Introduction to Mathematica® with Applications

274 Pages·2017·21.45 MB·English

by  Marian Mureşan (auth.)

Marian Mureşan
Introduction to
Mathematica®
with
Applications
r
Introduction to Mathematica with Applications
Marian Mures¸an
Introduction to
r
Mathematica
with Applications
123
MarianMures¸an
FacultyofMathematicsandComputerScience
Babes¸-BolyaiUniversity
Cluj-Napoca,Romania
ISBN978-3-319-52002-5 ISBN978-3-319-52003-2 (eBook)
DOI10.1007/978-3-319-52003-2
LibraryofCongressControlNumber:2016963590
©SpringerInternationalPublishingAG2017
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
To my wife Viorica, always with love
Foreword
The book (written by Professor Marian Mures¸an) is intended to present the
Mathematica system in a manner in which the reader will find it easy to acquire
a part of the considerable number of mathematical instruments offered by the
producer, based on a large variety of examples taken from different scientific
branches.
The importance of the symbolic calculus is incontestable, since nowadays the
scientific calculus is not reduced to the numerical one. Modern mathematical
problems, or those appearing through modeling natural phenomena, lead to such
complicated symbolic expressions that we need the help of computers to process
them. The fast development of the computers’ technical capacity has enabled the
genesis of a new domain—the symbolic (or formal) calculus—through which a
genuinehuman–machinecollaborationisachieved.Thesymboliccalculussystems
are aimed at automating difficult calculi. They provide simple access to many
sophisticated mathematical instruments, using a simple yet flexible language, that
allows avoidance of writing interminable subroutines (similarly to how it happens
withutilizingtraditionallanguages).
Itismainlymathematicalaspectsthatarecoveredinthisbook,sothatthereader
canunderstandaspreciselyaspossiblehowtohandletheproblem.Atthesametime,
theuseriswarnedwithregardtothetypeofproblemswhichcanbesolvedbymeans
of the computer and the necessity of the analysis and verification of the results.
Special emphasis is given to the fact that a session of Mathematica is typically
interactive.Oftentherearemultiplewaystoapproachproblems,thechoiceofone
ofthembeingmadeonthespot,accordingtotheanswerofthesystem.Thereader
iscarefullyguidedtothemostsuitablechoice.
Thebookcontains11chaptersonspecificaspectsoftheMathematicalanguage
or branches of mathematics in general, such as ordinary differential equations, to
which Chap.7 is dedicated. It is worth mentioning that part of the applications
presented in the book have been elaborated by the author in recent years, being
thesubjectofcertainscientificarticlesmentionedinthebibliography.Inthisbook,
standardlinguisticconstructionsareused.
vii
viii Foreword
The book is at an incredibly high scientific level, and it is useful to many
categoriesofusers;forresearchers,professors,andstudentsinscientificfields,the
knowledge of such a symbolic calculus system is nowadays indispensable. Math-
ematics researchers can use the programs for testing some conjectures or even for
provingsomeresultsthatneedabigamountofcalculation(indifferentialgeometry,
number theory, combinatorics, theory of functions of complex variable, numeric
calculus,differentialequations).Thebookisusefulforengineers,economists,and
IT specialists, who can therefore benefit from easy access to a volume of vast
mathematicalknowledge.Iwouldmentionthatpartsofthecontentofthisbookhave
been presented during the scientific seminars held at the Faculty of Mathematics
andComputerScience,havingbeenhighlyappreciated.Thankstoalloftheabove
mentioned,IstronglyrecommendProf.MarianMures¸an’sbook.
Cluj-Napoca,Romania Prof.ValeriuAnisiu
August2016
Preface
This book is the output of our work over several years. Being involved in
calculus of variations and optimal control problems, we have realized that an
exact calculation and a suggestive visualization are very useful, making the ideas
addressed many times only in an "-ı language clearer. Then we have chosen
Mathematicaforcomputationsandvisualizationsoftheideas.Whydidouroption
go to Mathematica? The answer is simple: because we had noticed the wonderful
resultsofProf.J.Borweinandhiscolleaguesregardingthedecimalsofnumber(cid:2).
TheirapproachwasbasedonanextensiveuseofMathematica.
Wolfram Research, located at Champaign, IL, USA, is the company which has
beendevelopingMathematica.
Mathematicaiscontinuouslydeveloping.WeusedMathematica10.3.Itisvery
likelytherewillbenewerversionswithextrafacilitiesinthefuture.
WehaveintroducednotionsandresultsinMathematicainourlecturestomaster
studentsattheFacultyofMathematicsandComputerScienceoftheBabes¸-Bolyai
UniversityinCluj-Napoca,Romania.WedidthesamethingwithourPhDstudents
at three summer schools organized in the framework of the grant “Center of
Excellence forApplications ofMathematics”supported byDAAD,Germany.The
summer schools have been organized in Struga (Macedonia, FYROM), Sarajevo
(BosniaandHerzegovina),andCluj-Napoca(Romania).
This book is not very large, but it collects many examples. In the first part of
thebook,theexamplesarediscussedindetailhelpingthereadertounderstandthe
reasoning in and with Mathematica. Later on, the reader is led to use the benefit
of the Help and other sources freely offered by Wolfram Research. We take into
account mainly the Wolfram community forum as well as the video training and
conferencesgenerouslyofferedbyWolframResearch.
Awell-motivatedcaseforvisualizationinmathematicsiscontainedin [58].
Here is the right place to express my gratitude to the following colleagues of
minefromtheFacultyofMathematicsandComputerScienceoftheBabes¸-Bolyai
University for their support: Anca Andreica, Valeriu Anisiu, Paul Blaga, Virginia
ix
x Preface
Niculescu, Adrian Petrus¸el, and Adrian Sterca. The existence and development of
theMOS(Modeling,Optimization,andSimulation)ResearchCenterofourfaculty
wasarealhelpforusinthepreparationofthisbook.
Cluj-Napoca,Romania Prof.MarianMures¸an
August2016
Contents
1 AboutMathematica......................................................... 1
1.1 Introduction........................................................... 1
1.1.1 Warning ..................................................... 3
2 FirstStepstoMathematica................................................. 5
2.1 TheIntroductoryTechniquesforUsingMathematica.............. 5
2.1.1 Numbers..................................................... 5
2.1.2 BracketinginMathematica ................................ 9
2.1.3 SetorSetDelayedOperator................................ 9
2.1.4 SomeSimpleSteps......................................... 10
3 BasicStepstoMathematica................................................ 13
3.1 ProblemsinNumberTheory,SymbolicManipulation,
andCalculus.......................................................... 13
3.1.1 ProblemsinNumberTheory............................... 13
3.1.2 SymbolicManipulations ................................... 17
3.1.3 Texts......................................................... 19
3.2 Riemann(cid:3) Function.................................................. 20
3.3 SomeNumericalSequences ......................................... 20
3.3.1 TheFirstSequence ......................................... 20
3.3.2 TheSecondSequence ...................................... 21
3.3.3 TheThirdSequence ........................................ 22
3.4 Variables.............................................................. 23
3.5 Lists................................................................... 24
3.5.1 OperationswithLists....................................... 24
3.5.2 OperationswithMatrices................................... 28
3.5.3 InnerandOuterCommands................................ 31
3.5.4 AgainontheThirdSequence .............................. 34
4 SortingAlgorithms......................................................... 35
4.1 Introduction........................................................... 35
4.2 SortingMethods...................................................... 36
xi