animate Selected Graphics, 117, 182 Animation', 117, 182 animationGraphics, 189 Animations, 116 Apart, 49, 53 Append, 150, 155-156 Appending

elements to lists, 156,288 expressions to files, 392

AppendTo, 288 Apply, 146--147 Applying

functions to parts of expressions, 232 functions to the elements of lists, 147 transformation rules, 254

Approximations numerical, 378 of data by functions, 202, 205 of numbers, 18,24 of products, 200 of real numbers by rational numbers, 18, 24 of sums, 200

Arbitrary-precision numbers, 376 ArcCos, 24,213 ArcCosh, 24, 213 ArcCot, 24, 213 ArcCoth, 24,213 ArcCsc, 24, 213 ArcCsch, 24, 213 Arcs, 176 ArcSec, 24, 213 ArcSech, 24, 213 ArcSin, 24, 213 ArcSinh, 24, 213 ArcTan, 24, 213 ArcTanh, 24, 213 Arg, 195,197,213,348,354 Arguments, 353

complex, 197 keeping unevaluated, 266 of Mathematica functions, 20, 23 sequences of, 247 unevaluated form, 276 using elements of lists as, 147 variable number of, 245, 247

Arithmetic interval, 378 sequences, 138, 142

ArithmeticGeometricMean, 213 Array, 140,142 Arrays, 176

of graphics, 102, 105, 176 ASCII, 321, 323, 331, 357 AspectRatio, 82, 110, 112, 133,135 Assignments, 44, 256 Associated

definitions, 198,260,262,341 messages, 327

Associative functions, 264, 266 Asterisk, 17 AtomQ, 239,242 Attributes, 263, 265

locking, 266 Needs is called, 266 pure functions with, 265-266

Attributes, 263-265 Attributes, 305 Automatic, 83 Axes, 82, 110, 112,129 AxesEdge, 110,120 AxesLabel, 82,88, 110, 112 AxesOrigin, 82, 112 AxesStyle, 82, 110, 112 Axes, labels for, 84, 88

B

Background, 82,110,112 Background jobs, 389 Backspace, 322 Bar charts, 192 BarChart, 193 Bars, error, 192 Base of numbers, printing, 314 BaseForm,313-314 Batch jobs, 391 Beep, 14 Begin, 302-304,307 BeginPackage, 302-306 Bending angle, 84, 88 BernoulliB,213

401

402

Bessel functions, 16,88 derivatives of, 67

Bessell, 88,213 BesselJ, 88, 213 BesselK, 89, 213 BesselY, 89, 213 Beta, 213 Bifurcation, 117 Binary, saving Mathematica, 221-222 Binomial,213 Blanks, 68,233, 236

three, 245 two, 245

Block, 295-297 Blue, 177 Blue, 181, 183 Boundary layers, 366 Bounding box, 120, 129

ratio of edges, 120 Boxed, 110, 129 BoxRatios, 110,120 BoxStyle,110 Braces, 23, 50 Brackets, 23

double, 50 Break,289 Brightness, 112, 120 Browser, 14 Bugs, 58 Built-in functions, changing, 260 Byte, 136 ByteCount, 223, 225 Bytes

c

in ReadList, 136 maximum number, 225

C,51,268,311,368 format for, 312

C,74 Calculations

numerical, 17, 368 symbolic, 44

Calculus, 66 Calculus'

DSol ve ' , 76, 79

Ellipticlntegrate',79 FourierTransform', 209, 212 LaplaceTransform', 211-212, 337 Master', 190 PDSolvel ',77,79

Cancel, 48, 53 Cancelling, 53 Capital letters, 45 Carriage return, 322 Cases, 235,237 Catalan, 213 Catch,289 Ceiling, 213 Cells, 9

containing several lines, 9 in graphics arrays, 176 initialization, 349

CForm, 311-312 CGS, 28 Changing built-in functions, 260 Character, 136 Characters

control, 322 from strings, 323 in ReadList, 136 lists of, 323 meta, 322 sequence of, in ReadList, 136 space, 17 special,321-322 wildcard, 15 zero or more, 16

Characters, 321,323 Charts

bar, 192 pie, 192

ChebyshevT, 213 Chebyshevu,213 Check, 325-327 checkForError, 325 Checking for messages, 327 Chop, 378 Cipher text, 33 Circle, 173,176 Circles, 176 Clear, 44, 52, 262, 265,350 ClearAll,265

Index

ClearAttributes, 264-265 Clearing

attributes, 265 definitions, 45, 52, 262

ClebschGordan, 213 ClipFill, 110 Close, 388-389 Closing streams, 389 Codes, ASCII, 321, 323, 357 Coefficient, 215 CoefficientList, 214-215 Coefficients, in polynomials, 214-215 Collect, 46, 53 Collecting powers, 53 ColorFunction, 110, 112,120 ColorOutput, 82,110,112 Colors

brightness, 112, 120 definitions for, 183 functions for, 112, 120 RGB, 120, 177 saturation, 112, 120

Colors', 179, 183, 186 ColumnForm, 312,315 Column, formatting as, 312 Combining

lists, 105, 151, 156 parts of the output, 309 strings, 120

Commands, running external, 392 Comments, 307, 309-310 Common denominators, 49, 53 Communication, 391 Commutative functions, 264, 266 comp, 371 Compilation, 368

of numerical functions, 381 Compile, 369-371,375 Compiled, 82,110,112,381 Compiled functions, 375 CompiledFunction, 369,375 Compiler, 368, 375 Complement, 151,156 Complements of lists, 156 Complete Selection, 14 Complex

arguments, 197

conjugates, 197 expressions, expanding, 52 numbers, 195, 197 numbers, heads of, 241 numbers, in compilation, 370 numbers, patterns for, 243, 247 symbols, 195, 197-198

403

Complex, 238, 241, 247 ComplexExpand, 49, 52, 195,197,208 Complicated patterns, 242 ComposeList, 286, 288 Composing

functions, 286, 288 polynomials, 216

CompoundExpression, 183 Computational geometry, 137 Concatenating

lists, 151, 156 strings, 120

Conditionals, 277, 287 with local variables, 292, 296

Conditions, 155,237 in patterns, 240 initial, 77, 332

Configuration custom, 219 space, 123

Conjugate, 195,197-198,213 Conjugates, complex, 197 Constant, 266 Constants

Euler's, 21, 335 local,292 mathematical, 213 of integration, 70, 74, 78

ConstrainedMax, 217 ConstrainedMin,217 Constraint-propagation, 337 Context, 298-300,306 Contexts, 26, 297

absolute, 298 current, 307 currently open, 299 in packages, 303, 306 list of, 32 list of all, 306 mechanism, 303

404

of symbols, 306 path, 307 relative, 298 removing names from, 306 sub, 302

Contexts, 29,32,299,306 Continue, 289 cont i nue, 12 Continuous Fourier transforms, 209, 212 Contour graphics object, 175 Contour plots, 111, 119-120

of data, 135 ContourGraphics,175 ContourLines, 112 ContourPlot, 111-112,119 ContourPlot3D, 113 Contours, 112-113,120 ContourShading, 112 ContourSmoothing, 112-113,120 ContourStyle, 112 Control

by messages, 327 characters, 322 over flow, 289 structures, 279, 288-289 structures, evaluation of, 275 variables, 288

Conventions, 2 Convergence

in numerical products, 383 in numerical sums, 383

Convert, 27-28 Convert Clipboard, 87 Convert to, 87 Converting

dates, 224 expressions to input form, 175 expressions to strings, 120,323 graphics, 116, 119 real into rational numbers, 18, 24 strings to expressions, 323 time, 224 units, 32

convertString,357 ConvertTernperature,28 Coordinates

for the triple pendulum, 90

scaled, 177 strained, 365

Coprocessor, 369 Copy, 11 Copy Output from Above, 310 CopyDirectory,219 CopyFile,219 Copying

directories, 219 files, 219

Cos, 24, 213 Cosh, 24,213 Cos Integral, 213 Cot, 24,213 Coth, 24, 213 Count, 235,237 Counting patterns, 237 Courier font, 90 CPU time, 42, 224 CreateDirectory,218 Creating

directories, 218 lists, 138

Cross, 152, 154 cross, 238 Cross product, 152 CrossProduct',152 Cryptosystem

public key, 33 RSA,34

csc, 24,213 Csch, 24,213 Cubes, 176 Cuboid, 173, 176 Current

context, 307 directory, 218

Curves in space, 122, 127, 129 Custom

formats, 318 messages, 327 output formats, 320

Cyclic permutations, 152, 157 Cyclotomic, 216 Cyclotomic polynomial, 216

Index

D

D,66-70,78 Damping factors, 381 DampingFactor, 381 Dashing, 177 Dashing, 174,177,206 Data, 203

contour plots of, 135 density plots of, 135 discrete, 130 discrete, plots of, 135 external, 130 fits, 202, 205 nonlinear fits, 205 of type, reading from file, 136 reading external, 131, 136 surface plots of, 135 three-dimensional, general, plots of, 137 types, 340

Data Encryption Standard, 33 Database, key, 35 Date, 224 Dates, 224

converting, 224 modification, of files, 219

Debugging, 342, 344, 347 Decimal points, 18 DeclarePackage, 220, 301,306 Decoding with RSA, 36 Decompose, 215-216 Decompositions

Jordan, 217 LV, 217 partial fraction, 49,53 QR,217 Schur,217

Decrement, 288 DefaultColor, 82, 110, 112 DefaultFont, 83,111-112 Defaults for optional arguments, 247 Definite integrals, 71, 78 Definition, 390,392 Definitions, 44, 52, 255

and ... Form commands, 140 associated, 198,260,262,341 clearing, 45, 52,262

delayed, 69, 78, 255, 261 evaluation of, 269, 274 for symbols, 392 immediate, 69, 78, 255, 261 of attributes, 265 of functions, 68, 78, 234 protecting, 266 saving to a file, 105 tracing the evaluation of, 343 with memory, 258

Degree, 21, 25,213 Degrees, 25

Centigrade, 28 Fahrenheit, 28

Delaunay triangulation, 137 Delayed

definitions, 69, 78, 255, 261 transformation rules, 253

Delete, 150,156 DeleteCases, 236-237 DeleteContents, 218 DeleteDirectory, 218 DeleteFile, 218-219 Deleting

according to patterns, 237 directories, 218 elements oflists, 149-150, 156 files, 219

Delimiters in ReadList, 137 Denominators

common, 49, 53 expanding, 48, 52 patterns for, 243

Density graphics objects, 175 Density plots, 114, 119

of data, 135 DensityGraphics, 175 DensityPlot, 114,119 Depth,232 Depth of recursions, 335 Derivative, 316,318 Derivatives, 66, 78

full form of, 318 higher, 66, 78 partial, 70, 315, 319 partial, format, 316 total,67

405

406

zero, attribute for, 266 DES, 33 Det, 158,163 Determinants, 158, 163 Developing programs, 342 Diagonal matrices, 158, 163 DiagonalMatrix, 158,163, 168 Dialog, 342, 344, 346-347 Dialogs, 344, 347 Differential equations, 73, 359

numerical solutions of, 77, 79, 380 partial, 77, 79 partial, formats for, 316 series expansions, 359 solutions of, 79 systems of, 74, 79

Differentials, total, 67, 78 DigitBlock,314 DigitQ,239 Digits

printed, 313 significant, 376, 378

Dimensions, sequence in tables, 142 Dirac function, 210 DiracDelta,21O Direction, 73, 79 Directives, 120, 173, 177 Directories, manipulation of, 217-218 Directory, 218 Discrete

data, 130 data, plots of, 135 Fourier transforms, 207, 212

DiscreteMath' ComputationalGeometry', 135, 137

Disk, 173,176, 184 Disks, 176 Dispatch, 253-254 Dispatch tables, 253-254 DISPLAY, 104 Display, 87-88, 105,389 Display of graphics, 87 DisplayFunction, 83,103,105,111-112,

203 Division, 17, 24 Divisors, 213 DivisorSigma,213

Divisor, greatest common, 37, 42 Do, 155,275,279-280,288,331 Documentation, 14, 16,327

of functions, 307 of standard packages, 190

DOS, 26 Dots, 246 Double pendulum, I, 106 DownValues, 258, 262 Downvalues, 260, 262

evaluation of, 269 Drop, 149,156 Dropping

characters of strings, 323 elements of lists, 156

DSolve, 74-77, 79 Dt,67,78 Dummy variables, 294 Dump, 221-222

E

E, 21,25,213 e,25 EdgeForm, 122,130,177 Edges, 130, 177 Edit, 11 EditDefinition,lJ Editln,lJ Editing, 310

input, 11 EDITOR, 11 Editor, 11 Effective potential, 107 Efficiency of list manipulations, 155 Eigensystem, 164,171 Eigenvalues, 164, 170-171 Eigenvalues, 164, 169,171 Eigenvectors, 164, 170-171 Eigenvectors, 164,171 Electronic mail, 5 Elementary functions, 20 Elements

of a list, 50, 52, 155 passing a test, 237 testing for, 241

Eliminate, 61, 64

Index

Elimination of variables, 61, 64 Ellipses, 176 Elliptic integrals, 79 EllipticE,213 EllipticExp,213 EllipticF,213 EllipticK,213 EllipticLog,213 EllipticNomeQ,239 EllipticPi,213 EllipticTheta,2l3 Encoding with RSA, 36 End, 302-304,307 EndPackage, 302-305,307 Energy

kinetic, 91, 107 potential, 90, 107

Enforcing evaluations, 273, 276 Engineering notation, 314 EngineeringForm, 313-314 Enter Dialog, 344 Enter Inspector Dialog, 344 Enter key, 9 epEff,108 Epilog, 82, 110, 112 EPS files, 87 Equal,64 Equal, 229, 340 Equations, 54, 64

algebraic, systems of, 60, 64 differential, 73, 79, 359 differential, constants in, 74 differential, numerical solutions of, 77, 79 differential, series expansions, 359 differential, systems of, 74, 79 elimination of variables, 61, 64 Euler-Lagrange, 90, 107, 125, 145 generation of a list of rules, 65 linear, 216-217 matrix, 217 numerical solutions of, 61, 65, 380 partial, differential, 77, 79,316 polynomial, 62, 65 polynomial, numerical solutions of, 381 solutions of, 54, 64 substitution of solutions into, 55 transcendental, 61, 339

transformation to algebraic rules, 65 Equilibrium

positions, of a double pendulum, 107 position, of the oscillator, 184 stable, 108 unstable, 108

Erf,213 Erfc,213 Error bars, 192 Error messages, 309, 323 Errors, 58, 342

absolute, 377-378 relative, 377-378 roundoff, 377

407

Euler-Lagrange equations, 90, 107, 125, 145 EulerE,213 EulerGamma,213 EulerPhi,213 Euler's

constant, 21, 335 theorem, 34 totient function, 34

Evaluate, 99, 105, 273, 276 Evaluate Selection, 9 Evaluation, 268

holding, 271, 276 immediate, 105, 273, 276 multiple, 252 non-standard, 275 of control structures, 275 of definitions, 274 of iteration functions, 275 of logical expressions, 275 ofloops, 275 of pure functions, 275 of transformation rules, 274 order, 257 preventing, 271, 276 special, 271, 275 standard, 268 stopping after maximum recursions, 335 tracing, 268, 270, 347 using rules, 52

Even numbers, testing for, 241 EvenQ, 239, 241 exi t, 12 Exit Dialog, 345

408

Exp, 24, 213 Expand, 46-47, 52 ExpandAll,52 ExpandDenominator, 49,52 Expanding

all parts, 52 complex expressions, 52 denominators, 48, 52 numerators, 49, 52 polynomials, 45 products and powers, 52 trigonometric functions, 47,52

ExpandNumerator, 49,52 Expansions

perturbation, 359 straight forward, 365 Taylor, 199-200

ExplntegralE,213 ExplntegralEi,213 Exponent, 314 Exponent, 214--215 ExponentFunction, 314 Exponential functions, 24 Exponentials of matrices, 158, 163 Exporting

expressions to other programs, 311 graphics, 87-88 names, 302

Expression, 136 Expressions, 229

appending to files, 392 applying functions to parts of, 232 compiling, 368 complex, expanding, 52 converting to strings, 120, 323 corresponding to strings, 323 currently being evaluated, 347 depth of, 232 full form of, 230 heads of, 230 in ReadList, 136 levels in, 231-232 logical, evaluation of, 275 partial, 232 parts of, 231 patterns for given heads, 241 positions of parts of, 232

sequence of, 183 structure of, 232 tree form of, 230 writing to files, 392

ExtendedGCD,213 External

data, 130 programs, 391-392

Extracting elements oflists, 149, 155-156 Extrapolation

F

in numerical products, 383 in numerical sums, 383

fac, 257 FaceForm, 177 FaceGrids, 110 Faces, 177 facFast, 259 Factor, 46-47, 53, 216 Factorial,213 Factorial function, 24, 252 Factoring, 53

integers, 42 numerical, 53 over integers, 46 polynomials, 46, 216 trigonometric functions, 47, 53

. Factorlnteger, 39,42,213 FactorList, 216 FactorSquareFree, 216 FactorSquareFreeList, 216 Factors, damping, 381 FactorTerms, 49,53,216 FactorTermsList, 216 False, 240-241 FFT,207 fibl,333 Fibonacci numbers, 259, 262, 333 Fields, vector, 190, 195 Figures, Lissajous, 130 File, 12,26 FileByteCount, 219 FileDate, 219 FileNames, 218-219

Index

Files, 219 appending expressions to, 392 closing, 389 initialization, 220-221 loading, 306 manipulation of, 217, 219 opening for writing, 389 reading, 31 reading data from, 136 reading expressions from, 389 reading Mathematica commands from, 306 saving definitions in, 105 types, 219 writing expressions to, 392

FileType, 219 Find in Function Browser, 14 findColorOfPoint, 371 Finding patterns, 235 FindMinimum, 124,130,339,380

options for, 382 FindRoot, 61, 65, 284, 374, 380

options for, 381 First, 155 Fit, 202, 205 Fits, 202

nonlinear, 205 Fixed points, 287 FixedPoint, 284,287, 369 FixedPointList, 284,288 Flat,264,266,268 Flatten, 98, 105, 157 Flattening lists, 105, 157 Floor, 213 Flow control, 289, 331-332

by messages, 325, 327 Fold,285,288,357 Folding functions, 285, 288 FoldList, 285,288 FontForm, 174, 177 Fonts

Courier, 90 used in this book, 3

For, 281,288,331 Formal parameters in pure functions, 147 Format

custom, 318, 320 for C, 312

for columns, 312 for Fortran, 312 for input, 312 for matrices, 312 for numbers, 313 for output, 312 for partial derivatives, 315, 319 for tables, 312 for TeX, 312 full,312 of partial derivatives, 316 options for formatting functions, 314 output, 312 padded,314 text, 312 tree, 312

Format, 318-320, 340 Formatted text, 177 Formatting

for input, 310 lists, 139, 142 packages, 305 strings, 174, 177 text, 174, 177

Forms, 310 Form, short, 57, 64 Fortran,51,268,311,331,368

format for, 312 FortranForm, 311-312 Fourier

series, 209 transforms, continuous, 209, 212 transforms, discrete, 207, 212

Fourier, 207,212 FourierTransform, 210 FourierTransform', 209, 212 FourierTrigSeries, 210-211 Fractal, IV Fractions, partial, decomposition, 49, 53 Frame, 82, 102,105, 112 FrameLabel, 83, 112 FrameMaker,5 FrameStyle, 83, 112 Frames, around graphics, 102, 105 FrameTicks, 83, 112 FreeQ, 239, 241-242 FromCharacterCode, 321,323

409

410

FromDate, 224 Frontend,3 ftp,5 Full notation of derivatives, 318 FullDefinition, 390,392 FullForm, 229-230, 271, 312 FullGraphics, 175 FullOptions,88 Function, 144,147,265-266,369 Function browser, 14 Functional programming, 277, 335 Functions

arguments of, 20, 23 associative, 264, 266 Bessel, 16,88 Bessel, derivatives of, 67 built-in, precision, 378 changing built-in, 260 color, 112, 120 commutative, 264, 266 compiled,375 composing, 286, 288 definitions of, 68, 78, 234 different notations of, 23 Dirac, 210 documentation of, 307, 327 elementary, 20 exponential, 24 exporting, 302 factorial, 24, 252 finding maxima of, 124 finding minima of, 124, 130 folding, 285, 288 for complex numbers, 197 for tests, 239, 241 gradient, 382 hyperbolic, 24 implicitly defined, 121 interpolating, 77, 204, 206 inverse hyperbolic, 24 inverse trigonometric, 24 iteration, evaluation of, 275 Lagrange, 107, 145 local,291 mathematical, 20, 213 names of, 20 nesting, 281-283, 287

G

numerical,380 numerical, compilation of, 381 numerical, controlling accuracy, 381 numerical, options for, 381 numerical, precision of, 381 plots of, 81, 87 pure, 75,144-145,147,282,368 pure, evaluation of, 275 pure, short notation for, 144 rational, 48 totient,34 trapdoor one-way, 34 trigonometric, 20, 24 trigonometric, expanding, 47, 52 trigonometric, factoring, 47, 53 with memory, 261 with optional arguments, 245, 247 with options, 246 with several variables, 69

Gamma, 213 Gaussian integers, 46, 53 Gaussianlntegers, 46, 53,213 GaussPoints, 382 GCD,42,213 GegenbauerC,213 generateD,356 GenerateE,37 genera teE, 356 generateGraphics, 188 generatePrimes,388 generateTable,372 Generator, random, 37, 42 Generic solutions, 60 Geometry, computational, 137 Get, 26,31,97,306 Get Info, 118 Global

maximum, 217 minimum, 217 symbols, 290

Global', 29,298-300 GoldenRatio,213 GoodRSAPrimeQ,384-386 gopher, 5

Index

GoTo,289 Gradient, 382 Gradient functions, 382 Graph, 86, 113, 117, 122 Graphics, 9,81

animation, 116 arcs, 176 arrays, 105, 176 bounding box, 120 circles, 176 color definitions, 183 colors, 177 contour, 111, 119, 175 contours, 120 converting between different types of, 116,

119 dashing, 177 density, 114, 119, 175 directives, 120, 173, 177 disks, 176 display of, 87 edges, 130, 177 ellipses, 176 explicit options of, 175 exporting, 87-88 frames, 102, 105 from a remote host, 104 gray-levels, 177 hue, 177 lines, 176 objects, 175 options for, 173 output, 389 point size, 177 points, 176 polygon faces, 177 polygons, 176 primitives, 173, 176 printing, 86, 88 programming language, 172 rectangles, 176 rows of, 105 styles for points and lines, 206 surface, 108, 119 surface, objects, 176 text, 176 thickness of lines, 177

three-dimensional, 175 tools for, 195

411

two-dimensional, 81, 87,108,119,175 view point, 120

Graphics, 173,175 objects, 173

Graphics', 192, 195 Animation', 117, 182 Colors' , 179, 183, 186 ContourPlot3D', 113, 121 Graphics', 192, 195 Graphics3D', 135, 137,201 Impl ici tPlot' , 113, 121 PlotField', 190, 195 PlotField3D', 190, 195 Polyhedra', 178

Graphics3D, 122, 173,175 Graphics3D', 135, 137,201 GraphicsArray, 102,105,176 Gray-levels, 120, 177 GrayLevel, 120, 177 Greater, 64 Greatest common divisor, 37, 42

of polynomials, 216 Greek symbols, 90 Green, 177 Green, 181, 183 Grid, 84, 88 GridLines, 83,88 Grouping, 23

H

Head, 230 Heads, 229-230

evaluation of, 269 inspecting, 346 of integers, 241 of numbers, 241

Height of plots, 135 HeldPart,276 Help, 14 Help, 14 HermiteH,213 HiddenSurface,11O Hold, 271, 276 HoldAll, 266, 271

412

HoldFirst, 266, 271 HoldForm, 272,276 Holding unevaluated, 266 HoldRest, 266,271 Home directory, 218 HomeDirectory,218 Hue, 177 Hue, 112,120,177 Hyper cube, 173 Hyperbolic functions, 24

inverse, 24 HypergeometricOF1,213 HypergeometriclFl,213 Hypergeometric2Fl,213 HypergeometricPFQ,239 HypergeometricU,213

I

1,88 I,21,25,213 Identity

matrices, 158, 163 testing for, 241

Identity, 103,105 IdentityMatrix, 158,163 If, 275, 277-278,287, 331 Im,195,197-198,213 Image, binary, of Mathematica, 222 Imaginary

parts, 197 parts, patterns for, 243 unit, 21, 25

Immediate definitions, 69, 78, 255, 261 evaluation, 105, 273, 276 transformation rules, 253-254

Implicitly defined functions, 121 surfaces, 121

ImplicitPlot,113 In,9 Increments, 288 Indefinite integrals, 78 Indices, 140, 142 Inequalities, 216--217 Infinite recursions, 51

Infinity, 21, 25 Infinity, 21,25,213 Infix, 317-318 Infix notation, 23, 317-318 Information, 14, 16 in it. m, 31, 220-221, 301 Initial conditions, 77, 332 Initialization

cells, 349 file, 220-221

Initialization Cell, 305 Inner, 163 Inner products, 163 Input

containing multiple lines, 10 editing, 11 formatting for, 310, 312 form, converting to, 175 from external programs, 392

InputForm, 172,175,310,312 Insert, 150,156 Inserting

characters in strings, 323 elements in lists, 156

inspect, 12,344 Install, 392 Installing external programs, 392 Integer, 238,241 IntegerDigits,213 IntegerQ, 239, 241 Integers, 18

factoring, 42 Gaussian, 46, 53 heads of, 241 testing for, 241

Integrals, 70, 78 constants of integration, 70, 74, 78 definite; 71, 78 elliptic, 79 indefinite, 78 line, 381 multiple, 71 numerical evaluation of, 72, 78, 381

Integrate, 70-73, 78 Integrator, 248 Interpolating

functions, 77

Index

polynomials, 203, 205 Interpolat ingFunct ion, 204, 206 InterpolatingPolynomial,205 Interpolation, 132,204,206

forms of data in, 206 options for, 206

InterpolationOrder,204,206 Interpolations, 203-206 Interrupt Calculation, 12,344 Interrupting Mathematica, 12 Intersection, 151,156 Intersections

of intervals, 378 of lists, 156

Interval, 378 IntervalIntersection,378 IntervalMemberQ, 239,378 Intervals, 378

intersections of, 378 unions of, 378

IntervalUnion,378 Inverse

hyperbolic functions, 24 matrices, 158, 163 pseudoo, 217 series, 200 trigonometric functions, 24

Inverse, 158,163, 169 InverseFourier, 207,212 InverseFourierTransform,210 InverseJacobiSN,213 InverseLaplaceTransform,211 InverseSeries,200 Irrational numbers, 18 Iteration functions, evaluation of, 275 Iterations, 277

maximum number of, 382 Iterative algorithms, 334 Iterators, 119,280

J J,88 JacobiAmplitude,213 Jacobian, 157, 160-161,382 Jacobian, 382 JacobiP,213

JacobiSN,2l3 Jacobi Symbol, 213 JacobiZeta,213 Jobs

background, 389 batch, 391

Join, 96, 98, 105, 151,156 Joining

lists, 151, 156 points, l35 strings, 323

Jordan decomposition, 217 JordanDecomposition,217

K

K,89 Kernel, 3 Key, 33

database, 35 Kinetic energy, 91, lO7

in a rotating system, 125

L

Label, 289 Labels, 84, 88, 192 lagEquations,94, 146 Lagrange function, 90, lO7, 145 lagSolRule,98 LaguerreL, 213 laplace, 336 Laplace transforms, 211-212, 336 LaplaceTransform, 211, 337 LaplaceTransform',211-212 Large primes, 384 Largest number, 24 Last, 155 Last result, 23 Layers, boundary, 366 LCM, 42, 213 Leaf Count, 225 Least common multiple, 42

of polynomials, 216 Least-squares fit, 202 LegendreP,213 LegendreQ, 239

413

414

Length of lists, 157 of polynomials, 215 of strings, 323

Length, 153,157,214-215 LerchPhi,2l3 Less, 64 Let terQ, 239 Letters, capital, 45 Level, 232 Levels

contour, 113 gray, 120 in expressions, 232 maximum number of, in tables, 142 of expressions, 231

Lighting, 110, 123,129 LightSources, 110 Limi t, 73, 79 Limiting

execution time, 222, 224 memory use, 225

Limits, 73, 79 Line, 173,176, 179 Line integrals, 381 Linear

algebra, 158,216-217 equations, 216-217 programming, 216-217

LinearAlgebra'CrossProduct',152 Linearization, 97 Linearization', 97, 355 Linearize, 97 LinearProgramming,217 LinearSolve,217 Lines, 176

contour, 113 style of, 206 thickness of, 177 zero, 113

LinkConnectedQ,240 LinkReadyQ, 240 linSolRule,98 Lissajous figures, 130 List, 238 Listable, 143,263,266,268 ListContourPlot,135,201

ListDensityPlot, 135, 201 ListPlot, 131-135, 159,201

options for, 135, 206 ListPlot3D, 135, 201 ListQ,239 Lists, 23, 50, 52,116,119,138,155-157

adding elements to, 150, 156 algebraic operations on, 143 appending elements to, 156, 288 applying functions, 147 combining, lSI, 156 complements of, 156 concatenating, 105, 151, 156 converting into parameters, 146-147 creating, l38 cyclic permutations of, 152, 157 deleting elements of, 149-150, 156 dropping elements of, 156 efficiency of manipulations, 155 elements of, 50, 52 extracting elements of, 149, 155-156 flattening, 105, 157 formatting of, see also: Tables, l39, 142 inserting elements, 156 intersections of, 156 joining, lSI, 156 length of, 157 manipulations of, 148, 155 manipulations of, efficiency, 155 multidimensional, 141 nested, 119, 150 of ASCII codes, 323 of characters, 323 of contexts, 306 of transformation rules, 251 partitions of, 157 plotting, 132, 135 prepending elements to, 288 replacing elements of, 150, 156 reversing, 151, 157 sorting, 151, 156 step size, 119 sublists, 149, 156 threading over, 162-163 unions of, 156 using as sets, 151

ListSurfacePlot3D, l35, 201

Index

ListToString,40 listToString, 358 Literal, 274,276 Loading

automatic, 301 packages, 26,31, 306

Local constants, 292 functions, 291 values, 295, 297 variables, 181, 183, 290, 296 variables in conditions, 292, 296

Localization, 290, 296 Locked,266 Locking attributes, 266 Log, 24, 213 Logarithmic plots, 192 Logarithms

base of, 24 natural,24

Logical and,64 expressions, evaluation of, 275 expressions, simplification of, 64 or, 64

Loglntegral,213 Loops,277,279-280, 288

evaluation of, 275 Lowercase, 45 LowerCaseQ, 239 LU decomposition, 217 LUBackSubstitution, 217 LUDecomposition,217

M

Machine-precision numbers, 376 MachineNumberQ,239 Macintosh, 1, 3 Mail, electronic, 5 Make Lines Thin, 113 Make Template, 14, 302 makeList, 358 makeListl,331 makeList2,332 Manipulations

of directories, 217-218

offiles, 217, 219 of lists, 148, 155 of strings, 321, 323

Mantissa, 314 MantissaExponent, 314 Manual,l Map, 143,147 MapAll,147 MapAt, 232 MapThread,147 Mass matrix, 167 massDisks, 184 Master packages, 190 Master.m,301 Matched and composite solutions, 366 MatchedSolutionList, 366 Matching

parts, testing for, 241 patterns, 237 testing for, 242

MatchLocalNameQ,239 MatchLocalNames, 347 MatchQ, 239,242 math,lO Mathematical functions, 20 mathgroup,5 MathLink,391-392 MathSource, 5 Matrices, 158

determinants of, 158, 163 diagonal, 158, 163 exponentials of, 158, 163 formatting, 142,312 identity, 158, 163 indexed, 140 inverse, 158, 163 Jacobian, 157, 160--161,382 Jordan decompositions of, 217 LU decompositions of, 217 mass, 167 multiplications of, 160 operations with, 163 powers of, 158, 163 products of, 160, 163 pseudoinverse, 217 QR decompositions of, 217 row-reduced,217

415

416

Schur decompositions of, 217 singular values, 217 sparse, 216 stiffness, 167 testing for, 241 transposition of, 158, 163

Matrix equations, 217 MatrixExp, 159,163 MatrixForm, 139,142,312

options for, 139,142 MatrixPower, 159,163 MatrixQ, 239, 241 Max, 24, 213 MaxBend, 83-84, 88 Maximum, 24

global, 217 number of bytes, 225 of a function, 124

MaxIterations,382 MaxMemoryUsed, 223, 225 MaxRecursion, 379,382 MaxSteps, 126,130,382 MaxStepSize,382 MemberQ, 239,241 Memory, 223

definitions with, 258 functions with, 261 limiting, 225 optimizing, 225

MemoryConstrained, 223,225 MemoryInUse, 223, 225 Mesh, 110 MeshStyle,110 Message, 324,327 Messages, 309, 323

associated, 327 checking for, 327 custom, 327 flow control with, 325, 327 place holders in, 324 printing, 327 suppressing, 327

Messages, 327 Metacharacters, 16, 322 Method, 382-383 Methods, 340 Min, 24, 213

Minimizing, linear programming, 217 Minimum, 24

global, 217 numerical, 380 of a function, 124, l30

MinRecursion,382

Minus, 17, 23 Miscellaneous'

PhysicalConstants',32 Units', 26, 32

MKS,28 Mod, 38,42,213 Modes, normal, 168 Modula-2, 279, 331 Modular powers, 42 Module, 181,183,290-296,331 Modules, 183 MoebiusMu,213 Movie, 117 MS-DOS, 26 Multidimensional tables, 141 Multinomial, 213 Multiple

integrals, 71 least common, 42

Multiplications, 17, 24 of lists, 160 with scalars, 160, 163

mylntegrate, 249, 319

N

N, 18,24,378 N[Solve [ ... ]],65 name, 340-341 Named patterns, 243 NameQ,239 Names

declaring, 220 exporting, 302 in contexts, 306 in packages, 305 of files, 219 of Mathematica functions, 20 removing, 306 shadowing, 301 unique, 293, 296

Index

nameSort,341 Naming, 45 Natural logarithm, 24 NDSolve, 77,79,98,126,130,191,380

options for, 382 Needs, 26, 32, 303,306 Negative, 240, 242,348 Negative numbers, testing for, 242 Nest, 287 Nested

functions, 281-283, 287 lists, 119, 150 transformation rules, 251

Nesting, maximum, in tracing, 348 NestList,282-283,287 Newline, 322 newtonLines,283,289 Newton's method, 282, 297, 336, 368, 374,

381 NeXT, 3 NIntegrate, 72, 78, 379,381

options for, 382 Non-standard evaluation, 275 Nonlinear

fits, 205 springs, 359

NonNegati ve, 240, 242, 348 Nonnegative numbers, testing for, 242 NonNega ti veQ, 348-353 NonNegativeO.m,349 Normal, 200 Normal modes, 168 Notation

accounting, 314 engineering, 314 full, 230 full, of derivatives, 318 infix, 23, 317-318 of Mathematica functions, 23 postfix, 19,23,317-318 prefix, 23, 317-318 scientific, 313 standard, 23, 230

Notebook, 3--4 cells, 9

Notebook front end, 3 animation, 117

exporting graphics, 87 help, 14 interrupt commands, 13 printing graphics, 86 quitting Mathematica, 12 starting, 9 suppressing lines, 122

Notes, 327 NProduct, 199-200,381

options for, 383 NProductExtraTerms,383 NProductTerms,383 NSol ve, 62, 65,381 NSum, 199-200,381

options for, 383 NSumExtraTerms,383 NSumTerms,383 Null, 278 Null space, 217 NullRecords,136 NullSpace,217 NullWords,136 Number, 136 NumberForm, 313 NumberFormat,314 NumberPadding,314 NumberPoint,314 NumberQ, 239,241 Numbers

arbitrary-precision, 376 base of, 314 complex, 195, 197 Fibonacci, 259, 262, 333 formatting, 313 heads of, 241 in ReadList, 136 integer, 18 irrational, 18 largest, 24 machine-precision, 376 precision of, 376 random, 42 rational, 18 real, 18 smallest, 24 substituting, 50 testing for, 241-242

417

418

NumberSeparator,314 Numbersigns,314 Numerators

expanding, 49, 52 patterns for, 243

Numerical

o

algorithms, precision and accuracy, 379 approximations, 18,24,378 calculations, 17, 368 factoring, 53 functions, 380 functions, compilation of, 381 functions, controlling accuracy, 381 functions, options for, 381 functions, precision of, 378, 381 integrals, 72, 78, 381 products, 199,381 search for minima, 380 solutions of differential equations, 77, 79,

380 solutions of equations, 61, 65, 380 sums, 199-200,381

0,200 Object-oriented programming, 340 Objects, 340

graphics, 173, 175 Odd numbers, testing for, 241 OddQ, 239, 241 Off, 324,327 On, 324,327 OneIdentity, 264, 266 Open, 26 Open Function Browser, 14 OpenAppend, 387, 389 Opening files, 389 OpenRead,389 OpenWri te, 388-389 Operate, 148 Operating system commands, running, 392 Operators, 145

overloading, 340 Optimizing

memory use, 225 transformation rules, 253-254

Optional arguments, 245, 247 arguments, patterns for, 244, 247

OptionQ, 240, 246 Options, 46, 314

default values for, 82, 88 explicit, of graphics, 175 for contour plots, 120 for FindMinimum, 382 for FindRoot, 381 for graphics, 105, 173 for Interpolation, 206 for ListPlot, 135, 206 for MatrixForm and TableForm, 139,

142 forNDSolve, 130,382 for NIntegrate, 382 for NProduct, 383 for NSum, 383 for number formats, 314 for numerical functions, 381 for Plot, 88, 206 for plots, 82, 88 for ReadList, 136 for Show, 135 for surface plots, 120 for three-dimensional graphics, 129 for Trace, 347 functions with, 246

Options, 88, 246 Or, 64 Order

of evaluation, 257 testing for, 241

OrderedQ,240-241 Orderless, 264, 266, 268-269 Ordinary differential equations, 73, 79 Oscillator, III, 164, 184

equilibrium position of, 184 oscPts,187 Out, 9 Outer, 160,163 Outer products, 160, 163 Output

combining parts of, 309 custom formats for, 320 formats, 312

Index

of graphics, 389 of text, 308 streams, 389 suppressing, 57, 64 to streams, 389 writing to files, 389

Output Form, 3lO, 312,319 OutputStream,389 Overloading, 340

p

1t,2l,25 Packages, 297

context commands, 306 contexts in, 303 declaring names, 220 documentation, 307 for animation, 117 for colors, 183 for complex symbols, 198 for cross products, 152 for elliptic integrals, 79 for Fourier transforms, 212 for graphics, 195 for implicitly defined functions, 121 for implicitly defined surfaces, 121 for Laplace transforms, 212 for nonlinear differential equations, 79 for partial differential equations, 79 for plots of three-dimensional data, 137 for polyhedra, 178 for statistics, 206 for symbolic sums and products, 201 for the conversion of units, 32 for vector fields, 195 formatting, 305 loading, 26, 301, 306 master, 190 names of functions in, 305 . protecting functions in, 304 reading, 26, 31 standard, 26 using other packages, 303

Pac k age s, 26 Padded formatting of numbers, 314 PaddedForm,313-314

Padding with binary zeros, 378 Parameters, 229

using elements oflists as, 14~147 Parametric

curves, planar, 126, 129 plots, 122 surfaces, 129

419

ParametricPlot, 126-127,129, 191 ParametricPlot3D, 122-124, 127-129 Parent directory, 218 ParentDirectory,218 Parentheses, 23 Part, 156 Partial

derivatives, 70, 315, 319 derivatives, format of, 316 differential equations, 77, 79, 316 fraction decompositions, 49, 53

Partition, 157, 183 Partitions, 213 Partitions of lists, 157 PartitionsQ, 213, 239 Parts

imaginary, 197 of expressions, 232 of strings, 323 real, 197

Pascal, 51, 268, 279, 331 Paste, 11 Paste Template, 14 Path, 221

contexts, 307 Pattems,68,233,23~237

complicated, 242 counting, 237 deleting elements, matching, 237 finding, 235 for complex numbers, 243, 247 for given heads, 241 for optional arguments, 244, 247 for rational numbers, 243, 247 in compiled functions, 375 in strings, 322-323 in transformation rules, 252 matching, 235, 237, 278 named, 243 positions of, 237

420

repeated, 246, 248 restricting, 237-238, 240 unevaluated, 276 with alternatives, 244, 247 with conditions, 240 with tests, 238

Pause, 224 Pausing, 224 PC, 1,3 PDEForm, 319-320 pdeForm, 316, 319 Pendulum

double, I, 106 triple, II, 89, 145, 178

pendulumLines, 180-182 Permutations, cyclic, 152, 157 Perturbation expansions, 359 Perturbation', 365 PhysicalConstants',32 pi, 25, 213 Pie charts, 192 piScale, 193 Place holders, 322

in messages, 324 in strings, 309-310

Planar parametric plots, 126, 129 Play, 194 Plot, 81-87,99-103,272-273,275

options for, 88, 206 Plot3D, 108-110,119,275 PlotDivision, 83-84,88 PlotField' , 190, 195 PlotField3D', 190,195 PlotJoined, 132,135 PlotLabel, 83,88, 110, 112 PlotPoints, 83-84, 88, 110, 112-113 PlotRange, 83, 86,88, 100, 110, 112 PlotRegion, 83, 110, 112 Plots, 81

aliasing in, 84 animation, 116 arrays, 102, 105 bar, 192 bending angle, 84, 88 bounding box, 120, 129 contour, 111, 119-120 contour, of data, 135

data, 135 data, with correct scales, 159 density, 114, 119 density, of data, 135 display of, 87 error bars, 192 exporting, 87-88 frames, 102, 105 from a remote host, 104 grids in, 84, 88 initial number of points, 84, 88 labels, 84, 88, 192 lighting, 129 lists, 132, 135 logarithmic, 192 maximum number of subdivisions, 84, 88 of general three-dimensional data, 137 of lists, 132, 135 of several functions, 81, 87 options for, 82, 88, 135 options for three-dimensional, 129 parametric, 122 pie, 192 planar parametric curves, 126, 129 printing, 86, 88 range, 85, 88 ratio of height to width, 135 rows of, 105 shading, 129 space curves, 122, 127, 129 suppressing display, 103, 105 surface, 108, 119 surface, of data, 135 three-dimensional, 106 titles of, 83, 88 tools for, 195 two-dimensional, 81, 87, 108, 119 view point, 120

Plot Style, 83, 202,206 PlotVectorField,190 Plus, 23, 146 Pochhammer,213 Pocket calculator, 17-18 Point, 173,176 Points, 176

decimal,18 joining, in plots, 135

Index

sample, 88 size of, 177 style of, 206 view, 110, 120

PointSize,177 PolyGamma,213 Polygon, 173, 176 Polygons, 176

faces of three-dimensional, 177 without edges, 177

Polyhedra', 178 PolyLog,213 Polynomial equations, 62, 65

solutions of, 381 PolynomialGCD,216 PolynomialLCM,216 PolynomialMod,216 PolynomialQ, 239, 241 PolynomialQuotient,215-216 PolynomialRemainder, 215-216 Polynomials, 52

coefficients in, 214-215 composing, 216 cyclotomic, 216 equations with, 62, 65 equations with, numerical solutions of, 381 expanding, 45 factoring, 46, 216 greatest common divisor of, 216 interpolating, 203, 205 least common multiple of, 216 length of, 215 maximum power of, 215 modular reduction of, 216 quotient of, 216 remainder of division of, 216 resultant of, 216 structure of, 214-215 testing for, 241 variables of, 148,215

polyOrderList,360 polyOrderSolutionList,364 position, 231-232, 235,237 Positions

according to patterns, 237 in expressions, 232 in strings, 323

of the triple pendulum, II positive, 240, 242,348 Positive numbers, testing for, 242 Postfix, 317-318 Postfix notation, 19,23,317-318 PostScript, 87 Potential

effective, 107 energy, 90,107 in configuration space, 123 of a rotating double pendulum, I Wintner's,89

Power, 24 Power series, 359 PowerExpand, 49, 53 PowerMod, 38, 40,42,213,357 Powers, 18,24

collecting, 53 expanding, 52 maximum, in polynomials, 215 modular, 42 of matrices, 158, 163

Precision, 376, 378 arbitrary, 376 in b~ilt-in functions, 378 in numerical functions, 381 machine, 376 of calculations, 22

Precision, 376, 378 PrecisionGoal,382-383 prefix, 317-318 Prefix notation, 23, 317-318 Prepare Input, 14, 110 Prepend, 150, 155-156 Prepending elements to lists, 288 PrependTo, 288 Preventing the evaluation, 271, 276 Previous results, 19,23 Prime, 34, 36

factors, large, 384 mth,42

relatively, 34 test, 42

Prime, 36--37,42,213 PrimePi,213 PrimeQ,39,42,213,239,241

421

422

Primes large, 384 testing for, 42, 241

pri mes, 386 Primitives, 173, 176 Print, 270, 308, 342 Print, 86 Print Selection, 86 Print Form, 312 Printing, 270, 308

custom formats for, 320 graphics, 86, 88 messages, 327

p r i n t p s, 87-88 Problems, eigenvalue, 164, 170--171 Procedural programming, 277, 331 Product, 147-148, 198,200 Products, 148, 198,200

cross, 152 expanding, 52 inner, 163 matrix, 160, 163 numerical, 199,381 of the elements of lists, 146 outer, 160, 163 scalar, 160 symbolic, 199 with scalars, 160, 163

Programming functional, 277, 335 graphics, 172 linear, 216-217 methodologies, 331 object-oriented, 340 procedural, 277, 331 recursive, 332 rule-based, 336 with constraint-propagation, 337

Programs developing, 342 external,391-392

Prolog, 83, 110, 112 Protect, 261-262, 304 Protected, 262-263,266 Protecting

attributes, 266 definitions, 266

functions, in packages, 304 symbols, 262

Pseudolnverse, 217 Pseudoinverse matrices, 217 ps fi x, 87-88 PSPrint, 86, 88 Public key cryptosystem, 33 Pure functions, 75,144-145,147,282,335,

368 evaluation of, 275 formal parameters in, 147 short notation for, 144 with attributes, 265-266

Put, 388, 390,392 PutAppend, 388,392

Q

QR decompositions, 217 QRDecomposition, 217 Quit, 13 Quit, 12-13 qui t, 12 Quitting

external programs, 392 Mathematica, 12

Quotient, 213 Quotients of polynomials, 216

R

Radians, 21, 25 Random

generator, 37, 42, 130 numbers, 42

Random, 37,42, 130,213 Range, 138-139,142 Ranges

in Table, etc., 141 of plots, 85, 88

Raster, 173,176 RasterArray, 176, 372 ra sterps, 87-88 Ratio

of height to width, of plots, 135 of the edges of the bounding box, 120

Index

Rational functions, 48 numbers, 18 numbers, heads of, 241 numbers, patterns for, 243, 247

Rational, 241,243,247 Rationalize, 18,24 Re, 197-198, 213 Read, 389 Reading

data, 131, 136 files, 31, 306 from files, 389 packages, 26, 31

ReadList, 131,136,159,201 ReadProtected,266 Real

numbers, 18 parts, 197 parts, patterns for, 243

Real, 136, 238,241 Receiver, 33 Record, 136 RecordLists,136 Records

empty, in ReadList, 136 in ReadList, 136

RecordSeparators,136 Rectangle, 173, 176 Rectangles, 176 Recursions

infinite, 51 limiting, 335

Recursive programming, 332 Red, 177 Red, 181, 183 Red-green-blue color model, 120 Reduce, 60, 64 Reference manual, 1 ReIm', 196, 198,353 Relative

context names, 298 errors, 377-378

Relatively prime, 34 ReleaseHold, 272,276 Remainders, 42

of the division of polynomials, 216

Remote host, 104 Remove, 31, 300,306 Removing names, 306 RenameDirectory,219 RenameFile,219 Renaming

directories, 219 files, 219

Repeated patterns, 246, 248 Replace, 254 ReplaceAll,254 ReplaceHeldPart,276 ReplacePart, 150,156 ReplaceRepeated,254 Replacing

characters in strings, 310, 323 elements of lists, 150, 156 without evaluation, 276

Representation, tree, 230 ResetDirectory,218 Residue, 79 Residues, 73, 79 Resources, 222

memory, 223 Rest, 149,156 Restricting patterns, 237-238, 240 Result

last, 23 previous, 23 using previous, 19

Resultant, 216 Resultant, 216 Return, 279, 289-290, 347 Return,322

key, 9 Reverse, 151,157 Reversing

lists, 151, 157 strings, 323

RGB,I77 RGBColor, 120, 177 RiemannSiegelTheta,213 RiemannSiegelZ,213 Risch algorithm, 70, 250 rl ogi n, 104 Roots, 24

Newton's method, 282

423

424

Roots, 65 rot2D,291 RotateLabel, 83, 112 RotateLeft, 152-153,157 RotateRight, 152-153,157 rot Do U b 1 e Pen d . m, 117 Round, 24, 213 Rounding, 24 Roundoff errors, 377 Row-reduced matrices, 217 RowReduce,217 Rows of graphics, 105 RSA, 331, 355, 384

cryptosystem, 34 decoding, 36 encoding, 36

RSA', 37, 359, 384, 386 r s a - p rime s - 100. co n t ro 1,390-391 rsa-primes-100.log,391 RSADecode, 40, 359 rSADecode,358 RSAEncode, 40, 359 rSAEncode, 358 rSAEncodeNumber,357 RSAPrime,385-386 Rule, 162,254 Rule-based programming, 336 RuleDelayed,254 Rules, see also: Transformation rules, 50, 52,

65 generation from equations, 65 transformation, 251, 254 using in evaluation, 52

Run, 392 Running external programs, 392 RunThrough,392

s SameQ, 239-241 SameTest,287 Sample points

in numerical integrations, 382 in plots, 88

Saturation, 112, 120 Save, 97, 105,131 Save As Other, 305

Saving definitions, 105 Mathematica binary, 221-222

Scalar multiplication with, 160, 163 products, 160

Scaled coordinates, 177 data plots, 159 plots, 159

Scaled, 177 ScatterPlot3D, 135,201 Schur decomposition, 217 SchurDecomposition,217 sci. math. symbol ie, 5 Scientific notation, 313 ScientificForm,313 Scoping, 331 Search path, 221 Sec, 24, 213 Sech, 24, 213 SeedRandom,213 Select, 236-237, 339 Selecting with tests, 237 Semicolons, 57, 64, 279-280 Sender, 33 Separators in ReadList, 136 SequenceForm, 309-310, 315 Sequences

arithmetic, 138, 142 of arguments, 247 of expressions, 183

Series, 200 expansions for the solutions of differential

equations, 359 Fourier, 209 power, 359 truncating, 200

Series, 199-200 Session, 2 SessionTime,224 Set, 261 SetAccuracy, 22, 377-378 SetAttributes, 263, 265, 269 Set Delayed, 261 SetDirectory,218 setenv,104

Index

SetFileDate, 219 SetPrecision, 22, 377-378 Sets, see also: Lists, 151

complements of, 156 intersections of, 156 unions of, 156

Shading, 110 Shading of surfaces, 129 Shadowing of names, 301 Shallow, 57, 64 Share, 224-225 Shell variable, 11 Short

form, 57, 64 notation for pure functions, 144

Short, 57, 64 Show, 83, 87, 119

options for, 135 show,12 Show Lines, 122 ShowAnimation, 117, 182, 189 showFractal,373 Sr,28 Sign, 213 Signature, 213 Significant digits, 376, 378 SignPadding, 314 SiliconGraphics, 11 0 Simplification, 46, 53

of equations, 64 Simplify, 46-48, 53 Simulation, 126 Sin, 24,213 Singular values, 217 Singularities in integrals, 71 SingularityDepth,382 Singularvalues,217 Sinh, 24, 213 Sinlntegral,213 SixJSymbol,213 Size

of points, 177 of steps, 119

S KP a c ka ge s, 349 SKPackages'

AbsArg' , 354--355 Linearization', 97, 355

NonNegativeQ', 349 Perturbation',365 RSA' , 37, 359, 384, 386

Smallest number, 24 smallLagSolRule, 103 smallLinSolRule,103 Software, 5 Solutions

generic, 60 matched and composite, 366

425

numerical, of algebraic equations, 61, 65, 381

numerical, of differential equations, 380-381

numerical, of polynomial equations, 381 of algebraic equations, 54, 64 of differential equations, 79

Solve, 54-55,64,216,338 solveOrder, 361 Sort, 151,156,341 Sorting, 341

lists, 151, 156 Sound,9,194 Space

characters, 17 configuration, 123 curves, 122, 127, 129 in tables, 142

Sparse matrices, 216 Special

characters, 321-322 evaluation, 271

SphericalHarmonicY, 213 SphericalRegion,110 springLines, 185 Spring, nonlinear, 359 Sqrt,24 Square roots, 24 Stable equilibrium, 108 Stack,347 Stack,346-347 Standard

evaluation, 268 notation, 23, 230 packages, 26

Starting external programs, 392

426

Starting Mathematica, 9 from text-based interface, 10 notebook front end, 9

StartingStepSize, 382 Statistical analysis, 205 Statistics, 206 Statistics', 205-206 Step size, 119 Stiffness matrix, 167 StirlingSl,2l3 StirlingS2,213 Straight forward expansions, 365 Strained coordinates, 365 StrainedSolutionList, 365 Streams, 389 String, 136, 321 StringDrop, 323 StringForm, 309-310 Stringlnsert, 321,323 StringJoin, 120, 323 StringLength,323 StringMatchQ, 239, 322-323 StringPosition, 323 StringQ,239 StringReplace, 323 StringReverse,323 Strings, 309-310, 321, 323

concatenating, 120 converting to expressions, 323 converting to lists of characters, 323 corresponding to expressions, 120,323 dropping characters of, 323 formatting, l74, 177 inserting characters, 323 joining, 323 metacharacters in, 322 parts of, 323 patterns in, 322-323 place holders in, 309-310 positions in, 323 replacing characters, 323 reversing, 323 special characters in, 322

StringTake, 323 StringToList,40 stringToList,357

Structure of expressions, 229, 232 of polynomials, 214-215

Stub, 266, 301 Style, 305 Style for points and lines, 206 Subcontexts, 302 Subdivisions

in numerical integrations, 382 in plots, maximum number of, 84, 88

Subexpressions, 231 testing for, 242

Sublists, 149, 156 in ReadList, l36

Subscript, 315-316,318 Subscripted, 315,317 Subscripts, 315, 317 Subsessions, 344 Substitutions, 50

of solutions into equations, 55 Subtractions, l7, 23 Sum, 147-148, 198,200,275 Sums, 148, 198

numerical, 199-200, 381 of the elements of lists, 146 symbolic, 199,201

Sun, 5 Superscript, 315,318 Superscripts, 315, 318 Suppressing

lines, 122 messages, 327 output, 57, 64 plots, 103, 105

Surface graphics objects, 176 implicitly defined, 121 parametric, 129 plots, 108, 119 plots of data, 135 shading, 129

SurfaceGraphics, 176 Switch, 275, 278-279, 287, 331 Symbol,238 Symbolic

calculations, 44 Fourier transforms, 212

Index

Laplace transforms, 212 products, 199 sums, 199,201

SymbolicSum', 199,201 Symbols

complex, 195, 197-198 contexts of, 306 definitions for, 392 exporting, 302 global,290 Greek,90 protecting, 262 removing, 306 with associated definitions, 260

SyntaxQ, 239 System', 29, 299 Systems

T

of algebraic equations, 60, 64 of algebraic equations, numerical solutions

of, 62, 65 of differential equations, 74, 79

Table, 116,119, 130, 138,141,275 TableAlignments, 142 TableDepth, 142 TableDirections, 142 TableForm, 139,142,312

options for, 139,142 TableHeadings, 139,142 Tables, 116, 119, 141

alignment in, 142 dispatch, 253-254 formatting, 142,312 levels of, 142 multidimensional, 141 sequence of dimensions in, 142 spaces in, 142 titles in, 142

TableSpacing, 142 Tabs, 322 Take, 149,156 Tan,24,213 Tanh, 24, 213 TargetFunctions, 196,198 Taylor expansions, 199-200

Template, 14 Temporary, 266 Tensors, 158

indexed, 140 inner products of, 163 outer products of, 160, 163

Tests, 236, 241, 278 for elements, 237 for primes, 42 for vectors, 157 functions for, 239 in patterns, 238

TeX, 311, 319 format for, 312

TeXForm, 311-312, 319 Text, 9, 176, 309

cipher, 33 converting, 323 format, 312 formatted, 177 output, 308 sending to external programs, 392 subscripts in, 315, 317 superscripts in, 315, 318

Text, 173, 176 Text-based interface

animation in, 117 exporting graphics from, 87 printing graphics in, 86 quitting Mathematica, 12 starting Mathematica, 10 suppressing lines in, 122

Text Form, 312 Thickness, 174,177,180 Thickness of lines, 177 Thread,148, 162-163, 184 Threading

automatic, attribute for, 266 overlists, 162-163

Three-dimensional data, general plots of, 137 graphics, 175 plots, 106

ThreeJSymbol,213 Through, 145,148 Throw,289 Ticks, 83, 111-112

427

428

Time, 224 absolute, 224 converting, 224 limiting, 224 resources, 222

TimeConstrained, 222, 224 Times, 23-24, 147 TimeUsed,224 Timing, 5, 38,42,224 Timings, 42 Titles, 9

in tables, 142 of plots, 83, 88

tmp, 118 ToCharacterCode, 321,323 ToDate, 224 ToExpression, 321-323 Together, 49,53 Token words, in ReadList, 137 TokenWords, 137 Tools for graphics, 195 ToRules,65 Torus, 121 torus, 123 ToString, 116,120,321,323 Total

derivatives, 67 differentials, 67, 78

Trace,268,270,325,342-344,347 evaluation of, 275 options for, 347

TraceAbove, 347 TraceBackward,348 TraceDepth, 348 TraceDialog, 344-345,347 TraceForward, 348 TraceOff,348 TraceOn, 348 TraceOriginal, 348 Tracing, 268, 270 Transcendental equations, 61, 339 Transformation rules, 251, 254

applying, 254 delayed, 253 evaluation of, 274 immediate, 253-254 multiple evaluation of, 252

nested,251 optimizing, 253-254 patterns in, 252

Transforms Fourier, 207, 212 Fourier, continuous, 209, 212 Laplace, 211-212, 336

Transpose, 158-159,163, 169 Transposition, 158, 163 Trapdoor one-way functions, 34 Tree representation, 230 TreeForm, 229-230, 311-312 Triangulation, 137 Trig, 47,52-53 Trigonometric functions, 20, 24

expanding, 47,52 factoring, 47, 53 inverse, 24

Triple pendulum, 89,145,178 positions of, II

triplePendulum.m,97,182 True, 240-241 True I False, 375 TrueQ, 239-240 Truncating series, 200 Two-dimensional

graphics, 175 plots, 81, 87, 108, 119

Types

u

data, 340 file, 219

Unequal,64 Unevaluated, 271, 276

form of arguments, 276 keeping arguments, 266 patterns, 276

Unevaluated,276 Uninstall,392 Union, 151,156 Unions

of intervals, 378 of lists, 156

Unique, 293, 296 Unique names, 293, 296

Index

Uni ts' , 26, 32 Units, converting, 32 Unit, imaginary, 21, 25 UNIX, 1,26,118 Unknowns, 44 Unprotect, 261-262, 304 Unprotecting symbols, 262 UnsameQ,239-241 Unstable equilibrium, 108 Uppercase, 45 UpperCaseQ, 239 Upvalues,262 Upvalues, 260, 262

evaluation of, 269 Usenet,5

v ValueQ, 240,242 Values

absolute, 24, 197 local, 295, 297 testing for, 242

Variables control, 288 dummy, 294 elimination of, 61, 64 global, 290 in compiled functions, 375 loca~ 181, 183,290,296 local, evaluation of, 275 of polynomials, 148,215 optional, 245, 247 shell, 11

Variables, 143,148,215 Vector fields, 190, 195 vectorQ, 153,157,240-241 Vectors, 158

cross product of, 152 indexed, 140, 142 inner products of, 163 multiplications of, 160 outer products of, 160, 163 scalar products of, 160 testing for, 157,241

VerifyConvergence,383 Version, 5

View point, 120 ViewCenter, 111 ViewPoint, 110--111,120 ViewVertical,l11 VMS, 1

w Warnings, 323 Which, 275, 278, 287,331 While, 280,288, 331-332 Why the Beep?, 14 Width of plots, 135 Wildcard characters, 15-16 Windows, 3 Wintner's potential, 89 with, 292-293,296 Wolfram, Stephen, 14 Word, 136 Word separators in ReadList, 136 Words

empty, in ReadList, 136 token, in ReadList, 137

wordSeparators,136-137 WorkingPrecision, 379, 381 Write, 388-389 WriteString,388-389 WynnDegree, 383

x X, 104 XlI,3 xhost,104

y

Y,89 Yellow, 183

z Zero

lines, 113 Newton's method, 282

Zeta, 213

429