Essentials of Mathematica Nino Boccara Essentials of Mathematica With Applications to Mathematics and Physics Springer University of Illinois at Chicago Department of Physics (M/C 273) 845 West Taylor Street Chicago, IL 60607 USA boccara@uic.edu Library of Congress Control Number: 2006936428 ISBN-10: 0-387-49513-4 ISBN-13: 978-0-387-49513-2 e-ISBN-10: 0-387-49514-2 e-ISBN-13: 978-0-387-49514-9 Printed on acid-free paper © 2007 Springer Science+Business Media, LLC All rights reserved This work may not be translated or copied in whole or in part without the written permission of the pubhsher (Springer Science-i-Business Media, LLC, 233 Spring Street, New York, NY 10013, 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 in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights springer.com Preface This book consists of two parts Part I describes the essential Mathematica commands illustrated with many examples and Part II presents a variety of applications to mathematics and physics showing how Mathematica could be systematically used to teach these two disciplines The book is based on an introductory course taught at the University of Illinois at Chicago to advanced undergraduate and graduate students of the physics department who were not supposed to have any prior knowledge of Mathematica Mathematica is a huge mathematical software developed by Wolfram Research Inc It is an interactive high-level programming language that has all the mathematics one is likely to need already built-in Moreover, its interactivity allows testing built-in and user-defined functions without difficulty thanks to numerical, symbolic and graphic capabilities All these features should encourage students to look at a problem in a computational way, and discover the many benefits of this manner of thinking For instance, when studying a new problem, Mathematica makes it easy to test many examples that might reveal unsuspected patterns The reader is advised to first study Chapter of Part I entitled A Panorama of Mathematica which presents an overview of the most frequently used commands The following chapters—dealing with Numbers, Algebra, Analysis, Lists, Graphics, Statistics and Programming—go into more details The reader would probably make the most of the book browsing, as soon as possible, Part II, devoted to Applications to Mathematics and Physics, coming back to Part I to go deeper into specific commands and their various options This book is intended for beginners who want to be able to write a small efficient Mathematica program in order to solve a given problem Having this in mind, we made every effort to follow the same technique: first the problem is broken up into its different component parts, then each part of the problem is vi Preface solved using either a built-in or a user-defined Mathematica function, checking carefully that this function does exactly what it was supposed to do, and the program is finally built up by grouping together all these functions using a standard structure N o t e concerning the figures Most figures have been generated using colors as indicated by their Mathematica code but are represented in the book using only various shades of grey However all the figures can be found in color in the accompanying CD-ROM which also contains all the Mathematica cells that appear in the book Nino Boccara Contents Preface List of Figures v xix Part I Essential Commands A Panorama of Mathematica 1.1 Notebooks and Cells 1.2 Basic Syntax 1.3 Basic Operations 1.4 Mathematica as a Functional Language 1.5 Getting Help 10 1.6 Logical Operators 12 1.7 Elementary Functions 14 1.8 User-Defined Functions 15 1.9 Rules and Delayed Rules 18 1.10 Built-in Nonelementary Functions 21 1.11 Plotting 21 1.11.1 2D plots 21 1.11.2 3D plots 22 1.12 Solving Equations 1.12.1 Exact Solutions 23 23 viii Contents 1.12.2 Numerical Solutions 23 1.13 Derivatives and Integrals 24 1.13.1 Exact Results 24 1.13.2 Numerical Integration 26 1.14 Series Expansions and Limits 27 1.15 Discrete Sums 29 1.16 Ordinary Differential Equations 30 1.16.1 Symbolic Solutions 30 1.16.2 Numerical Solutions 31 1.17 Lists 32 1.18 Vectors and Matrices 36 1.19 Clear, ClearAll, and Remove 40 1.20 Packages 42 1.21 Programming 43 1.21.1 Block and Module 43 1.21.2 Collatz Problem 47 1.21.3 Generalizing the Collatz Problem 49 Numbers 55 2.1 Characterizing Numbers 55 2.2 Real Numbers 56 2.3 Integers 58 2.4 Prime Numbers 61 2.5 Combinatorial Functions 62 2.5.1 Factorial 62 2.5.2 Binomial CoefRcients 63 2.6 Rational Numbers 66 2.7 Complex Numbers 67 2.8 Different Bases 68 2.9 Calendars 70 Contents 2.10 Positional Number Systems 71 2.11 Zeckendorf s Representation 73 Algebra 77 3.1 Algebraic Expressions 77 3.2 Trigonometric Expressions 82 3.3 Solving Equations 86 3.3.1 Solving Polynomial Equations Exactly 86 3.3.2 Numerical Solutions 89 3.4 ix Vectors and Matrices 95 Analysis 103 4.1 Differentiation 103 4.1.1 103 4.2 Partial Derivative Total Derivative 105 4.3 Integration 106 4.3.1 Indefinite Integrals 106 4.3.2 Definite Integrals 107 4.3.3 Numerical Integration 109 4.3.4 Multiple Integrals 112 4.4 4.5 Differential Equations 113 4.4.1 Solving nonelementary ODE 114 4.4.2 Numerical Solutions 114 4.4.3 Series Solutions 117 4.4.4 Differential Vector Equations 119 Sum and Products 122 4.5.1 Exact Results 122 4.5.2 Numerical Results 123 4.6 Power Series 125 4.7 Limits 126 4.8 Complex Functions 130 X Contents 4.9 Fourier Transforms 136 4.9.1 Discrete Fourier Transform 136 4.9.2 Fourier Transform 137 4.10 Fourier Series 139 4.11 Laplace Transforms 142 4.12 Recurrence Equations 144 4.13 Z Transforms 145 4.14 Partial Differential Equations 146 Lists 151 5.1 Creating Lists 151 5.2 Extracting Elements 155 5.3 Adding Elements 159 5.4 Finding, Grouping, and Counting Elements 162 5.5 Mathematical Operations on Lists 164 5.6 Rearranging Lists 167 5.7 Listability 169 Graphics 173 6.1 2D Plots: Function Plotting 173 6.1.1 Parametric Plots 174 6.1.2 Polar Plots 174 6.1.3 Implicit Plots 176 6.1.4 Color 176 6.1.5 Dashing 178 6.1.6 Text 178 6.1.7 Axes, Ticks and Labels 180 6.1.8 Graphics Array 182 6.1.9 Plot Range 185 6.2 More 2D Plots 186 6.2.1 186 Plotting Lists .. .Essentials of Mathematica Nino Boccara Essentials of Mathematica With Applications to Mathematics and Physics Springer University of Illinois at Chicago Department of Physics (M/C... consists of two parts Part I describes the essential Mathematica commands illustrated with many examples and Part II presents a variety of applications to mathematics and physics showing how Mathematica. .. 6.30 Bar chart of a list of 20 random integers between and 190 6.31 Pie chart of a list of 20 random integers between and 190 6.32 Histogram of a list of 20 random integers between and 191 6.33