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The Maple Handbook Maple V Release 3 Darren Redfern The e a---- Handbook Maple V Release 3 With 16 illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo HongKong Barcelona Budapest Darren Redfern Practical Approach P.O. Box 1007 Stratford, ON N5A 6W4 Canada Library of Congress Cataloging-in-Publication Data Redfern, Darren. The Maple handbook / Darren Redfern. --2nd ed. p. cm. Includes bibliographical references and index. ISBN-I3: 978-0-387-9433 I -2 e-ISBN-I3: 978-1 -4684-0229-2 DOl: 10. I 007/978-I -4684-0229-2 1. Maple (Computer file) 2. Mathematics --Data processing. 1. Title. QA76.95.R43 1994 510'.285'53 --dc20 93-25796 Maple is a registered trademark of Waterloo Maple Software. Printed on acid-free paper. @1993, 1994 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Karen Phillips; manufacturing supervised by Vincent Scelta. Photocomposed copy prepared from the author's u\TI¥' file. 987654321 Contents Introduction ........................................... 1 Getting Started with Maple ........................... 9 Calculus ............................................. 35 Linear Algebra ...................................... 69 Solving Equations .................................. 111 Polynomials and Common Transformations ...... 145 Geometry ........................................... 189 Combinatorics and Graph Theory ................. 231 Number Theory .................................... 265 Statistics ............................................ 299 Standard Functions and Constants ................. 331 Expression Manipulation .......................... 363 Plotting ............................................. 383 Programming and System Commands ............ 415 Miscellaneous ...................................... 461 Index ............................................... 515 Introduction How to Use This Handbook The Maple Handbook is a complete reference tool for the Maple language, and is written for all Maple users, regardless of their dis cipline or field(s) of interest. All the built-in mathematical, graphic, and system-based commands available in Maple V Release 3 are detailed herein. Please note that The Maple Handbook does not teach about the mathematics behind Maple commands. If you do not know the meaning of such concepts as definite integral, identity matrix, or prime integer, do not expect to learn them here. As well, while the introductory sections to each chapter taken together do provide a basic overview of the capabilities of Maple, it is highly recom mended that you also read a more thorough tutorial such as In troduction to Maple by Andre Heck or First Leaves: A Tutorial Introduction to Maple V. Overall Organization One of the main premises of The Maple Handbook is that most Maple users approach the system to solve a particular problem (or set of problems) in a specific subject area. Therefore, all commands are organized in logical subsets that reflect these different cate gories (e.g., calculus, algebra, data manipulation, etc.) and the com mands within a subset are explained in a similar language, creating a tool that allows you quick and confident access to the information necessary to complete the problem you have brought to the system. This design goes hand in hand with the fact that there are many Maple commands that behave differently when faced with the vary ing data types and constructs inherent in different subject areas. When this is the case, the commands have separate entries in each relevant subset. A good example of this is the op command (which 2 Introduction deals with the internal structure of data objects). The results of the op command are sufficiently different when dealing with expres sions, matrices, plots, and procedures to warrant an entry in each corresponding section, as well as one in the section on manipulat ing data structures. The overall philosophy is to make you travel as little as possible to access information necessary to your work. In addition, because there is much information about Maple that is very difficult to express on a purely command-by-command basis, each subject is prefaced with a short introductory section, written to represent an active Maple session where input, output, text, and graphics are combined. There is also an introductory session, titled Getting Started with Maple, which is intended to get those readers not already familiar with Maple off to a quick start. Cross Referencing One of the most important goals of The Maple Handbook is to provide pointers to appropriate information so that you are able to solve your problems quickly and efficiently. The Maple Handbook is rife with valuable references, presented in such a way to be ac cessible yet not clutter the infoITIlation to which they are attached. There are three types of references present within The Maple Hand book. Firstly, each command listing has a See also section that points you to commands within that section, or in other sections, that contain related information. Secondly, in case you know the name of a command but are unsure to which subject area it belongs, there is a complete alphabetical index of all Maple commands at the end of the handbook. Finally, most command listings contain essential page references to supplementary information or illumi nating examples contained in the official Maple manuals (First Leaves: A Tutorial Introduction to Maple V, Maple V Language Reference Manual, Maple V Library Reference Manual). Individual Command Entries While the information contained in the entries for each individ ual command is unique, the format in which the information is presented is identical across all entries. The following fictitious example illustrates the various elements that are used throughout this book. Introduction 3 acommand(expr, var) Performs a command on expr with respect to var. Output: If expr contains only numeric values, an expression se quence of integers is returned. Otherwise, a list of symbolic values is returned. Argument options: (expr. var=a. . b) to limit the computation to var ranging from a to b. a must be greater than b. • ([exprl • ...• exprn]. [varl •...• yarn]) to perform the calculation on exprl through exprn with respect to varl through varn , respectively. An expression sequence of n lists is returned. Additional information: If var is previously assigned to anything other than its own name, an error message is returned. See also: bcommand. cpackage[dcommand]. anothercommand = = = FL 12-14 LA 43 LI 386 The command call, acommand(expr. var), gives the command name as well as its most common type of parameter sequence. The command name can be of several different forms but always repre sents how the command can be typed into Maple to be immediately executed: acommand-a command that is in the standard library apackage[acommand]-a command that is in a specialty pack age. To access these commands they must either be entered in their long form (as given) or they must follow an appropriate with com mand. Unfortunately, in Maple V Release 3, there are still a few packages where the long form (i.e., apackage[acommandJ) does not provide access to the command. In these cases, the acommand call must be preceded by with(apackage) or with(apackage. acommand). The individual command entries always list the long form of the command name. If that doesn't appear to work, con sult the command entry for the entire package that precedes the individual entries. acommand(aparameter)-a command that has a unique function when combined with a particular type of parameter (including spe cial optional arguments). 'acommand/bcommand'-used for specialized second-level commands that combine the functionality of acommand and bcom mand. Backquotes ensure that special character I is not translated as a division sign. While there are many, many such second-level commands in Maple, very few of them should be used directly by you and are therefore not detailed in this book. 4 Introduction The command call's parameter sequence (as well as the parameter sequences found in othet elements of a command entry) normally represents placeholders for the actual input you would use when calling the command. For example, a placeholder of int could be replaced with the integer 2, 754, or -11. When a parameter se quence contains an element that appears in italics, for example nu meric, it means that the word numeric is to be entered as is in the command, not replaced with some other value. Such elements most frequently occur with prede:fmed options and input values. Whenever possible, the expected data type of a parameter is spec ified with one of the following abbreviations. A an array data type boolean true or false complex a complex value or expression eqn an equation expr an expression exprseq an expression sequence float, # a floating-point value filename a file name fnc a function fnc(var) a function in variable var ineq an inequation int an integer list, [] a list data type list[typed a list with elements of type typel n, m, i, j, posint a positive integer name a name to be assigned a value num, a, b a numeric value M a two-dimensional array, matrix option one of a set of predefined options poly a polynomial expression proc a procedure pt a point (e.g., [a,b]) rat a rational expression or value ratpoly a rational polynomial RootOf a root notation s, series a series expression set, {} a set data type set{typel} a set with elements of type typel subexpr a subexpression T a table data structure V a one-dimensional array, vector var an unassigned variable Introduction 5 The above abbreviations deal with parameter types that are encoun tered across all disciplines covered by The Maple Handbook. There are also dozens of other data types that are specific to individual ar eas; these are detailed in the introductions to each chapter. [Note: when a sequence of parameters is represented with ... , for exam ple, exprl, ... , exprn, do not confuse this with Maple's ellipsis operator .. -One is a short-hand representation, while the other is a language structure.] Following the command call is a short description of how the com mand works on the given parameter sequence. This is meant to give you enough information to use the command in most instances. If more information is needed, the Argument options and Additional information sections should be read. The Output listing gives some idea of what type of output (i.e., what data types) to expect from the most common calling structure. This is extremely helpful when you either want to dissect the answer for further use or correctly include the command within another command or a procedure. The Argument options listing provides valid variations to the pa rameter sequence and brief explanations of their functioning. If any alternate parameter sequence is of paramount importance, there is an individual command listing to discuss it. • characters appear in this section to separate multiple parameter sequences. The Additional information section lists just that-additional in formation about the command. This could include, among other things, special pointers to other command entries, brief descrip tions of algorithms used to compute the command, or warnings about dangerous combinations of parameters. • characters ap pear in this section to distinguish separate items. The See also section gives pointers to other command listings within that section (in normal typeface) or in other sections (in italic type face) which contain related information or work in conjunction with the initial command. When searching for a command in an other section, it is best to consult the index at the back of this book for an exact page location. The Manual Cross References at the end of each command listing provide page numbers from the Maple manuals where more infor mation related to the command is available. FL = First Leaves, LA = Language Reference Manual, LI = Library Reference Manual. 6 Introduction Maple's On-line Help System All versions of Maple come with an exhaustive on-line help facility containing hundreds of pages of detailed descriptions. These help pages have been updated during the many years of Maple devel opment and contain much information that is valuable and much that is esoteric. On many platforms, there are on-line help topic browsers that allow you to navigate easily between various help pages. The Maple Handbook adds to this facility. By trimming down the information presented to you, it allows you to save many hours of needless reading. The on-line help pages for Maple were written by many different researchers/developers and lack a common voice and a completely unifying style. By putting all the information in one voice, with consistent handling of similar situations through out, The Maple Handbook furthers comprehension. Please keep in mind that on-line help is always available to you when you are running Maple. The Maple Handbook is meant to offer you fast and efficient access to information about Maple the on-line help pages can be used to broaden that knowledge at your leisure. To view the on-line help for any Maple command, simply enter the command name prepended with a question mark character. For example, the command ?factor displays the on-line help page for the factor command. Where to Go for More Information The range of Maple books, courseware, and third-party applica tions is constantly growing. Apart from the standard Maple man uals (First Leaves, etc.) there are books on using Maple in subject areas from linear algebra to calculus to differential equations; and there are several more volumes currently being written by authors in the academic and commercial fields. The next few years will see an explosion in the number and variety of Maple materials. In addition, there have been many scholarly papers, reports, and theses written around the Maple system. For more information on these or other Maple materials, contact the vendor who sold you Maple. They have access to much information that will be of inter est to you.