Table Of ContentGraduate Texts in Mathematics
Richard Beals
Roderick S. C. Wong
Explorations in
Complex
Functions
Graduate Texts in Mathematics 287
Graduate Texts in Mathematics
Series Editors
Sheldon Axler
San Francisco State University, San Francisco, CA, USA
Kenneth Ribet
University of California, Berkeley, CA, USA
Advisory Editors
Alejandro Adem, University of British Columbia
David Eisenbud, University of California, Berkeley & MSRI
Brian C. Hall, University of Notre Dame
Patricia Hersh, University of Oregon
J. F. Jardine, University of Western Ontario
Jeffrey C. Lagarias, University of Michigan
Eugenia Malinnikova, Stanford University
Ken Ono, University of Virginia
Jeremy Quastel, University of Toronto
Barry Simon, California Institute of Technology
Ravi Vakil, Stanford University
Steven H. Weintraub, Lehigh University
Melanie Matchett Wood, Harvard University
Graduate Texts in Mathematics bridge the gap between passive study and
creative understanding, offering graduate-level introductions to advanced topics
in mathematics. The volumes are carefully written as teaching aids and highlight
characteristic features of the theory. Although these books are frequently used as
textbooks in graduate courses, they are also suitable for individual study.
More information about this series at http://www.springer.com/series/136
Richard Beals Roderick S. C. Wong
(cid:129)
Explorations in Complex
Functions
123
Richard Beals Roderick S.C. Wong
Department ofMathematics Department ofMathematics
Yale University City University of HongKong
NewHaven, CT,USA Kowloon, HongKong
ISSN 0072-5285 ISSN 2197-5612 (electronic)
Graduate Textsin Mathematics
ISBN978-3-030-54532-1 ISBN978-3-030-54533-8 (eBook)
https://doi.org/10.1007/978-3-030-54533-8
MathematicsSubjectClassification: 30-01,33-01,30D35,33E05,11M06
©SpringerNatureSwitzerlandAG2020
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, expressed or implied, with respect to the material contained
hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard
tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
Afriendofoneoftheauthorshaswrittenabookwiththe(ironic)titleIntheMidst
of Plenty. This would be an apt, if unusual, choice of title for this book. A first
course incomplex analysisintroduces keysthatunlockmany doors. One suchkey
is the residue theorem, in its many forms; another is analytic continuation. The
doorsopenontomanysubjectsofinterest.Toomanysubjects,infact,tocoverina
single follow-up course.
This book assumes as background a standard first course in complex analysis.
Ourpurposeistoproviderelativelybrief,butself-contained,introductionstomany
ofthesubjectsalludedtoabove.Some ofthese subjects arewithinthemainstream
of complex analysis itself. Other topics provide tools that are widely used within
pure mathematics, or that have applications beyond mathematics, or both. Some
topicscomeupindifferentcontextsindifferentchapters.Forexample,therearetwo
proofsofPicard’s“little”theorem,andseveraldiscussionsoftheparametrizationof
algebraic curves.
Chapter1isasummary,withselectedproofs,ofmaterialfromabasiccoursein
complex analysis. Included are Cauchy’s theorem and consequences (integral for-
mula, series expansion, residue theorem, maximum modulus principle, reflection
principle). Also included are the basics of infinite products and analytic continua-
tion.ForlaterusethechaptercontainsintroductionstotheStieltjesintegralrelative
toajumpfunction,toHilbertspaces,andtoLp spaces(ascompletionsofspacesof
nice functions).
Chapter2introducestheRiemannsphereanditsautomorphismgroup,thegroup
of linear fractional transformations. The cross product and general mapping prop-
erties are covered, and theautomorphism groups ofthe half-plane and the disk are
identified.Theseconsiderationsleadnaturallytohyperbolicgeometryinthediskor
half-plane, which is the subject of Chapter 3.
Chapter4introducesharmonicfunctionsintheplane.TheDirichletproblemand
Poisson’s formula lead to the Weierstrass approximation theorems, and to the
Riesz-FischertheoremforFourierseries.TheSchwarzreflectionprincipleprepares
thewaytotheresultsonboundarybehaviorofconformalmapsthatarecoveredin
several later chapters.
v
vi Preface
Riemann’s mapping theorem, the Schwarz–Christoffel formulas, and univalent
functions are covered in Chapter 5. In Chapter 6 the Schwarzian derivative is
introduced as a measure of curvature, followed by the proof that mappings to
curvilinear polygons are quotients of solutions of Fuchsian equations. Particular
cases of this are mappings to triangles, or to regular polygons, shown to be quo-
tients of hypergeometric functions.
Chapter7coversanalyticcontinuation,Riemannsurfacesoffunctions,algebraic
curves, and compact Riemann surfaces. The chapter concludes with a very brief
introduction to surfaces of higher genus, exemplified by the Bolza surface. This
chapterleads,inaway,tothefollowingtwochapters,aswellastolaterchapterson
elliptic functions.
TheWeierstrassproducttheoremandHadamard’sproductformulaforfunctions
of finite order are the focus of Chapter 8. Application is made to Riemann’s xi
function, and to an eigenvalue problem. Chapter 9 introduces Nevanlinna’s value
distributiontheoryforentiremeromorphicfunctions,startingwithJensen’stheorem
and the Nevanlinna and Ahlfors–Shimizu characteristics. Nevanlinna’s second
fundamental theorem is shown to have applications to two theorems of Picard.
Chapter 10 introduces Euler’s two definitions of the gamma function, the beta
function, the reflection formula, and Legendre’s duplication formula. Included is a
far-reaching extension, due to Stieltjes, of Stirling’s asymptotic approximation,
which is important for the study of the Riemann hypothesis in Chapter 13.
Chapters 11 through 13 make up an introduction to analytic number theory.
Riemann’szetafunctionandxifunctioninChapter11leadtoDirichletL-functions
andDirichlet’stheoremonprimesinarithmeticprogressionsinChapter12.Chapter
13 treats theprimenumbertheoreminthecontextof theRiemannhypothesis, and
alsotherelationoftheRiemannhypothesistotheaccuracyofGauss’sapproximate
formula for the distribution of primes.
Chapters14through16introduceellipticfunctionsandthreeapproachestotheir
construction. The general theory and construction by means of theta functions, are
covered in Chapter 14. Chapters 15 and 16 are independent of each other. The
pendulum equation leads to the Jacobi elliptic functions in Chapter 15.
Weierstrass’s direct construction, starting with the period lattice, is covered in
Chapter 16.
TheWeierstrasstheoryofChapter16leadstothestudyofthemodularfunction,
Picard’s theorems, and a glance into automorphic functions and the J function in
Chapter 17. (An appendix notes the connection to “moonshine” and the monster
group.)
Chapter 18 introduces approximate identities, Schwartz functions, and the
CauchyandHilberttransforms.TheCauchytransformleadsnaturallytotheFourier
transform in L1ðRÞ and in L2ðRÞ. This, in turn, prepares the way for the following
two chapters, which are independent of each other.
Preface vii
Chapter 19 treats the Phragmén–Lindelöf principle. This principle is applied in
Hardy'scharacterizationoftheGaussianprobabilitydistribution,intheproofofthe
Paley–Wiener theorem, and in the proof of a theorem of Hardy concerning func-
tions of exponential type.
AtheoremofWiener,anditsgeneralizationbyLévy,areprovedinChapter20.
These theorems are used in a version, due to Gohberg and Krein, of the Wiener–
Hopf approach to equations of convolution type on the half-line.
Chapters 21 through 23 are generally independent of each other and of earlier
chapters (other than Chapter 1). Chapter 21 treats some tauberian theorems, from
TauberandHardythroughKaramataandWiener.AtheoremofMalliavinprovides
an error estimate that is applicable to distribution of eigenvalues. The section on
Wiener’s theorem has some dependence on results from Chapter 18.
Chapter 22 introduces the method of steepest descent. Applications include
asymptotics of the Airy integral, and the Hardy–Ramanujan theorem on asymp-
totics of the partition function. Chapter 23 sketches the complex interpolation
method,interpolationofLp spaces,andtheRiesz–Thorintheorem,withapplication
to Fourier series.
We have tried to make the various presentations self-contained. This has led us
to some short excursions into real analysis, functional analysis, and algebra. In
particular we have included expositions of irreducibility for polynomials in two
variables, and of the character theory offinite abelian groups.
We have also tried to make the various chapters as independent as possible.
Charts that show the principal dependence relations follow this preface.
The first author used selections of drafts of a number of these chapters in two
versionsofasecond-semestercomplexvariablesclassatYalein2018and2019.He
is grateful to the students for their indulgence and attentiveness. Their pertinent
questionsandcomments ledtoanumberofcorrectionsandclarifications.Thefirst
author is also grateful to the staff of the Liu Bei Ju Center of City University of
Hong Kong, as well as Dr. Huang Xiaomin and Dr. Wang Xiangsheng for their
assistance and technical help in June 2019. The second author also thanks
Dr.HuangXiaominforherconsiderablehelpasapostdocin2019.Bothauthorsare
grateful to Alberto Guzman for critical reading of much of the manuscript, and to
their wives, Nancy and Edwina, for their unfailing moral support.
New Haven, CT, USA Richard Beals
Kowloon, Hong Kong Roderick S. C. Wong
May 2020
viii Preface
Dependence relations among chapters: 2 charts
Basics
Linearfractional Harmonic Elliptic
transformations functions functions
hyperbolic Conformal Jacobielliptic Weierstrass
geometry mapping functions ellipticfcns
Schwarzian Automorphic
derivative functions
Basics
Gamma,Beta Entire Riemann
functions functions surfaces
Zeta Valuedistribution
function theory
L–functions Riemann
andprimes hypothesis
Preface ix
Dependence relations among chapters: one more chart
Basics
Integral Tauberian Steepest Complex
transforms theorems descent interpolation
Phragmen–Lindelo¨f; Wiener–Hopf
Paley–Wiener method
