Advanced Analytic Methods in Applied Mathematics, Science, and Engineering

124 Pages·0.564 MB·English

by  Cheng Hung.

Advanced Analytic Methods
in Applied Mathematics, Science,
and Engineering

By Hung Cheng

LuBan Press, 2007
6.125 x 9.25, 504 pp.
Hardcover
ISBN: 0975862510
$69.95

THIS IS A SAMPLE COPY, NOT
TO BE REPRODUCED OR SOLD

This sample includes: Table of Contents; Preface; Chapters 2 and 7; and Index

Please see Table of Contents for a listing of this book’s complete content.

PUBLICATION DATE: August 2006

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SAMPLE: Advanced Analytic Methodsin Applied Mathematics, Science, and Engineering

by Hung Cheng ISBN: 0975862510

Advanced Analytic Methods
in Applied Mathematics, Science, and Engineering
SAMPLE: Advanced Analytic Methodsin Applied Mathematics, Science, and Engineering

by Hung Cheng ISBN: 0975862510

Contents
1 The Algebra of Operators 1
A. Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
B. OrdinaryDifferentialEquations . . . . . . . . . . . . . . . . . . 7
2 Complex Analysis 35
A. ComplexNumbersandComplexVariables . . . . . . . . . . . . 35
B. AnalyticFunctions . . . . . . . . . . . . . . . . . . . . . . . . . 41
C. TheCauchyIntegralTheorem . . . . . . . . . . . . . . . . . . . 47
D. EvaluationofRealIntegrals . . . . . . . . . . . . . . . . . . . . 59
E. BranchPointsandBranchCuts . . . . . . . . . . . . . . . . . . 70
F. FourierIntegralsandFourierSeries . . . . . . . . . . . . . . . . 87
G. TheLaplaceTransform . . . . . . . . . . . . . . . . . . . . . . . 105
3 First-Order Partial Differential Equations 117
A. TrivialExample . . . . . . . . . . . . . . . . . . . . . . . . . . 118
B. LinearHomogeneousPDEs . . . . . . . . . . . . . . . . . . . . 119
C. Quasi-LinearPDEs . . . . . . . . . . . . . . . . . . . . . . . . . 127
D. GeneralCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4 Second-Order Partial Differential Equations 145
A. TheLaplaceEquation . . . . . . . . . . . . . . . . . . . . . . . 145
B. TheWaveEquation . . . . . . . . . . . . . . . . . . . . . . . . . 158
C. TheHeatEquation . . . . . . . . . . . . . . . . . . . . . . . . . 161
5 Separation of Variables 175
A. TheLaplaceEquation . . . . . . . . . . . . . . . . . . . . . . . 176
B. TheWaveEquationwithTwoSpatialVariables . . . . . . . . . . 186
C. TheSchrödingerEquation . . . . . . . . . . . . . . . . . . . . . 192
vii
SAMPLE: Advanced Analytic Methodsin Applied Mathematics, Science, and Engineering

by Hung Cheng ISBN: 0975862510

viii Contents

6 SingularPointsofOrdinaryDifferentialEquations201
A. TaylorSeriesSolutions . . . . . . . . . . . . . . . . . . . . . . . 201
B. Frobeniusmethod . . . . . . . . . . . . . . . . . . . . . . . . . 208
C. SolutionsNearanIrregularSingularPoint . . . . . . . . . . . . . 220
Appendix: TheGammaFunction . . . . . . . . . . . . . . . . . . . . 235
7 The WKB Approximation 239
A. WKBintheZerothandtheFirstOrder . . . . . . . . . . . . . . 239
B. SolutionsNearanIrregularSingularPoint . . . . . . . . . . . . . 246
C. Higher-OrderWKBApproximation . . . . . . . . . . . . . . . . 252
D. TurningPoints . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
8 Asymptotic Expansions of Integrals 269
A. IntegralRepresentation . . . . . . . . . . . . . . . . . . . . . . . 269
B. TheLaplaceMethod . . . . . . . . . . . . . . . . . . . . . . . . 273
C. MethodofStationaryPhase . . . . . . . . . . . . . . . . . . . . 293
D. TheSaddlePointMethod . . . . . . . . . . . . . . . . . . . . . 309
AppendixA:GaussianIntegrals . . . . . . . . . . . . . . . . . . . . . 337
AppendixB:InfiniteContours . . . . . . . . . . . . . . . . . . . . . . 337
9 Boundary Layers and Singular Perturbation 347
A. RegularPerturbation . . . . . . . . . . . . . . . . . . . . . . . . 347
B. BoundaryLayerTheory . . . . . . . . . . . . . . . . . . . . . . 349
C. TurningPoints . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
D. TurningPointatanEndpoint . . . . . . . . . . . . . . . . . . . . 373
E. InteriorTurningPoints . . . . . . . . . . . . . . . . . . . . . . . 380
F. OtherProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
10Small Nonlinear Oscillations 405
A. SummingLeadingTerms . . . . . . . . . . . . . . . . . . . . . . 405
B. RenormalizedPerturbation—TheImprovedPoincareMethod . . 414
C. TheTwo-ScaleMethod . . . . . . . . . . . . . . . . . . . . . . . 430
D. TheRenormalizedTwo-ScaleMethod . . . . . . . . . . . . . . . 443
E. TheRenormalizationGroup . . . . . . . . . . . . . . . . . . . . 453
Appendix of Useful Formulae 461
Bibliography 471
Index 487
SAMPLE: Advanced Analytic Methodsin Applied Mathematics, Science, and Engineering

by Hung Cheng ISBN: 0975862510

Preface
For almost four decades, the Department of Mathematics at MIT has offered a
course called “Advanced Analytic Methods in Science and Engineering.”1 The
purposeofthiscourseistostrengthenthemathematicalbackgroundofallentering
graduate students, sothey may be better prepared for theirrespective coursework
andspecialties.
During the past ten years I’ve been teaching this course I occasionally wrote
notes for the studentsin my class, intendedonly to supplementthe course’s text-
books. However, at the end of one recent semester several students suggested I
makemymaterials,whichwere bythattimemorethansimply“notes,”accessible
to studentsbeyond MIT. Thistextbookis theresult of expansionof and revisions
tothatmaterial.
Thebackgroundofstudentstakingthecourseisusuallyfairlydiverse. Manyof
them lacksome of thefundamentalsthatwouldprepare them for agraduate math
course. Thefirstfivechaptersowetheirorigintotheneedforhelpingsuchstudents,
bringingthemuptospeed. Thelastfivechapterscontainmoreadvancedmaterials.
Teachersandstudentswillthusfindthatthisbook’scontentisflexibleenough
tomeettheneedsofavarietyofcoursestructures. Foraone-semestercoursewith
emphasisonapproximatemethods,ateachermayjustskimoverthefirstfivechap-
ters, leaving the students to read in more detail the parts they need most. Such
a plan wouldbe especiallyuseful for graduate studentsentering a Ph.D. program
in engineering, science, or applied mathematics. But if this book is adopted for
a course in advanced calculus for undergraduate engineering, science, or applied
mathematics students, then Chapters 1–6 should be emphasized. Chapters 1–5,
plusafewselectedlaterchapters,wouldbesuitableforagraduatecourseforMas-
ter’s degree students. In addition,Chapters 3, 4, and5 may be usedas partof the
materialsforacourseonpartialdifferentialequations.
1ThecoursewascreatedbyProfessorHarveyGreenspan. In1978CarlM.BenderandSteven
A.Orszag,twolecturersofthiscourse,authoredatextbook,AdvancedMathematicalMethodsfor
ScientistsandEngineers(New York, McGraw-Hill, Inc.; reprintedbySpringer-VerlagNew York,
Inc.,1998).
ix
SAMPLE: Advanced Analytic Methodsin Applied Mathematics, Science, and Engineering

by Hung Cheng ISBN: 0975862510

x Preface

Whilemostgraduatestudentsandupper-classundergraduatestudentshaveal-
ready had a full semester of ordinary differential equations, some of them may
need a refresher. Therefore, Chapter 1includesa very brief summary of ordinary
differential equations. This first chapter is also a convenient place to reintroduce
theelementarybutpowerfuloperatormethodtostudents.Theoperatormethoden-
ablesonetomorequicklyproducetheparticularsolutionsofcertainlinearordinary
differentialequationsaswellaspartialdifferentialequations,anditalsofacilitates
manyothercalculations.2
Chapter 2 is for students who need a quick summary of some of the rele-
vant materials in complex analysis. The important but often neglected subjects
of branchpointsandbranchcutsare included,aswellasa shortdiscussionof the
Fourierintegral,theFourierseries,andtheLaplacetransform.
Many of the analytic methods discussed in this book arose from the need to
solve partial differentialequations. To help thereader see thatconnection, Chap-
ters3,4,and5addresspartialdifferentialequations.
Because many problems encountered in real life are often not solvable in a
closed form, it will benefit a student to learn how to do approximations. Chap-
ter6presentsthemethodsofseriessolutions.Afewwell-knownspecialfunctions
are used as examples in order to help students gain some familiarity with these
functions while learning the methods of series solutions. I will address the topic
of irregular singularpointsof an ordinary differentialequation, which is notusu-
allycoveredinstandardtextbooksonadvancedcalculus,suchasF.B.Hildebrand,
Advanced CalculusforApplications,Prentice Hall, 1976. The series solutionex-
pandedaroundanirregularsingularpointofanintegralrankisgenerallydivergent
and leads naturallytothe concept of asymptoticseries, whichwe’ll cover insub-
sequentchapters.
Chapter 7 discussesthe WKB method. This method gives good approximate
solutionsto many linear ordinary differentialequationswith a large parameter or
those with coefficients that are slowly varying. It is also helpful for yielding so-
lutionsnear anirregular singularpointof alinear differentialequation. Whilethe
lowest-order WKB solutionsare obtained by solving nonlinear differential equa-
tions, the higher-order WKB approximationsare obtained by iterating linear dif-
ferential equations. The last sectionof thischapter discussesthe solutionsnear a
turningpoint.
Chapter 8 addresses theLaplace method, the methodof stationaryphase, and
the saddle point method, which are useful for finding the asymptotic series of
2Whilethismethodhasbeenroutinelyusedinfieldtheories,particularlywiththederivationof
variousGreenfunctions,ithasnotbeenadequatelycoveredinmostundergraduatetextbooks,with
thenotableexceptionofDifferentialEquationsbyH.T.H.Piaggio,G.BellandSons,Ltd.,London,
1946(reprintedintheU.S.byOpenCourtPublishingCompany,LaSalle,Illinois,1948).
SAMPLE: Advanced Analytic Methodsin Applied Mathematics, Science, and Engineering

by Hung Cheng ISBN: 0975862510

xi

integrals with a large parameter. In the saddle point method, we deviate from
therigorousapproachoffindingthepathofsteepestdescent. Instead,weadvocate
findingjustapathofdescent,asthismaysomewhatreducethesolution’schores.
Chapters 9 and 10 address the subjects of regular perturbation and singular
perturbation. Chapter9isdevotedtothetopicofboundarylayers,andChapter10
coversthetopicofsmallnonlinearoscillations.
Throughout this book I emphasize a central theme rather than peripheral de-
tails. For instance, before discussinghow to solve a class of advanced problems,
I relate it to the basics and, when possible, make comparisons with similar but
more elementary problems. As I demonstratea methodtosolve a certainclass of
problems,Istartwithasimpleexample beforepresentingmore difficultexamples
to challengethe mindsof thestudents. Thisprocess givesstudentsa firmer grasp
of thesubject,enablingthem toacquirethe keyideamore easily. Hopefully,ours
willmake itpossibleforthemtodomathematics withouttheneedofmemorizing
alargenumberofformulae. IntheendIhopethattheywillknowhowtoapproach
a generalproblem; thisisa skillthatleaves studentsbetterprepared totreatprob-
lemsunrelatedtotheonesgiveninthisbook,whichthey’lllikelyencounterintheir
futureacademicorprofessionallives.
Duringmyclassroomlectures,Iemphasizeinteractionwiththestudents. Iof-
ten stoplecturing for a few minutes to pose a questionand ask everyone to work
throughit. Ibelievethismethodhelpstoencouragestudentstolearninamorethor-
oughwayandtoabsorbconceptsmoreeffectively,andthisbookreflectsthatinter-
activeapproach;many“ProblemsfortheReader”arefoundthroughoutthetext. To
deepentheirunderstandingofthethemesthatthey’relearning,studentsareencour-
agedtostopandworkontheseproblemsbeforelookingatthesolutionsthatfollow.
This book also passes on to learners some of the problem-solving methods
I’ve developed through the years. In particular, parts of Chapters 9 and 10 offer
techniques, which I hope will benefit students and researchers alike. Indeed, I
believe that the renormalization methods given in Chapter 10 are more powerful
thanothermethodstreatingproblemsofnon-linearoscillationssofaravailable.
I am indebted to the group of students who encouraged me to publish this
book. Several studentshavereadthefieldtestversionofthisbookandhavegiven
me theirveryhelpfulsuggestions. TheyincludeMichael Demkowicz,JungHung
Lee, Robin Prince, and Mindy Teo. Also, Dr. George Johnston read Chapter 7
andgavemeveryusefulcomments. IwanttothankProfessorT.T.WuofHarvard
University,whointroducedmetothesaddlepointmethodseveraldecadesagowith
a depthIhad neverfathomedasa graduatestudent. I thankMr. DavidHufor the
graphs in Chapters 2 and 8. Special thanksare due to Dr. DionisiosMargetis for
graphs in Chapter 9 and the compilationof an extensive bibliography,and to Mr.
NikosSavva for graphsin Chapters 3 and 9. I alsoam trulygrateful to Professor
JohnStrainforhisinexhaustibleeffortsinreadingthroughallofthechaptersinthe
first draft. I am greatlyindebtedtoDr. H. L. Huforthemany graphshetirelessly
drewforthisbook.
SAMPLE: Advanced Analytic Methodsin Applied Mathematics, Science, and Engineering

by Hung Cheng ISBN: 0975862510
SAMPLE: Advanced Analytic Methodsin Applied Mathematics, Science, and Engineering

by Hung Cheng ISBN: 0975862510

Chapter 2
Complex Analysis
A. Complex Numbers and Complex Variables
In this chapter, I give a short discussionof complex numbers and the theory of a
functionofacomplexvariable.
Before we get to complex numbers, let me first say a few words about real
numbers.
All real numbers have meanings in the real world. Ever since the beginning
of civilization, people have found great use of real positive integers, say 2 and
30, which came up in conversations such as “my neighbor has two pigs, and I
have thirtychickens.” Theconcept of a negative real integer, say−5, isnot quite
as easy, but it became relevant when a person owed another person five copper
coins. It was also natural to extend the concept of integers to rational numbers.
For example, when six personsshare equallya melon, the number describingthe
fraction of melon each of them has isnot an integer but the rational number 1/6.
Whenweadd,subtract,multiply,ordivideintegersorrationalnumbers,theresult
isalwaysanintegerorarationalnumber.
But theneed for other real numbers came upas mathematicians ponderedthe
lengthofthecircumference of,say,acircularcitywall. Toexpressthislength,the
real number π must be introduced. This real number is neither an integer nor a
rationalnumber,andiscalledanirrationalnumber. Anotherwell-knownirrational
numberfoundbymathematiciansistheconstante.
Each of thereal numbers, be it positiveor negative, rationalor irrational,can
be geometrically represented by a point on a straight line. The converse is also
true: apointonastraightlinecanalwaysberepresentedbyarealnumber.
When we add, subtract, multiply, or dividetwo real numbers, the outcome is
alwaysarealnumber. Thustherootofthelinearequation
35
SAMPLE: Advanced Analytic Methodsin Applied Mathematics, Science, and Engineering

by Hung Cheng ISBN: 0975862510

36 COMPLEXANALYSIS

ax+b = c,

with a, b, and c real numbers, is always a real number. That is to say that if we
make nothingbut linear algebraic operations of real numbers, what comes out is
invariablyarealnumber. Thustherealnumbersformaclosedsystemunderlinear
algebraicoperations.
But as soon as we get to nonlinear operations, the system of real numbers
alonebecomes inadequate. Asweallknow, thereare noreal numbersthatsatisfy
thequadraticequation
x2 = −1.
Thus we use our imagination and denote i as a root of this equation. While we
have gotten to be comfortable with the imaginary number i, the concept of the
imaginary number was not always easy. Indeed, even Gauss once remarked that
the“truemetaphysics”ofiwas“hard.”
Thenumber
α = a+ib,
whereaandbarerealnumbers,iscalledacomplexnumber. Thenumbersaandb
arecalledtherealpartandtheimaginarypartofα,respectively.
Whilecomplexnumbersmighthaveonceappearedtohavenodirectrelevance
in the real world, people have since found that the use of complex numbers en-
ablesthemtohandlemoreeasilymanyphysicalproblemsinclassicalphysics. For
example, electrical engineers use the imaginarynumber i extensively, except that
they call it j. And at the turn of the twentiethcentury, complex numbers became
almostindispensablewiththeinventionofquantummechanics.
Letusenterthenever-neverlandofthecomplexvariablez denotedby
z = x+iy,
wherexandy arerealvariablesand
i2 = −1.
Thecomplexconjugateofz willbedenotedas
z∗ = x−iy.
Thevariablezcanberepresentedgeometricallybythepoint(x,y)intheCartesian
two-dimensionalplane. In complex analysis, this same two-dimensionalplane is
calledthecomplexplane. Thexaxisiscalledtherealaxis,andthey axisiscalled
theimaginaryaxis. Letr andθ bethepolarcoordinates. Thenwehave