A Guide to Complex Variables

202 Pages·2008·1.042 MB·English

by  Krantz S.G.

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A Guide
to
Complex Variables
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© 2008 by
The Mathematical Association of America (Incorporated)
Library of Congress Catalog Card Number 2008927550
Print Edition ISBN 978-0-88385-338-2
Electronic Edition ISBN 978-0-88385-914-8
Printed in the United States of America.
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TheDolcianiMathematicalExpositions
NUMBERTHIRTY-TWO
MAAGuides#1
A Guide
to
Complex Variables
Steven G. Krantz
Washington University, St. Louis
®
PublishedandDistributedby
TheMathematicalAssociationofAmerica
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DOLCIANIMATHEMATICALEXPOSITIONS
CommitteeonPublications
JamesDaniel,Chair
DolcianiMathematicalExpositionsEditorialBoard
UnderwoodDudley,Editor
JeremyS.Case
TevianDray
RobertL.Devaney
JerroldW.Grossman
VirginiaE.Knight
MarkA.Peterson
JonathanRogness
JoeAlynStickles
JamesS.Tanton
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The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical
AssociationofAmericawasestablishedthroughagenerousgifttotheAssociation
fromMaryP.Dolciani,ProfessorofMathematicsatHunterCollegeoftheCityUni-
versityofNewYork.Inmakingthegift,ProfessorDolciani,herselfanexceptionally
talentedandsuccessfulexpositorofmathematics,hadthepurposeoffurtheringthe
idealofexcellenceinmathematicalexposition.
TheAssociation,foritspart,wasdelightedtoacceptthegraciousgestureinitiat-
ingtherevolvingfundforthisseriesfromonewhohasservedtheAssociationwith
distinction,bothasamemberoftheCommitteeonPublicationsandasamemberof
theBoardofGovernors.ItwaswithgenuinepleasurethattheBoardchosetoname
theseriesinherhonor.
Thebooksintheseriesareselectedfortheirlucidexpositorystyleandstimu-
latingmathematicalcontent.Typically,theycontainanamplesupplyofexercises,
manywithaccompanyingsolutions.Theyareintendedtobesufficientlyelementary
fortheundergraduateandeventhemathematicallyinclinedhigh-schoolstudentto
understandandenjoy,butalsoto beinterestingandsometimeschallengingto the
moreadvancedmathematician.
1. MathematicalGems,RossHonsberger
2. MathematicalGemsII,RossHonsberger
3. MathematicalMorsels,RossHonsberger
4. MathematicalPlums,RossHonsberger(ed.)
5. GreatMomentsinMathematics(Before1650),HowardEves
6. MaximaandMinimawithoutCalculus,IvanNiven
7. GreatMomentsinMathematics(After1650),HowardEves
8. MapColoring,Polyhedra,andtheFour-ColorProblem,DavidBarnette
9. MathematicalGemsIII,RossHonsberger
10. MoreMathematicalMorsels,RossHonsberger
11. Old and New Unsolved Problems in Plane Geometry and Number Theory,
VictorKleeandStanWagon
12. ProblemsforMathematicians,YoungandOld,PaulR.Halmos
13. ExcursionsinCalculus:AnInterplayoftheContinuousandtheDiscrete,Robert
M.Young
14. The WohascumCountyProblemBook, George T. Gilbert, Mark Krusemeyer,
andLorenC.Larson
15. LionHuntingandOtherMathematicalPursuits:ACollectionofMathematics,
Verse,andStoriesbyRalphP.Boas,Jr.,editedbyGeraldL.Alexandersonand
DaleH.Mugler
16. LinearAlgebraProblemBook,PaulR.Halmos
17. FromErdo˝stoKiev:ProblemsofOlympiadCaliber,RossHonsberger
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18. WhichWayDidtheBicycleGo?...andOtherIntriguingMathematicalMyster-
ies,JosephD.E.Konhauser,DanVelleman,andStanWagon
19. InPo´lya’sFootsteps:MiscellaneousProblemsandEssays,RossHonsberger
20. DiophantusandDiophantineEquations,I.G.Bashmakova(UpdatedbyJoseph
SilvermanandtranslatedbyAbeShenitzer)
21. LogicasAlgebra,PaulHalmosandStevenGivant
22. Euler:TheMasterofUsAll,WilliamDunham
23. TheBeginningsandEvolutionofAlgebra,I.G.BashmakovaandG.S.Smirnova
(TranslatedbyAbeShenitzer)
24. MathematicalChestnutsfromAroundtheWorld,RossHonsberger
25. CountingonFrameworks:MathematicstoAidtheDesignofRigidStructures,
JackE.Graver
26. MathematicalDiamonds,RossHonsberger
27. ProofsthatReallyCount:TheArtofCombinatorialProof,ArthurT.Benjamin
andJenniferJ.Quinn
28. MathematicalDelights,RossHonsberger
29. Conics,KeithKendig
30. Hesiod’s Anvil: falling and spinning through heaven and earth, Andrew J.
Simoson
31. AGardenofIntegrals,FrankE.Burk
32. AGuidetoComplexVariables(MAAGuides#1),StevenG.Krantz
MAAServiceCenter
P.O.Box91112
Washington,DC20090-1112
1-800-331-1MAA FAX:[phone]
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To
Paul Painleve´
(1863–1933)
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Preface
MosteverymathematicsPh.D.studentmusttakeaqualifyingexamincom-
plex variables.The taskis a bitdaunting.Complexvariablesis oneof the
oldest areas in mathematics, itis beautifuland compelling, and there is a
plethoraofmaterial.Itsliteratureisvastanddiverse.Thereareagreatmany
textbooksinthesubject, buteach has a differentpointofview and places
differentemphasesaccordingtothetastesoftheauthor.
Thusitisabitdifficultforthestudenttofocusonwhataretheessential
parts of this subject. What must one absolutely know for the qualifying
exam?Whatwillbeasked?Whattechniqueswillbestressed?Whatarethe
keyfacts?
Thepurposeofthisbookistoanswerthesequestions.Thisisdefinitely
nota comprehensivetextbooklike[GRK].Itisratheran entreetothedis-
cipline.Thebookcanalsobeconsideredasarefresherforthosewhohave
not seen this lovely subject for a while. It will tellyou the key ideas in a
first-semester graduate course in the subject, map outthe importanttheo-
rems,andindicatemostoftheproofs.By“indicate”wemeanthat(i)ifthe
proofisshortthenweincludeit,(ii)iftheproofisofmediumlengththen
weoutlineit,and(iii)iftheproofislongthenwesketchit.
Thisbookhasplentyoffigures,plentyofexamples, copiouscommen-
tary, and even in-text exercises. Since it is not a formal textbook, it does
nothave exercise sets, nora TableofNotation.We do,however, includea
thoroughGlossaryinordertofacilitatethereader’srapidacclimatizationto
thesubject.
This is meant to be a breezy book that you could read at one or two
sittings, just to get the sense of what this subject is about and how it fits
together.Itisquitedifferentfromatypicalmathematicstextormonograph.
After reading this book (or even while reading this book), you may want
topickupa moretraditionalandcomprehensive tomeandworkyourway
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