STEPHEN WOLFRAM THE B O O K T H E D IT IO N T H E D IT IO N Published by Wolfram Media LATEX filename frontmatter tex 4 20 p m July 1, 2003 iv Mathematica Version 5 0 for Linux (June 15, 2003) Library of Congress Cataloging in Publication Data Wolfram, Stephen, 1959 – Mathematica book Stephen Wolfram — 5th ed p cm Includes index ISBN 1–57955–022–3 (hardbound) 1 Mathematica (Computer file) 2 Mathematics—Data processing I Title QA76 95 W65 2003 510 2855369—dc21 03–53794 CIP Comments on this bo.
STEPHEN WOLFRAM Published by Wolfram Media TH EDITION BOOK THE Library of Congress Cataloging in Publication Data Wolfram, Stephen, 1959 – Mathematica book / Stephen Wolfram — 5th ed p cm Includes index ISBN 1–57955–022–3 (hardbound) Mathematica (Computer file) Mathematics—Data processing I Title QA76.95.W65 2003 510'.285'5369—dc21 03–53794 CIP Comments on this book will be welcomed at: comments@wolfram.com In publications that refer to the Mathematica system, please cite this book as: Stephen Wolfram, The Mathematica Book, 5th ed (Wolfram Media, 2003) First and second editions published by Addison Wesley Publishing Company under the title Mathematica: A System for Doing Mathematics by Computer Third and fourth editions co published by Wolfram Media and Cambridge University Press Published by Wolfram Media, Inc Copyright c 1988, 1991, 1996, 1999, 2003 by Wolfram Research, Inc All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the copyright holder Wolfram Research is the holder of the copyright to the Mathematica software system described in this book, including without limitation such aspects of the system as its code, structure, sequence, organization, “look and feel”, programming language and compilation of command names Use of the system unless pursuant to the terms of a license granted by Wolfram Research or as otherwise authorized by law is an infringement of the copyright The author, Wolfram Research, Inc and Wolfram Media, Inc make no representations, express or implied, with respect to this documentation or the software it describes, including without limitations, any implied warranties of merchantability, interoperability or fitness for a particular purpose, all of which are expressly disclaimed Users should be aware that included in the terms and conditions under which Wolfram Research is willing to license Mathematica is a provision that the author, Wolfram Research, Wolfram Media, and their distribution licensees, distributors and dealers shall in no event be liable for any indirect, incidental or consequential damages, and that liability for direct damages shall be limited to the amount of the purchase price paid for Mathematica In addition to the foregoing, users should recognize that all complex software systems and their documentation contain errors and omissions The author, Wolfram Research and Wolfram Media shall not be responsible under any circumstances for providing information on or corrections to errors and omissions discovered at any time in this book or the software it describes, whether or not they are aware of the errors or omissions The author, Wolfram Research and Wolfram Media not recommend the use of the software described in this book for applications in which errors or omissions could threaten life, injury or significant loss Mathematica, MathLink and MathSource are registered trademarks of Wolfram Research J/Link, MathLM, MathReader, NET/Link, Notebooks and webMathematica are trademarks of Wolfram Research All other trademarks used are the property of their respective owners Mathematica is not associated with Mathematica Policy Research, Inc or MathTech, Inc Printed in the United States of America Acid free paper 15 14 13 12 11 10 Author’s website: Other books by Stephen Wolfram: www.stephenwolfram.com Cellular Automata and Complexity: Collected Papers (1993) Author’s address: A New Kind of Science (2002) email: s.wolfram@wolfram.com mail: c/o Wolfram Research, Inc 100 Trade Center Drive Champaign, IL 61820, USA www.wolfram.com vii About the Author Stephen Wolfram is the creator of Mathematica, and a wellknown scientist He is widely regarded as the most important Following his scientific work on complex systems research, Wolfram in 1986 founded the first research center and first innovator in technical computing today, as well as one of the journal in the field Then, after a highly successful career in world’s most original research scientists academia—first at Caltech, then at the Institute for Advanced Study in Princeton, and finally as Professor of Physics, Math- Born in London in 1959, he was educated at Eton, Oxford and ematics and Computer Science at the University of Illinois— Caltech He published his first scientific paper at the age of fifteen, and had received his PhD in theoretical physics from Caltech by Wolfram launched Wolfram Research, Inc the age of twenty Wolfram’s early scientific work was mainly Wolfram began the development of Mathematica in late 1986 in high energy physics, quantum field theory and cosmology, and included several now classic results Having started to use The first version of Mathematica was released on June 23, 1988, and was immediately hailed as a major advance in com- computers in 1973, Wolfram rapidly became a leader in the emerging field of scientific computing, and in 1979 he began puting In the years that followed, the popularity of Mathematica grew rapidly, and Wolfram Research became established the construction of SMP—the first modern computer algebra as a world leader in the software industry, widely recognized system—which he released commercially in 1981 for excellence in both technology and business Wolfram has been president and CEO of Wolfram Research since its incep- In recognition of his early work in physics and computing, tion, and continues to be personally responsible for the overall Wolfram became in 1981 the youngest recipient of a MacArthur Prize Fellowship Late in 1981, Wolfram then set out design of its core technology on an ambitious new direction in science: to develop a gen- Following the release of Mathematica Version in 1991, eral theory of complexity in nature Wolfram’s key idea was to use computer experiments to study the behavior of simple Wolfram began to divide his time between Mathematica development and scientific research Building on his work computer programs known as cellular automata And in 1982 he made the first in a series of startling discoveries about the from the mid 1980s, and now with Mathematica as a tool, Wolfram made a rapid succession of major new discoveries By origins of complexity The publication of Wolfram’s papers on the mid 1990s his discoveries led him to develop a fundamentally cellular automata led to a major shift in scientific thinking, and laid the groundwork for a new field of science that Wolfram new conceptual framework, which he then spent the remainder of the 1990s applying not only to new kinds of questions, but named “complex systems research” also to many existing foundational problems in physics, biology, computer science, mathematics and several other fields Through the mid 1980s, Wolfram continued his work on complexity, discovering a number of fundamental connections After more than ten years of highly concentrated work, between computation and nature, and inventing such concepts as computational irreducibility Wolfram’s work led to a Wolfram finally described his achievements in his 1200 page book A New Kind of Science Released on May 14, 2002, the wide range of applications—and provided the main scientific foundations for the popular movements known as complexity book was widely acclaimed and immediately became a bestseller Its publication has been seen as initiating a paradigm theory and artificial life Wolfram himself used his ideas to shift of historic importance in science develop a new randomness generation system and a new approach to computational fluid dynamics—both of which are In addition to leading Wolfram Research to break new ground now in widespread use with innovative technology, Wolfram is now developing a series of research and educational initiatives in the science he has created ix About Mathematica Mathematica is the world’s only fully integrated environment for technical computing First released in 1988, it has had a profound effect on the way computers are used in many technical and other fields It is often said that the release of Mathematica marked the beginning of modern technical computing Ever since the 1960s individual packages had existed for specific numerical, algebraic, graphical and other tasks But the visionary concept of Mathematica was to create once and for all a single system that could handle all the various aspects of technical computing in a coherent and unified way The key intellectual advance that made this possible was the invention of a new kind of symbolic computer language that could for the first time manipulate the very wide range of objects involved in technical computing using only a fairly small number of basic primitives When Mathematica Version was released, the New York Times wrote that “the importance of the program cannot be overlooked”, and Business Week later ranked Mathematica among the ten most important new products of the year Mathematica was also hailed in the technical community as a major intellectual and practical revolution At first, Mathematica’s impact was felt mainly in the physical sciences, engineering and mathematics But over the years, Mathematica has become important in a remarkably wide range of fields Mathematica is used today throughout the sciences—physical, biological, social and other—and counts many of the world’s foremost scientists among its enthusiastic supporters It has played a crucial role in many important discoveries, and has been the basis for thousands of technical papers In engineering, Mathematica has become a standard tool for both development and production, and by now many of the world’s important new products rely at one stage or another in their design on Mathematica In commerce, Mathematica has played a significant role in the growth of sophisticated financial modeling, as well as being widely used in many kinds of general planning and analysis Mathematica has also emerged as an important tool in computer science and software development: its language component is widely used as a research, prototyping and interface environment The largest part of Mathematica’s user community consists of technical professionals But Mathematica is also heavily used in education, and there are now many hundreds of courses— from high school to graduate school—based on it In addition, with the availability of student versions, Mathematica has become an important tool for both technical and non technical students around the world The diversity of Mathematica’s user base is striking It spans all continents, ages from below ten up, and includes for example artists, composers, linguists and lawyers There are also many hobbyists from all walks of life who use Mathematica to further their interests in science, mathematics and computing Ever since Mathematica was first released, its user base has grown steadily, and by now the total number of users is above a million Mathematica has become a standard in a great many organizations, and it is used today in all of the Fortune 50 companies, all of the 15 major departments of the U.S government, and all of the 50 largest universities in the world At a technical level, Mathematica is widely regarded as a major feat of software engineering It is one of the largest single application programs ever developed, and it contains a vast array of novel algorithms and important technical innovations Among its core innovations are its interconnected algorithm knowledge base, and its concepts of symbolic programming and of document centered interfaces The development of Mathematica has been carried out at Wolfram Research by a world class team led by Stephen Wolfram The success of Mathematica has fueled the continuing growth of Wolfram Research, and has allowed a large community of independent Mathematica related businesses to develop There are today well over a hundred specialized commercial packages available for Mathematica, as well as more than three hundred books devoted to the system x Features New in Mathematica Version Mathematica Version introduces important extensions to the Mathematica system, especially in scope and scalability of numeric and symbolic computation Building on the core language and extensive algorithm knowledge base of Mathematica, Version introduces a new generation of advanced algorithms for a wide range of numeric and symbolic operations Symbolic Computation Solutions to mixed systems of equations and inequalities in Reduce Complete solving of polynomial systems over real or complex numbers Solving large classes of Diophantine equations Numerical Computation Major optimization of dense numerical linear algebra New optimized sparse numerical linear algebra Support for optimized arbitrary precision linear algebra Generalized eigenvalues and singular value decomposition LinearSolveFunction for repeated linear system solving ForAll and Exists quantifiers and quantifier elimination Representation of discrete and continuous algebraic and transcendental solution sets FindInstance for finding instances of solutions over different domains Exact constrained minimization over real and integer domains p norms for vectors and matrices Integrated support for assumptions using Assuming and Refine Built in MatrixRank for exact and approximate matrices RSolve for solving recurrence equations Support for large scale linear programming, with interior point methods Support for nonlinear, partial and q difference equations and systems New methods and array variable support in FindRoot and FindMinimum Full solutions to systems of rational ordinary differential equations FindFit for full nonlinear curve fitting Support for differential algebraic equations Constrained global optimization with NMinimize CoefficientArrays for converting systems of equations to tensors Support for n dimensional PDEs in NDSolve Support for differential algebraic equations in NDSolve Support for vector and array valued functions in NDSolve Programming and Core System Highly extensive collection of automatically accessible algorithms in NDSolve Integrated language support for sparse arrays Finer precision and accuracy control for arbitrary precision numbers EvaluationMonitor and StepMonitor for algorithm monitoring Higher efficiency big number arithmetic, including processor specific optimization Enhanced timing measurement, including AbsoluteTiming Enhanced algorithms for number theoretical operations including GCD and FactorInteger Optimization for 64 bit operating systems and architectures Direct support for high performance basic statistics functions New list programming with Sow and Reap Major performance enhancements for MathLink Support for computations in full 64 bit address spaces xi Interfaces Notebook Interface Support for more than 50 import and export formats Enhanced Help Browser design High efficiency import and export of tabular data Automatic copy/paste switching for Windows PNG, SVG and DICOM graphics and imaging formats Enhanced support for slide show presentation Import and export of sparse matrix formats AuthorTools support for notebook diffs MPS linear programming format Cascading style sheets and XHTML for notebook exporting Preview version of NET/Link for integration with NET Standard Add on Packages Statistical plots and graphics Algebraic number fields New in Versions 4.1 and 4.2 Enhanced pattern matching of sequence objects High efficiency CellularAutomaton function Enhanced optimizer for built in Mathematica compiler J/Link MathLink based Java capabilities Enhanced continued fraction computation MathMLForm and extended MathML support Greatly enhanced DSolve Extended simplification of Floor, Erf, ProductLog and related functions Additional TraditionalForm formats Efficiency increases for multivariate polynomial operations Support for import and export of DXF, STL, FITS and STDS data formats Full support for CSV format import and export Support for UTF character encodings Extensive support for XML, including SymbolicXML subsystem and NotebookML Native support for evaluation and formatting of Nand and Nor Integration over regions defined by inequalities Integration of piecewise functions Standard package for visualization of regions defined by inequalities ANOVA standard add on package Enhanced Combinatorica add on package AuthorTools notebook authoring environment xii The Role of This Book The Scope of the Book times the definitions may actually modify the behavior of This book is intended to be a complete introduction to Mathematica It describes essentially all the capabilities of Mathematica, functions described in this book In other cases, the definitions may simply add a collection of new functions that are not and assumes no prior knowledge of the system described in the book In certain applications, it may be primarily In most uses of Mathematica, you will need to know only these new functions that you use, rather than the standard ones described in the book a small part of the system This book is organized to make it easy for you to learn the part you need for a particular calculation In many cases, for example, you may be able to This book describes what to when you interact directly with the standard Mathematica kernel and notebook front set up your calculation simply by adapting some appropriate examples from the book end Sometimes, however, you may not be using the standard Mathematica system directly Instead, Mathematica may be an embedded component of another system that you are You should understand, however, that the examples in this book are chosen primarily for their simplicity, rather than to using This system may for example call on Mathematica only for certain computations, and may hide the details of those correspond to realistic calculations in particular application areas computations from you Most of what is in this book will There are many other publications that discuss Mathematica from the viewpoint of particular classes of applications In some only be useful if you can give explicit input to Mathematica If all of your input is substantially modified by the system you cases, you may find it better to read one of these publications first, and read this book only when you need a more general are using, then you must rely on the documentation for that system perspective on Mathematica Additional Mathematica Documentation Mathematica is a system built on a fairly small set of very powerful principles This book describes those principles, but by no means spells out all of their implications In particular, while the book describes the elements that go into Mathematica programs, it does not give detailed examples of complete programs For those, you should look at other publications For all standard versions of Mathematica, the following is available in printed form, and can be ordered from Wolfram Research: Getting Started with Mathematica: a booklet describing installation, basic operation, and troubleshooting of Mathematica on specific computer systems The Mathematica System Described in the Book This book describes the standard Mathematica kernel, as it Extensive online documentation is included with most versions of Mathematica All such documentation can be accessed from exists on all computers that run Mathematica Most major the Help Browser in the Mathematica notebook front end supported features of the kernel in Mathematica Version are covered in this book Many of the important features of the In addition, the following sources of information are available on the web: front end are also discussed www.wolfram.com: the main Wolfram Research website Mathematica is an open software system that can be customized documents.wolfram.com: full documentation for Mathematica in a wide variety of ways It is important to realize that this book library.wolfram.com/infocenter: the Mathematica Information Center—a central web repository for information on Mathe- covers only the full basic Mathematica system If your system is customized in some way, then it may behave differently from what is described in the book The most common form of customization is the addition of various Mathematica function definitions These may come, for example, from loading a Mathematica package Some- matica and its applications 1450 Record, MathLink stream, MLCreateMark(), 693, 1341 of input, $Echo, 705, 1328 of session, 703 Record, 646, 1259 RecordLists, 644, 646, 1259 Records, null, NullRecords, 648, 1227 searching, FindList, 651, 1147 RecordSeparators, 646, 647, 648, 651, 1259 Recoupling coefficients, 760 Rectangle, 492, 516, 1260 Rectangular form, 813 Rectangular parallelepiped, Cuboid, 524, 1116 Recurrence equations, RSolve, 96, 891, 1269 Recurring decimals, RealDigits, 755, 1258 Recursion, 309, 315, 714 infinite, 369 tracing of, Trace, 359, 1310 Recursion equations, RSolve, 96, 891, 1269 $RecursionLimit, 369, 390, 715, 1048, 1337 Recursive functions, 316 Red, 500 Red variable names, 448 Redirection, of input and output, 204 of interactive output, 705 Redisplay graphics, Show, 139, 487, 1280 Redoing calculations, 48 Redraw plot, Show, 139, 487, 1280 Reduce, 90, 92, 830, 835, 839, 841, 843, 845, 1070, 1261 implementation of, 1070 Reduce in APL, Apply (@@), 243, 1081 Reducing expressions to standard form, 325 Reducing inequalities, Reduce, 835, 1261 Reducing radicals, RootReduce, 826, 1266 Reducing space, 454 Reduction, modular of polynomials, PolynomialMod, 804, 1246 of complex powers, ExpToTrig, 812, 1141 of trigonometric functions, TrigReduce, 71, 811, 1314 Reduction formulas, FunctionExpand, 792, 1159 Redundant linear equations, NullSpace, 910, 1227 Reference counts, 1066 References, iv, 1066 in hyperlinks, 201 Referring to symbols by name, 402 Refine, 815, 816, 1261 Reflection formula, for gamma function, FullSimplify, 792, 1159 Reflectivity of surfaces, 546, 547 Region, Interval, 740, 1184 Polygon, 492, 1245 Regions, and inequalities, Reduce, 835, 1261 defined by inequalities, 836 Register link, LinkCreate, 680, 1197 \[RegisteredTrademark] ( ), 996, 1389 Registers, in compiled code, 377 Regression analysis, 109 Record — ReturnExpressionPacket Regular expressions, 58, 411 Reinterpolation, FunctionInterpolation, 935, 1160 Rejection of matches, 266 Relational operators, 86 input of, 1033 mixing of, 86 Relations, as equations, 84 involving special functions, FullSimplify, 792, 1159 Relative coordinates, 505, 531 Relative error, Precision, 727, 1249 Relative positions, of graphics objects, Offset, 507, 1230 Relative primality, GCD, 749, 1161 Release (Version function), see Evaluate, 1402 ReleaseHold, 339, 369, 1261 $ReleaseNumber, 717, 1337 Releasing held arguments, Evaluate, 1047, 1136 Remainder, in polynomial division, PolynomialRemainder, 803, 1247 Mod, 31, 749, 1213 Remembering previously computed values, 314 Remote computers, 682, 690 Remove, 59, 395, 397, 403, 404, 1052, 1261 Removing assignments, Unset (=.), 304, 1316 Removing elements from lists, Delete, 125, 288, 1119 Drop, 123, 287, 1128 Removing elements that match a pattern, DeleteCases, 262, 1120 Removing files, DeleteFile, 641, 1120 Removing functions, Clear, 110, 1103 Removing objects, 1052 Removing sublists, Sequence, 258, 1275 Removing symbols, Clear, 304, 1103 Removing values, Unset (=.), 39, 1316 RenameDirectory, 641, 1262 RenameFile, 641, 1262 Renaming, of variables, 385 RenderAll, 555, 556, 1262 Rendering, low-level, 553 order of graphics, 494 suppressing for graphics, 491 Reordering, of rules, 310 of rules explicitly, 323 Reordering terms, Collect, 71, 1106 Repeat, Do, 112, 348, 1127 Repeated ( ), 277, 1028, 1262 Repeated rule application, ReplaceRepeated (//.), 300, 1263 RepeatedNull ( ), 277, 1028, 1262 Repeating expressions, Table, 116, 1299 Repetitive operations, 112, 241 Replace, 301, 1262 ReplaceAll (/.), 64, 65, 299, 1029, 1263 ReplaceList, 263, 271, 274, 302, 1263 Index Replacements, compared with assignments, 65, 66 creating functions to perform, 312 in programs, With, 380, 1323 of parts, ReplacePart, 125, 288, 1263 of parts without evaluation, ReplacePart, 339, 1263 of substrings, StringReplace, 410, 1292 Rule (->), 64, 1269 ReplacePart, 125, 235, 288, 339, 1263 ReplaceRepeated (//.), 300, 1029, 1263 Replaying input, 48, 702 Replaying sound, Show, 173, 1280 Replicating expressions, Table, 116, 1299 Report style definitions, 602 Representation, continued fraction, ContinuedFraction, 754, 1112 internal, 220 Representations, for expressions, 425 Reproducing examples from book, xv Requesting input, Input, 478, 1178 Resampling, FunctionInterpolation, 935, 1160 Reserved memory, in MathLink, 674 Reset medium, SetOptions, 635, 1278 ResetDirectory, 636, 1264 ResetMedium (Version function), see SetOptions, 1402 Resetting, symbols, 404 Resetting parts of lists, 42 Reshape in APL, Partition, 128, 292, 1240 Residuals, minimization of, FindFit, 929, 1146 Residue, Mod, 749, 1213 Residue, 895, 1264 Resizing, constraints on, AspectRatioFixed, 616, 1084 Resizing graphics, 51 Resolution, of graphics, ImageResolution, 569, 1174 Resolve, 848, 1264 implementation of, 1070 Resource constraints, 712 Rest, of arguments in pure functions, SlotSequence (##), 250, 1285 Rest, 123, 1264 Restriction, of patterns, Condition (/;), 265, 1111 \[VerticalSeparator] ( ), 1001 Restrictions on transformation rules, Condition (/;), 265, 1111 Resultant, 803, 805, 1264 Results, in notebooks, 38 previous, Out (%), 38, 1234 tracing generation of, 362 Ê ØÙƯỊ, 48 Return, 353, 354, 366, 707, 1265 Return characters, 1037 in MS-DOS, DOSTextFormat, 1054 Return to directory, ResetDirectory, 636, 1264 ReturnExpressionPacket, 684 Index \[ReturnIndicator] ( ), 1009, 1389 \[ReturnKey] ( ), 1009, 1389 ReturnPacket, 685, 700 Returns, non-local, Throw, 354, 1304 soft, 460, 611 ReturnTextPacket, 683, 700 Reverse, strings, StringReverse, 407, 1293 Reverse, 127, 236, 1265 \[ReverseDoublePrime] ( ), 996, 1389 \[ReverseElement] ( ), 1004, 1389 \[ReverseEquilibrium] ( ), 1007, 1389 \[ReversePrime] ( ), 996, 1389 \[ReverseUpEquilibrium] ( ), 1007, 1390 Reversion of power series, InverseSeries, 887, 1188 Reviewing results, 38, 48 Revisions, in current version, x since earlier editions, 1402 Rewinding, after MathLink errors, 697 Rewrite rules, ReplaceList, 302, 1263 Rewriting, of strings, StringReplace, 410, 1292 Rewriting algebraic expressions, 67 Rewriting rule (transformation rules), 299 \rfloor (TEX), \[RightFloor] ( ), 1002 RGBColor, 499, 563, 1265 \[Rho] ( Ρ ), 175, 990, 1390 Riccati equations, DSolve, 873, 1129 Riemann hypothesis, 961 Riemann integrals, 867 Riemann P function, Hypergeometric2F1, 780, 1172 Riemann sheets, 762 Riemann zeta function, Zeta, 772, 1324 Riemann-Siegel formula, Zeta, 1068, 1324 Riemann-Siegel functions, 772 RiemannSiegelTheta, 772, 1265 RiemannSiegelZ, 772, 1265 Right, 442, 450 \right (TEX), 456 Right aligned, TextAlignment, 610, 1303 Right justification, TextJustification, 610, 1303 Right shift, RotateRight, 127, 291, 1267 Right-handed coordinate system, 531 \[RightAngle] ( ), 996, 1390 \[RightAngleBracket] ( ), 1002, 1390 \[RightArrow] ( ), 985, 1006, 1390 \Rightarrow (TEX), \[DoubleRightArrow] ( ), 1006 \rightarrow (TEX), \[RightArrow] ( ), 1006 \[RightArrowBar] ( ), 1006, 1390 \[RightArrowLeftArrow] ( ), 1006, 1390 \[RightBracketingBar] ( ), 1002, 1390 \[RightCeiling] ( ), 1002, 1390 \[RightDoubleBracket] ( ), 1002, 1390 \[RightDoubleBracketingBar] ( ), 1002, 1391 \[RightDownTeeVector] ( ), 1007, 1391 \[RightDownVector] ( ), 1007, 1391 \[RightDownVectorBar] ( ), 1007, 1391 \[ReturnIndicator] — Rules \[RightFloor] ( ), 1002, 1391 \[RightGuillemet] ( » ), 996, 1391 \rightharpoonup (TEX), \[RightVector] ( ), 1007 \[RightModified] ( ), 1009, 1391 \[RightSkeleton] ( ), 1009, 1391 \[RightTee] ( ), 191, 1001, 1007, 1391 \[RightTeeArrow] ( ), 1006, 1391 \[RightTeeVector] ( ), 1007, 1392 \[RightTriangle] ( ), 191, 1005, 1392 \[RightTriangleBar] ( ), 1005, 1392 \[RightTriangleEqual] ( ), 1005, 1392 \[RightUpDownVector] ( ), 1007, 1392 \[RightUpTeeVector] ( ), 1007, 1392 \[RightUpVector] ( ), 1007, 1392 \[RightUpVectorBar] ( ), 1007, 1392 \[RightVector] ( ), 1007, 1392 \[RightVectorBar] ( ), 1007, 1392 Rigid selection, StructuredSelection, 615, 1293 Ring, \[SmallCircle] ( ), 1002 Ring functions, LegendreP, 778, 1192 Rings, 73, 817, 839 Risch algorithm, Integrate, 862, 1070, 1182 Rising factorial, Pochhammer, 770, 1244 \rm (TEX), FontWeight, 444, 612, 1152 rm Unix command, DeleteFile, 641, 1120 rmdir command, DeleteDirectory, 641, 1120 Robust statistics, 924 Roman fonts, 444, 558, 612 character encodings, 421 Root, primitive, MultiplicativeOrder, 752, 1214 Root, 88, 821, 1070, 1265 Root finding, complex, 961 numerical, FindRoot, 104, 960, 1149 starting points for, 961 RootReduce, 826, 1266 Roots, choice of, 763 expansion of, PowerExpand, 798, 1248 field extensions by, 809 formatting of, RadicalBox, 445, 1254 input of, 178 number of in polynomial equations, 823 simplification of, PowerExpand, 72, 1248 simplification of, Simplify, 72, 1282 simplifying, RootReduce, 826, 1266 square, Sqrt, 31, 1288 Roots, 89, 819, 1266 Roots of complex numbers, 763 Roots of equations, Solve, 87, 820, 1285 Roots of negative numbers, 763 Roots of polynomials, numerical, NSolve, 104, 959, 1226 RootSum, 827, 862, 1266 Rot, Curl, 97 RotateLabel, 514, 1266 RotateLeft, 127, 130, 291, 941, 1266 RotateRight, 127, 130, 291, 1267 1451 Rotation, of three-dimensional coordinates, SphericalRegion, 536, 1287 of three-dimensional object, ViewVertical, 534, 1319 Rotation group, 768 coupling coefficients in, 760 Rotational coupling coefficients, 760 Round, 31, 745, 1267 Round brackets, \[RawLeftParenthesis] ( ( ), 1010 Roundabout indicator, \[CloverLeaf] ( ), 1009 \[RoundImplies] ( ), 1001, 1006, 1392 Rounding, 741 Roundoff errors, 33, 731 Chop, 730, 1103 \[RoundSpaceIndicator] ( ), 1009, 1393 Routines, 110 compilation of, Compile, 376, 1109 exiting, Return, 354, 1265 MathLink library, 658 Row, 441 Row form, SequenceForm, 434, 1275 Row-major output order, 441 RowAlignments, 449, 1267 RowBox, 445, 1267 RowLines, 446, 449, 1268 RowMinHeight, 449, 1268 RowReduce, 907, 1268 implementation of, 1069 Rows, assigning values to, 899 of matrices, Part, 118, 1238 resetting, 126, 286 RowsEqual, 449, 1268 RowSpacings, 449, 1268 RPC, 690 \rq (TEX), \[RawQuote] ( ’ ), 1010 RSolve, 96, 891, 892, 893, 1071, 1269 implementation of, 1071 Rule (->), 64, 299, 314, 1029, 1052, 1269 implementation of, 1066 scoping in, 386 \[Rule] ( ), 182, 985, 1006, 1393 Rule 30, CellularAutomaton, 942, 1101 Random, 1067, 1254 Rule numbers, CellularAutomaton, 944, 1101 RuleDelayed (:>), 299, 314, 341, 1029, 1052, 1269 \[RuleDelayed] ( ), 1006, 1393 Ruler function, IntegerExponent, 725, 749, 1181 RulerBar toolbar element, 622 Rules, 65 applying, ReplaceAll (/.), 299, 1263 applying to whole expressions, Replace, 301, 1262 automatic application of, 303 conditions in, 345 delayed, RuleDelayed (:>), 314, 341, 1269 evaluation in, 340 1452 explicit reordering of, 323 generated by Solve, 88 immediate, Rule (->), 314, 1269 immediate and delayed, 1052 in sparse arrays, 297 lists of, 65, 299, 322 naming sets of, 66, 302 ordering of, 310 organization of, 316 patterns in, 259 preventing evaluation in, RuleDelayed (:>), 314, 1269 putting constraints on, Condition (/;), 265, 1111 reordering of, 310 scanning lists of, 302 scoping of, 385 tracing of, Trace, 360, 1310 transformation, 64, 299 use of in evaluation, 325 variables in, 385 Rulings, on surfaces, Mesh, 151, 539, 1210 Run, 629, 1269 Run length encoding, Split, 128, 292, 1288 Run-length encoding, Split, 292, 1288 Run-time, TimeUsed, 710, 1306 Runge-Kutta methods, 961 NDSolve, 979, 1216 Runge’s phenomenon, 933 Running external programs, Install, 659, 1180 Running Mathematica, 26 Runs, uniqueness of symbols between different, 383 Runs of arguments, Sequence, 258, 1275 RunThrough, 630, 1270 \S (TEX), \[Section] ( ), 996 \[SadSmiley] ( ), 995, 1393 Same expressions, testing for, Equal (==), 85, 1135 SameQ (===), 268, 346, 347, 1028, 1270 \[Sampi] ( ), 990, 1393 Sample points, for inequalities, FindInstance, 838, 1147 Sampled sounds, 172 SampleDepth, 566, 1270 SampledSoundFunction, 566, 1270 SampledSoundList, 566, 1270 SampleRate, 172, 1270 Sampling, of Mathematica, Sampling in plots, PlotPoints, 137, 1243 Sandwich naming scheme, for executables, 677 SAT problem, FindInstance, 845, 1147 Satisfiability, FindInstance, 845, 1147 of inequalities, FindInstance, 838, 1147 Satisfies, \[DoubleRightTee] ( ), 1001 Satisfying criterion, picking elements, Select, 251, 1273 Saturation, of colors, 500 Rules — SelectedNotebook Save, automatic for notebooks, NotebookAutoSave, 618, 1222 Save, 204, 624, 625, 1271 Save notebook, NotebookSave, 591, 1225 Saving, definitions, Save, 204, 624, 1271 dependencies, 625 expressions, Put (>>), 624, 1253 intermediate values, 315 local variables, 383 the complete state of Mathematica, DumpSave, 627, 1129 \sb (TEX), SubscriptBox, 445, 1295 Scalable Vector Graphics, exporting, Export, 568, 1141 Scalar multiplication, 121 Scalar product, Dot (.), 118, 119, 901, 1127 Scalars, 118, 899 Scaled, 505, 531, 1271 Scaled coordinates, 505, 531 Scales, on axes, Ticks, 134, 512, 552, 1305 Scaling, of color values, ColorFunctionScaling, 517, 1106 Scan, 247, 1271 and Throw, 350 levels in, 1041 Scatter plot, ListPlot, 158, 1202 Schrodinger equation, in periodic potential, MathieuS, 789, 1207 SchurDecomposition, 915, 1271 implementation of, 1069 Scientific notation, 29, 34, 729, 1021 ScientificForm, 435, 1272 ScientificForm, 435, 1272 Scope, of assumptions, Assuming, 818, 1084 of Throw, 351 Scoping, 112 dynamic, Block, 391, 1091 in pure functions, 385 lexical, Module, 391, 1213 of graphics directives, 499 of names, 378, 392 of pattern variables, 263 of variables, 378 Scoping constructs, 386 tracing, 364 Scratch files, 629 Scratch mark (#), Slot, 249, 1284 Scream, \[FreakedSmiley] ( ), 995 Screen density, GrayLevel, 499, 1165 ImageResolution, 569, 1174 Screen entry, 176 Screen sizes, 620 ScreenStyleEnvironment, 197, 602, 618, 1272 Script boxes, 445 Script files, 38, 623 ScriptA, 992 \[ScriptA] ( ), 993, 1393 ScriptBaselineShifts, 457, 1272 \[ScriptCapitalA] ( ), 993, 1393 \[ScriptCapitalE] ( ), 192, 992 \[ScriptCapitalH] ( ), 992 Index \[ScriptCapitalL] ( ), 992 \[ScriptCapitalZ] ( ), 993, 1393 \[ScriptL] ( ), 192, 992 ScriptMinSize, 457, 1272 Scripts, for front end, FrontEndTokenExecute, 593 Scripts (packages), 59 \scriptscriptstyle (TEX), ScriptSizeMultipliers, 457, 1272 ScriptSizeMultipliers, 457, 1272 \scriptstyle (TEX), ScriptSizeMultipliers, 457, 1272 \[ScriptZ] ( ), 993, 1393 Scroll notebook, SelectionMove, 582, 1275 Scrolling back, 38 sdb, 691 SDTS importing, Import, 208, 1176 Seams between polygons in three dimensions, EdgeForm, 528, 1130 Search, for files, FileNames, 638, 1145 for parts, Position, 124, 261, 1247 for substrings, StringPosition, 409, 1292 in notebooks, NotebookFind, 584, 1223 Search path, context, $ContextPath, 394, 1327 for files, $Path, 206, 637, 1335 Searching, for sublists, ReplaceList, 263, 302, 1263 in files, FindList, 207, 650, 1147 in strings, 654 IntervalMemberQ, 741, 1184 \searrow (TEX), \[LowerRightArrow] ( ), 1006 Sec, 761, 1273 Secant method, FindRoot, 1068, 1149 Sech, 761, 1273 Second, Date, 709, 1117 Timing, 711, 1306 Seconds, \[DoublePrime] ( ), 996 Section, Module, 378, 1213 \[Section] ( ), 996, 1393 Section style, 573 Sections, in notebooks, 51 numbering of, 202 Security, Protect, 321, 1044, 1252 ReadProtected, 330, 1258 Seeding of random generator, SeedRandom, 748, 1273 SeedRandom, 747, 748, 1273 Seek, in files, SetStreamPosition, 653, 1279 in MathLink, MLSeekMark(), 693, 1350 in notebooks, NotebookFind, 584, 1223 Segments, of circles, Disk, 496, 1125 Select, 251, 261, 284, 1273 Selectability, of windows, WindowClickSelect, 620, 1321 Selectable, 448, 607, 619, 1273 Selected notebook, setting, SetSelectedNotebook, 591, 1279 SelectedNotebook, 579, 1273 Index Selecting sublists — Signalling a process Selecting sublists, ReplaceList, 263, 302, 1263 Selection, highlighting, ShowSelection, 619, 1281 in notebooks, 581 in two-dimensional expressions, 180 SelectionAnimate, 588, 617, 1274 SelectionCreateCell, 588, 1274 SelectionEvaluate, 588, 1274 SelectionEvaluateCreateCell, 588, 1274 SelectionMove, 582, 1275 \[SelectionPlaceholder] ( ), 199, 587, 1008, 1393 Selector, part, Part, 117, 1238 Self-intersection, of polygons, 495 Self-similar, DigitCount, 756, 1123 Semi-algebraic sets, Reduce, 845, 1261 Semi-inverse, PseudoInverse, 914, 1252 Semi-log plots, LogPlot, 168 Semicolon, at end of line, 43, 75 CompoundExpression (;), 43, 111, 1029, 1111 Sending, notebooks to the front end, NotebookPut, 578, 1224 Sensitive dependence on initial conditions, 979 Sentences, reading, 647 Separation, of colors for printing, 564 Separator, vertical, \[VerticalSeparator] ( ), 1001 Separators, for records, RecordSeparators, 646, 1259 for words, WordSeparators, 646, 1323 in number output, NumberSeparator, 436, 1229 in tables, RowLines, 446, 1268 Sequence, of boxes, RowBox, 445, 1267 Range, 119, 1255 Sequence, 258, 671, 1275 preventing use, SequenceHold, 340, 1276 Sequence equations, RSolve, 891, 1269 Sequence of calculations, in notebooks, 51 Sequence of elements, extracting from lists, Part, 41, 1238 Take, 123, 287, 1301 Sequence specifications, 408, 1040 SequenceForm, 434, 1275 SequenceHold, 329, 340, 1045, 1276 Sequences, in patterns, 1050 List, 40, 1199 of commands, 43 of letters, CharacterRange, 413, 417, 1102 summation of, Sum, 83, 890, 1296 Sequent, \[RightTee] ( ), 1001 Sequential reading from files, Read, 650, 1257 Sequential searching, Find, 652, 1146 Serial lines, 27 Serial number, for streams, 631 for symbols, 381 of computer, $MachineID, 718, 1331 Series, solving for coefficients in, SolveAlways, 833, 1285 summation of, Sum, 890, 1296 Series, 94, 883, 884, 1276 implementation of, 1071 Series expansions, Series, 94, 883, 1276 SeriesCoefficient, 889, 1276 SeriesData, 885, 1276 Server, Mathematica, 47 Service loop, MLMain(), 679, 1347 Session, saving a complete, DumpSave, 627, 1129 $SessionID, 384, 716, 1337 Sessions, 26, 1055 communicating between, 680 controlling in MathLink, 685 in book, xv main loop in, 702, 1056 terminating, 706 uniqueness of symbols between different, 383 SessionTime, 710, 1277 Set (=), 39, 66, 303, 305, 307, 311, 1029, 1051, 1277 scoping in, 386 Set difference, \[Backslash] ( ), 1002 Set operations, input of, 183, 1002 SetAccuracy, 736, 1277 SetAttributes, 271, 328, 1277 SetDelayed (:=), 110, 311, 1029, 1051, 1278 SetDirectory, 206, 636, 1278 SetFileDate, 641, 1278 \setminus (TEX), \[Backslash] ( ), 1002 SetOptions, 144, 581, 635, 1040, 1278 SetPrecision, 736, 1278 setrgbcolor PostScript command, 563 Sets, complements of, Complement, 127, 1109 difference between, Complement, 127, 1109 intersection of, Intersection, 127, 1184 lists as, 126 union of, Union, 127, 1315 Sets of numbers, 73, 817, 839 SetSelectedNotebook, 591, 1279 SetStreamPosition, 653, 655, 1279 Setting up Mathematica, 26 Setting values, Set (=), 39, 1277 \sf (TEX), FontFamily, 444, 612, 1151 SGML names for characters, 174, 982, 1353 \[SHacek] ( sˇ ), 998, 1393 Shading, Lighting, 526, 1194 of surfaces, 156, 542 Shading, 151, 155, 1279 Shading models, 544 Shadowing of names, 395 in packages, 59, 396 Shallow, 431, 432, 1279 Shape, of arrays, Dimensions, 120, 916, 1124 of plots, AspectRatio, 134, 509, 1084 1453 of three-dimensional plots, 531 of two-dimensional plots, AspectRatio, 163, 1084 Shape fixed, AspectRatioFixed, 616, 1084 Share, 714, 1279 share directory, $BaseDirectory, 637, 1325 Shared data, 1066 Shared styles, 603 \[Sharp] ( ), 996, 1393 Sharp (#), Slot, 249, 1284 Sheets, Riemann, 762 Sheffer stroke, Nand, 87, 1215 \[VerticalBar] ( ), 1005 Shell, Unix, 204 Shell command, Run, 629, 1269 Shell escapes, 629, 1038 Shell type, $OperatingSystem, 717, 1334 Shell variables, Environment, 716, 1134 Shift, in box, BoxBaselineShift, 455 RotateLeft, 127, 291, 1266 Shift JIS, 421 Shift map, 979 Ë Ø¹ ĨỊØƯĨй ỊØ Ư, 181 Ë Ø¹ ỊØ Ư, 49, 181 Shifting graphics objects, Offset, 507, 1230 Shiny surfaces, 547 Short, 75, 431, 1280 short C type, 678 Short characters, 998 Short names, conventions for, 1353 Shortcuts, user-defined, InputAutoReplacements, 613, 1179 Shortening lists, Delete, 125, 288, 1119 Shortest form, Simplify, 68, 813, 1282 \[ShortLeftArrow] ( ), 1006, 1394 \[ShortRightArrow] ( ), 1006, 1394 Show, 139, 151, 173, 487, 1280 options for, 134, 151, 489 Show Expression menu item, 427, 572, 600 Show Ruler menu item, 605 ShowAnimation, 170 ShowAutoStyles, 613, 1280 ShowCellBracket, 604, 1280 ShowCellLabel, 607, 1280 ShowCellTags, 607, 1280 ShowContents, 455 ShowCursorTracker, 613, 1072, 1281 Showing files, 204, 623 ShowPageBreaks, 577, 618, 1281 ShowSelection, 619, 1281 ShowSpecialCharacters, 613, 1281 ShowStringCharacters, 613, 1281 Shriek, Factorial (!), 31, 757, 1143 Shut notebook, NotebookClose, 591, 1222 Side effects, of Random, $RandomState, 748, 1336 Sigma, versus summation sign, 185 \[Sigma] ( Σ ), 175, 990, 1394 Sign, 745, 813, 1281 Signal processing, ListConvolve, 937, 1200 ZTransform, 879, 1325 Signalling a process, $ProcessID, 716, 1336 1454 Signals, Fourier transforms of, 936 sound, 171 Signature, 757, 760, 920, 1282 Significance arithmetic, 740 Significant figures, 33, 727 in output, 435 Signing off from Mathematica, Quit, 28, 1057, 1254 SignPadding, 436, 1282 Signum, Sign, 745, 1281 Silent operation, 43, 74 \sim (TEX), \[Tilde] ( ), 1003 \simeq (TEX), \[TildeEqual] ( ), 1003 Similarity, \[Tilde] ( ), 1003 Similarity transformation, Eigenvectors, 121, 910, 1131 JordanDecomposition, 915, 1190 SchurDecomposition, 915, 1271 Simplest forms, Simplify, 68, 1282 Simplex algorithm, LinearProgramming, 1068, 1195 Minimize, 1070, 1212 Simplicity, definition of, ComplexityFunction, 815, 1110 Simplification, adding your own rules for, 281 algebraic, 63 algebraic, Simplify, 68, 813, 1282 as interpretation of evaluation, 324 of Boolean expressions, LogicalExpand, 87, 1203 of powers, PowerExpand, 72, 1248 of special functions, FullSimplify, 792, 1159 of square roots, PowerExpand, 72, 1248 of square roots, Simplify, 72, 1282 Simplify, 68, 69, 72, 73, 813, 815, 1282 implementation of, 1070 Simplifying radicals, RootReduce, 826, 1266 Simulated annealing, NMaximize, 1219, 1220 Simultaneous equations, 89, 828 eliminating variables in, Eliminate, 832, 1132 ways to present, 829 Sin, 31, 761, 1282 manipulation of, 811 Since, \[Because] ( ), 1001 Sine integral, SinIntegral, 774, 1284 Sine integrals, nested, 864 Sine transform, FourierSinTransform, 878, 1154 SingleLetterItalics, 613, 989, 1283 Singular eigenvalue problem, Eigenvalues, 913, 1131 Singular matrices, 904 Singularities, in integrals, 866 in NDSolve, 967 in numerical integration, 955 in plots, 131, 136 normal form for, GroebnerBasis, 805, 1168 SingularityDepth, 956 Signals — Sounds SingularValueDecomposition, 914, 1283 implementation of, 1069 SingularValueList, 913, 1283 SingularValues, 1069 Sinh, 761, 1283 SinhIntegral, 774, 1284 SinIntegral, 774, 1284 SixJSymbol, 760, 1284 \[SixPointedStar] ( ), 995, 1394 Size, of expressions, ByteCount, 714, 1095 of files, FileByteCount, 641, 1145 of font, FontSize, 444, 612, 1151 of graphic, ImageSize, 569, 616, 1174 of graphics, 51 of list, Dimensions, 120, 1124 of lists in MathLink, 670 of Mathematica, 224 of plots, PlotRegion, 509, 1244 of points, PointSize, 500, 1245 of text in graphics, 558 of windows, WindowSize, 620, 1322 Size of computation, limits on, 75 MaxMemoryUsed, 713, 1209 Skeleton, 75, 1284 Skeleton output, Short, 75, 431, 1280 \[SkeletonIndicator] ( ), 1009, 1394 \skew (TEX), AdjustmentBox, 455, 1078 Skewness, 794 Skip, data in files not between separators, RecordSeparators, 648, 1259 line, LineSpacing, 611, 1196 Skip, 649, 650, 653, 1284 Skip packet, MLNewPacket(), 697, 1347 \sl (TEX), FontSlant, 444, 612, 1151 Slant, of font, FontSlant, 444, 612, 1151 Slash, double, 233 \[RawSlash] ( / ), 1010 Slash-dot (/.), ReplaceAll, 65, 299, 1263 Slash-semi (/;), Condition, 265, 1111 Slash-slash-dot (//.), ReplaceRepeated, 300, 1263 Slashed characters, AdjustmentBox, 455, 1078 Slater’s Theorem, Integrate, 1070, 1182 Slave processes, in MathLink, LinkLaunch, 683, 1197 Sleep, Pause, 710, 1241 Slices, of lists, 286 Slot (#), 249, 1024, 1030, 1284 SlotSequence (##), 249, 250, 1030, 1285 Small letters, 413 Small screens, styles for, ScreenStyleEnvironment, 197, 1272 \[SmallCircle] ( ), 191, 1002, 1394 Smaller, Less (>), 624, 1253 in notebooks, NotebookWrite, 585, 1225 WWW, 211 Wynn epsilon method, 958 X Windows graphics format, exporting, Export, 568, 1141 importing, Import, 570, 1176 x-axis, range for, PlotRange, 137, 1243 x-height, 453 XBitmap format, exporting, Export, 568, 1141 importing, Import, 570, 1176 XCMDs, 657 XFUNs, 657 \[Xi] ( Ξ ), 175, 990, 1400 XML, exporting, Export, 212, 1141 graphics export in, Export, 568, 1141 importing, Import, 212, 1176 MathMLForm, 211, 1207 symbolic, 212 xo option for mcc, 677 Xor, bitwise, BitXor, 756, 1090 Xor, 87, 1324 argument evaluation in, 1046 \[Xor] ( ), 1001, 1400 XPI, 658 y-axis, range for, PlotRange, 137, 1243 \[YAcute] ( ), 998, 1401 Year, Date, 709, 1117 Yellow, 500 Yen, on keyboard, xv \[Yen] ( ), 994, 1401 Yields, \[RightTee] ( ), 1001 Yuan sign, \[Yen] ( ), 994 Zapf dingbats, 421 Zariski closure, Eliminate, 832, 1132 Zero, approximate, 734 rounding to, Chop, 730, 1103 Zero finding, numerical, FindRoot, 104, 960, 1149 Zero testing, numerical, 735 1463 Zero-origin arrays, Mod, 749, 1213 Zero-width space, \[InvisibleSpace], 454, 1008 Zeros, at end of number, IntegerExponent, 725, 749, 1181 ZeroTest, 908 ZeroWidthTimes, 613 Zeta, 772, 1068, 1324 implementation of, 1068 \[Zeta] ( Ζ ), 175, 990, 1401 Zeta function, generalized, Zeta, 772, 1324 Jacobi, JacobiZeta, 783, 1189 Riemann, Zeta, 772, 1324 Zip, Thread, 256, 1304 Zippel’s algorithm, PolynomialGCD, 1069, 1246 Zoom, in plot, PlotRange, 137, 1243 ZTransform, 879, 1325 $ symbols, 381 $ variables, in tracing, 364 $Aborted, 371, 1048, 1325 $Assumptions, 818, 1325 $BaseDirectory, 637, 1064, 1325 $BatchInput, 715, 1326 $BatchOutput, 715, 1326 $ByteOrdering, 717, 1326 $CharacterEncoding, 420, 421, 422, 1327 $CommandLine, 716, 1327 $Context, 393, 1015, 1327 $ContextPath, 394, 1015, 1327 $CreationDate, 717, 1328 $CurrentLink, 688, 1328 $Display, 554, 705, 1328 $DisplayFunction, 491, 1328 $Echo, 705, 1328 $Epilog, 706, 709, 1328 $ExportFormats, 208, 1328 $Failed, 623, 1329 $FormatType, 556, 1329 $FrontEnd, 592, 1329 $HistoryLength, 703, 1329 $HomeDirectory, 637, 1329 $IgnoreEOF, 706, 707, 1057, 1329 $ImportFormats, 208, 1329 $InitialDirectory, 637, 1330 $Input, 639, 705, 1330 $Inspector, 1330 $InstallationDate, 717, 1330 $InstallationDirectory, 637, 1061, 1330 $IterationLimit, 369, 1048, 1330 $Language, 483, 706, 1331 $LaunchDirectory, 637 $LicenseExpirationDate, 718 $LicenseID, 718 $LicenseProcesses, 718 $LicenseServer, 718 $Line, 702, 703, 1056, 1331 in dialogs, 708 $Linked, 1331 $MachineDomain, 718, 1331 $MachineEpsilon, 739, 1331 1464 $MachineID — $$Media (Version variable) $MachineID, 718, 1331 $MachineName, 718, 1331 $MachinePrecision, 728, 739, 1332 $MachineType, 717, 1332 $MATHEMATICA_BASE environment variable, 1055 $MATHEMATICA_USERBASE environment variable, 1055 $MATHINIT environment variable, 1055 $MATHKERNELINIT environment variable, 1055 $MaxExtraPrecision, 733, 1332 $MaxLicenseProcesses, 718 $MaxMachineNumber, 739, 1017, 1332 $MaxNumber, 739, 1332 $MaxPrecision, 736, 1332 $MessageList, 481, 709, 1333 $MessagePrePrint, 480, 706, 1333 $Messages, 705, 1333 $MinMachineNumber, 739, 1017, 1333 $MinNumber, 739, 1333 $MinPrecision, 736, 1333 $ModuleNumber, 381, 1333 $NetworkLicense, 718 $NewMessage, 482, 1334 $NewSymbol, 405, 1015, 1334 $Notebooks, 715, 1334 $NumberMarks, 730, 1334 $OperatingSystem, 717, 1334 $Output, 705, 1334 $Packages, 397, 1335 $ParentLink, 686, 716, 1335 $ParentProcessID, 716, 1335 $PasswordFile, 718 $Path, 206, 207, 637, 659, 1335 $Post, 703, 1335 $Pre, 703, 1335 $PrePrint, 703, 1336 $PreRead, 703, 1336 $ProcessID, 716, 1336 $ProcessorType, 717, 1336 $ProductInformation, 717, 1336 Index $RandomState, 747, 748, 1336 $RecursionLimit, 369, 390, 715, 1048, 1337 $ReleaseNumber, 717, 1337 $SessionID, 384, 716, 1337 $SoundDisplay, 705 $SoundDisplayFunction, 567, 1337 $SyntaxHandler, 703, 704, 1337 $System, 717, 1338 $SystemCharacterEncoding, 420, 717, 1338 $SystemID, 627, 677, 717, 1338 $TERM Unix environment variable, 628 $TextStyle, 556, 1338 $TimeUnit, 710, 711, 1338 $Urgent, 705, 1338 $UserBaseDirectory, 637, 1064, 1339 $UserName, 716, 1339 $Version, 717, 1339 $VersionNumber, 717, 1339 $$Media (Version variable), see Streams, 1402 ... Visualization with Mathematica Mathematica Notebooks Palettes and Buttons Mathematical Notation Mathematica and Your Computing Environment The Unifying Idea of Mathematica Mathematica as a Programming... start Mathematica the shell command to start Mathematica input for Mathematica (SHIFT-RETURN on some keyboards) exiting Mathematica Running Mathematica with a notebook interface In a “notebook”... command mathematica to start Mathematica When Mathematica starts up, it usually gives you a blank notebook You enter Mathematica input into the notebook, then type SHIFT-ENTER to make Mathematica