Complex Integration: A Compendium of Smart and Little-Known Techniques for Evaluating Integrals and Sums

254 Pages·2023·2.153 MB·English

by  Ron Gordon

Undergraduate Lecture Notes in Physics
Ron Gordon
Complex
Integration
A Compendium of Smart and
Little-Known Techniques
for Evaluating Integrals and Sums
Undergraduate Lecture Notes in Physics
SeriesEditors
NeilAshby,UniversityofColorado,Boulder,CO,USA
WilliamBrantley,DepartmentofPhysics,FurmanUniversity,Greenville,SC,USA
MatthewDeady,PhysicsProgram,BardCollege,Annandale-on-Hudson,
NY,USA
MichaelFowler,DepartmentofPhysics,UniversityofVirginia,Charlottesville,
VA,USA
MortenHjorth-Jensen,DepartmentofPhysics,UniversityofOslo,Oslo,Norway
MichaelInglis,DepartmentofPhysicalSciences,SUNYSuffolkCounty
CommunityCollege,Selden,NY,USA
BarryLuokkala ,DepartmentofPhysics,CarnegieMellonUniversity,Pittsburgh,
PA,USA
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Ron Gordon
Complex Integration
A Compendium of Smart and Little-Known
Techniques for Evaluating Integrals and Sums
RonGordon
Northborough,MA,USA
ISSN 2192-4791 ISSN 2192-4805 (electronic)
UndergraduateLectureNotesinPhysics
ISBN 978-3-031-24227-4 ISBN 978-3-031-24228-1 (eBook)
https://doi.org/10.1007/978-3-031-24228-1
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Preface
Inreal-worldmaths,computersdoalmostallthecalculating;
bycontrast,ineducationalmaths,
peopledoalmostallthecalculating.
—ConradWolfram,TheMath(s)Fix:AnEducationBlueprint
fortheAIAge.
Conrad Wolfram, in his book The Math(s) Fix: An Education Blueprint for the AI
Age,laysoutnotonlyacaseforabandoningthetraditionalmathcurriculuminwhich
we teach concepts involving hand calculations, but also a proposed all-computer
computation curriculum. Wolfram is related to Stephen Wolfram who gifted the
computationalsystemMathematicatousallsincethe1980s.
You might think this Preface and this book are meant to be a broadside against
Wolfram,andIamgoingtofurnishargumentsastowhyheisdeadwrongandwe
shouldkeepteachingtraditionalhandcomputationtoeveryone.Butitisn’t,andIam
not.Dr.Wolfram(theauthor)islargelycorrect:Weneedtoteachmathdifferentlyto
readystudentsfordealingwithconceptssuchasmachinelearning,Kalmanfilters,
graphs, and networks, which have become the backbone of the problems of our
societytoday.
SoamIsayingthatabookdedicatedtohandcalculationsisacompletewasteof
time.
No.Theopposite,actually.
The problems requiring hand calculation have never really gone away. We still
generatemodelsofthingslikeimagingdetailedobjectsinturbulentmedia,orreturn
on European put options, or knocking in an automotive engine. These problems
are solved, others will replace them. Mr. Wolfram will tell us simply to fire up
Wolfram|AlphaorMathematicaandwhateverintegralorsumordifferentialequation
thatneedstobesolvedwillbesolvedbythecomputer.And,again,heisn’twrong
aboutit.
Muchofthetime,anyway.
vii
viii Preface
The truth is the systems behind the computers we use as substitutes for hand
calculationsrelyonalgorithmswrittenbyhumanbeings.Humanbeingsthatmake
mistakes,alot.
Quiscustodietipsoscustodes?
Sowewillalwaysneedpeoplewhoknowhowtodothehandcalculations.Idoubt
Dr.Wolframwoulddisagree—otherwise,whywouldtheotherDr.Wolframneedto
hireallofthosepeopletokeepMathematicarunning?
Isupposethishasalwaysbeenmycalling.WhenIworkedforIBMMicroelec-
tronics,Iwasinagroupthatdidlithographicprocessdesignanddidsobyrunning
interminable simulations of the lithography process on vendor-provided software.
(Previously, I worked for such a vendor.) One of my projects was to design test
cases that we could model by hand, so we could evaluate the simulation results.
Thiswasmyfavoritepartofmyjob,andwewereabletofindnumerousissuesto
taketothesoftwarevendors.Myenthusiasmspilledouttomymanagement,andI
wasallowedtoteachashortcoursetotheengineersrunningthesimulationsabout
how to do simple hand calculations of the lithography models—a sort of Conrad
Wolfram-in-reverse,ifyouwill.
EvenMathematicaitself—andbyextension,Wolfram|Alpha—arenotperfect.I
myselfcanpointtoaspecificexampleMathematicagotwrong.Theproblemwasto
computethefollowingsum(whichwillbeconsideredinthebook):
(cid:2)∞ 1 13π3
(cid:3)√ (cid:4) =− √
n=1 n3sin 2πn 360 2
WhenIcheckedtheresult,Mathematicareturned− 13π√2 .π squared,notcubed.
360 2
WhenIconvincinglyestablishedthatthecubewascorrect,thegoodstaffatWolfram
Researcheventuallyfixedtheerror.Butittooksomeonewhodidhandcalculations
tofixapresumedbugintheircode.
Again, this is not to disparage the Dr. Wolfram’s point about math education.
In fact, I think he would agree with me that there is a fundamental need to have
peoplewhoknowthebasicsverywelltobemakingsurethecomputermodelswe
arerunningareaccurate.
Butthisisallaverylongwayofansweringthequestion,whydowewanttowork
outthesecrazyintegralsandsumswhenwehavegreatcomputersoftwarethatcando
itforus?Buteveniftherewerenoneedforcustodes,evenifthelikesofMathematica
wereperfectandneedednochecking,thereisamoreimportant,overrridingreason
toworkouttheseintegralsandsumsbyhand:
Itisfun!
Ifyouareperusingthroughthisbook,itisnotlikelybecauseyouhavecalculus
homeworkduetomorrow.Itisbecauseyouhaveacuriosityastohowreallydifficult
problemsincalculuscanbesolvedusingcomplexintegration.Itisbecauseyouwant
tolearnthesemethodsforyourself.Thereisapowerbehindthemthatisaddictive.
It is like the classic line, “amuse your friends, confound your enemies” taken in a
wholenewdirection.
Preface ix
WhyComplexIntegration?
Ithinkwehaveansweredthequestionofwhyevaluatetheseintegralsandsumsby
handingeneral.Butunansweredisthequestion,whyarewefocusingoncomplex
integrationtechniques?
TheoriginofthecomplexintegrationtechniquesisaresultofmyworkontheMath
StackExchangewebsite.Onthiswebsite,therearelotsofpeopleaskingotherpeople
likemehowtoevaluatevariousintegralsandsums.Infact,therewouldbeseveral
ofusworkingontheseproblemsatonce,whichwouldresultinverysimilar,good
answers being offered at roughly the same time. Most of the time, these answers
would use fundamentally real techniques—“Feynman’s trick” of differentiating a
parameterundertheintegralsign,Taylorexpansion,orthogonalfunctiondecompo-
sition,forexample.IstartedevaluatingintegralsusingtheResidueTheoremjustas
awaytoputdownadifferentanswer.
Itoccurredtome,however,thattheResidueTheoremasIhadlearneditinschool
was not really a great arrow in my quiver of integration techniques I could bring
tothetable—yet.TheexamplesIhadlearnedinschool—andthesameexamplesI
wasworkingoutintheStackExchangeproblems—wereextremelylimitedtoafew,
scientificallyselectedexamples.Forexample,wearetaughttoevaluatethefollowing
integralusinganalysisofbranchcutsinamultivaluedintegrand:
(cid:5)1
dxxp(1−x)q
0
It turns out, however, that the techniques we are taught only work when
p + q is an (cid:6)integer. So many times we would be taught to perform inte-
grals such as 1dxx−1/3(1 − x)−2/3, but not something more interesting like
(cid:6) 0
1dxx−1/3(1−x)−2/3logx.(Notethatthefactoroflogmaybetreatedasthelimit
0
ofthederivativeof xa asa → 0.)Thereasonwhythisissoandtheevaluationof
themoredifficultintegralusingcomplexintegrationwillbeprovidedinthisbook.
Howwouldwegoaboutattackingthesenewproblems?Bydoingthemcorrectly!
For example, inthe branch cut integrals above, the usual way to approach them is
toconsidersomethingcalleda“dog-bone”contourformedbythebranchcutsabove
and below the integration interval. We would then need to consider a “residue at
infinity” to handle the effect of integrating out to infinity—the residue at infinity
wassimplyaformuladerivedfromanintegralaboutalargecircle.Butwhatifthe
integralaboutthelargecircledidn’tconvergeastheradiusbecameinfinite?Whatif,
instead,therewasasingularpiecethathadtocancelanothersimilarpieceelsewhere
on the contour? By “the contour,” I no longer mean the dog-bone contour but the
logicalextensionofittothelargecircle.
The cancelations of divergences formed the impetus for attacking whole new
classes of problems using complex integration. This is one of the main themes
throughoutthisbook.
x Preface
Anotherthemeistheexpansionofourquiverofproblem-solvingtechniques.Once
we know a more complete form of problems that may be attacked using complex
integration techniques, we may attempt to transform difficult integrals into one of
thoseforms.Awell-knownexampleofthisisthefollowingintegral.
(cid:5)1 1(cid:7)1+x (cid:8)1+2x +2x2(cid:9) (cid:10)
dx log =4πCot−1 φ
x 1−x 1−2x +2x2
−1
whereφisthegoldenratio.Iattackedthisproblembytransformingtheintegralinto
thefollowingform.
(cid:5)1 (cid:7) (cid:8) (cid:9) (cid:5)∞
1 1+x 1+2x +2x2
dx log = dy p(y)logy
x 1−x 1−2x +2x2
−1 0
where p is a rational function of y. I did this because I knew that integrals of this
form, where the poles of p are determined exactly, could be evaluated using the
ResidueTheorem.Inthiscase,eventhoughthedenominatorof pwasdegreeeight,
Iwasabletofindthepolesbecauseofthehighdegreeofsymmetryof p.
Goals
What are the goals one should expect to achieve by reading this book? Generally,
to appreciate whole new types of integrals that may be evaluated using complex
integrationtechniques.Specifically,however,thereareseveralgoals.
1. Appreciatethatpolesandbranchpointsmaybetreatedinaunifiedmannerwhen
itcomestoevaluatingrealintegralsusingcomplexintegration.
2. Gainperspectiveonthetypesofintegralsthatmaybeevaluatedusingcomplex
integration.
3. Learnhowtoconstructevenreasonablysophisticatedcontoursinevaluatingreal
integralsusingcomplexintegration.
4. BecomecomfortablewithCauchyprincipalvaluesandcancelingdivergencesas
partofevaluatingintegralswithcombinedpole/branchpoints.
5. AppreciatetransformtechniquessuchasParseval’sTheoremandtheConvolution
Theoremforevaluatingintegrals.
6. Gainfacilitywithasymptotictechniquesandhowtheymaybeusedtoevaluate
realintegralsandsumsexactly.