Complex Analysis with Applications

501 Pages·2018·18.14 MB·English

by  Asmar

Undergraduate Texts in Mathematics
Nakhlé H. Asmar
Loukas Grafakos
Complex Analysis
with Applications
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics
Series Editors:
Sheldon Axler
San Francisco State University, San Francisco, CA, USA
Kenneth Ribet
University of California, Berkeley, CA, USA
Advisory Board:
Colin Adams, Williams College
David A. Cox, Amherst College
L. Craig Evans, University of California, Berkeley
Pamela Gorkin, Bucknell University
Roger E. Howe, Yale University
Michael E. Orrison, Harvey Mudd College
Lisette G. de Pillis, Harvey Mudd College
Jill Pipher, Brown University
Fadil Santosa, University of Minnesota
Undergraduate Texts in Mathematics are generally aimed at third- and fourth-
year undergraduate mathematics students at North American universities. These
texts strive to provide students and teachers with new perspectives and novel
approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation
ofinterrelations among different aspectsofthesubject.Theyfeature examples that
illustrate key concepts as well as exercises that strengthen understanding.
More information about this series at http://www.springer.com/series/666
é
Nakhl H. Asmar Loukas Grafakos
(cid:129)
Complex Analysis
with Applications
123
NakhléH.Asmar Loukas Grafakos
Department ofMathematics Department ofMathematics
University of Missouri University of Missouri
Columbia, MO,USA Columbia, MO,USA
ISSN 0172-6056 ISSN 2197-5604 (electronic)
Undergraduate Texts inMathematics
ISBN978-3-319-94062-5 ISBN978-3-319-94063-2 (eBook)
https://doi.org/10.1007/978-3-319-94063-2
LibraryofCongressControlNumber:2018947485
MathematicsSubjectClassification(2010): 97-XX,97I80
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Preface
Our goal in writing this book was to present a rigorous and self-contained intro-
duction to complex variables and their applications. The book is based on notes
fromanundergraduatecourseincomplexvariablesthatwetaughtattheUniversity
of Missouri and on the book Applied Complex Analysis with Partial Differential
Equations by N. Asmar (with the assistance of Gregory C. Jones), published by
Prentice Hall in 2002.
A course in complex variables must serve students with different mathematical
backgrounds from engineering, physics, and mathematics. The challenge in
teaching such a course is to find a balance between rigorous mathematical proofs
and applications. While recognizing the importance of developing proof-writing
skills, we have tried not to let this process hinder a student’s ability to understand
andappreciatetheapplicationsofthetheory.Thisbookhasbeenwrittensothatthe
instructorhastheflexibilitytochoosethelevelofproofstopresenttotheclass.We
have included complete proofs of most results. Some proofs are very basic (e.g.,
those found in the early sections of each chapter); others require a deeper under-
standing of calculus (e.g., use of differentiability in Sections 2.4, 2.5); and yet
otherspropelthestudentstothegraduatelevelofmathematics.Thelatterarefound
in optional sections, such as Section 3.5.
Thecorematerialforaone-semestercourseiscontainedinthefirstfivechapters
ofthebook.Aimingforaflexibleexposition,wehavegivenatleasttwoversionsof
Cauchy’stheorem, whichisthemostfundamentalresult contained inthisbook.In
Section 3.4 we provide a quick proof of Cauchy’s theorem as a consequence of
Green’s theorem which covers practically most applications. Then in Section 3.4
wediscussamoretheoreticalversionofCauchy’stheoremforarbitraryhomotopic
curves;thisapproachmaybeskippedwithoutalteringtheflowofthepresentation.
Thebookcontainsclassicalapplicationsofcomplexvariablestothecomputationof
definite integrals and infinite series. Further applications are given related to con-
formal mappings and to Dirichlet and Neumann problems; these boundary value
problemsmotivatetheintroductiontoFourierseries,whicharebrieflydiscussedin
Section 6.4.
v
vi Preface
Theimportancethatweattributetotheexercisesandexamplesisclearfromthe
spacetheyoccupyinthebook.Wehaveincludedfarmoreexamplesandexercises
than can be covered in one course. The examples are presented in full detail. As
with the proofs, the objective is to give the instructor the option to choose the
examples that are suitable to the class, while providing the students many more
illustrations to assist them with the homework problems.
The exercises vary in difficulty from straightforward ones to more involved
project problems. Hints are provided in many cases. Solutions to the exercises can
be provided upon request free-of-charge to instructors who use the text.
Complimentary solutions to every-other-odd exercise and other material related to
the book, such as errata and improvements, can be found at the Web site:
https://www.springer.com/us/book/9783319940625
We wish to thank Professors Tanya Christiansen and Stephen Montgomery-
Smithwhohaveusedthetextintheclassroomandhaveprovideduswithvaluable
comments. We also wish to thank the following individuals who provided us with
assistance and corrections: Dimitrios Betsakos, Suprajo Das, Hakan Delibas,
Haochen Ding, Michael Dotzel, Nikolaos Georgakopoulos, Rebecca Heinen, Max
Highsmith, Jeremy Hunn, Dillon Lisk, Caleb Mayfield, Vassilis Nestoridis,
GeorgiosNtosidis,AdisakSeesanea,Yiorgos-SokratisSmyrlis,SuzanneTourville,
Yanni Wu, and Run Yan.
We are thankful to all of Springer’s excellent staff, but especially to Elizabeth
Loew for being so helpful during the preparation of the book. Finally, we are
especially grateful for the support and encouragement that we have received from
our families. This book is dedicated to them.
Columbia, Missouri, USA Nakhlé H. Asmar
Loukas Grafakos
Contents
1 Complex Numbers and Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Polar Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Complex Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.5 Sequences and Series of Complex Numbers. . . . . . . . . . . . . . . . . 52
1.6 The Complex Exponential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
1.7 Trigonometric and Hyperbolic Functions . . . . . . . . . . . . . . . . . . . 75
1.8 Logarithms and Powers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.1 Regions of the Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.2 Limits and Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2.3 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
2.4 Differentiation of Functions of Two Real Variables . . . . . . . . . . . 123
2.5 The Cauchy-Riemann Equations . . . . . . . . . . . . . . . . . . . . . . . . . 130
3 Complex Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.1 Paths (Contours) in the Complex Plane . . . . . . . . . . . . . . . . . . . . 139
3.2 Complex Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3.3 Independence of Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
3.4 Cauchy’s Integral Theorem for Simple Paths . . . . . . . . . . . . . . . . 177
3.5 The Cauchy-Goursat Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 181
3.6 Cauchy Integral Theorem For Simply Connected Regions. . . . . . . 187
3.7 Cauchy’s Theorem for Multiply Connected Regions. . . . . . . . . . . 198
3.8 Cauchy Integral Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
3.9 Bounds for Moduli of Analytic Functions . . . . . . . . . . . . . . . . . . 218
vii
viii Contents
4 Series of Analytic Functions and Singularities . . . . . . . . . . . . . . . . . 227
4.1 Sequences and Series of Functions . . . . . . . . . . . . . . . . . . . . . . . 227
4.2 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
4.3 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
4.4 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
4.5 Zeros and Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
4.6 Schwarz’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
5 Residue Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
5.1 Cauchy’s Residue Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
5.2 Definite Integrals of Trigonometric Functions. . . . . . . . . . . . . . . . 302
5.3 Improper Integrals Involving Rational and Exponential
Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
5.4 Products of Rational and Trigonometric Functions . . . . . . . . . . . . 320
5.5 Advanced Integrals by Residues . . . . . . . . . . . . . . . . . . . . . . . . . 332
5.6 Summing Series by Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
5.7 The Counting Theorem and Rouché’s Theorem . . . . . . . . . . . . . . 350
6 Harmonic Functions and Applications . . . . . . . . . . . . . . . . . . . . . . . 367
6.1 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
6.2 Dirichlet Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
6.3 Dirichlet Problem and the Poisson Integral on a Disk. . . . . . . . . . 387
6.4 Harmonic Functions and Fourier Series . . . . . . . . . . . . . . . . . . . . 395
7 Conformal Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
7.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
7.2 Linear Fractional Transformations . . . . . . . . . . . . . . . . . . . . . . . . 411
7.3 Solving Dirichlet Problems with Conformal Mappings . . . . . . . . . 429
7.4 The Schwarz-Christoffel Transformation . . . . . . . . . . . . . . . . . . . 443
7.5 Green’s Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
7.6 Poisson’s Equation and Neumann Problems. . . . . . . . . . . . . . . . . 470
Appendix. .... .... .... .... ..... .... .... .... .... .... ..... .... 483
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 485
Chapter 1
Complex Numbers and Functions
√
Dismiss√ingmentaltortures,andmultiplying5+ −15
by 5− −15, we obtain 25−(−15). Therefore the
product is 40. ...and thus far does arithmetical sub-
tlety go, of which this, the extreme, is, as I have said,
sosubtlethatitisuseless.
-GirolamoCardano(orCardan)(1501–1576)
[First explicit use of complex numbers, which ap-
peared around 1545 in Cardan’s solution of the prob-
lem of finding two numbers whose sum is 10 and
whoseproductis40.]
Thischapterstartswiththeearlydiscoveryofcomplexnumbersandtheirrolein
solvingalgebraicequations.Complexnumbershavethealgebraicformx+iy,where
x,yarerealnumbers,buttheycanalsobegeometricallyrepresentedasvectors(x,y)
intheplane.Bothrepresentationshaveimportantadvantages;thefirstoneiswell-
suitedforalgebraicmanipulationswhilethesecondprovidessignificantgeometric
intuition.Thereisalsoanaturalnotionofdistancebetweencomplexnumbersthat
satisfies the familiar triangle inequality. Complex numbers also have a polar form
(r,θ) based on their distance r to the origin and angle θ from the positive real
semi-axis.Thisalternativerepresentationprovidesadditionalinsight,bothalgebraic
andgeometric,andthisisexplicitlymanifestedeveninsimpleoperations,suchas
multiplicationanddivision.
Complex analysis is in part concerned with the study of complex-valued func-
tionsofacomplexvariable.Themostimportantofthesefunctionsisthecomplex
exponentialez whichisusedinthedefinitionofthetrigonometricandlogarithmic
functions.Sincewecannotplotthegraphsofcomplex-valuedfunctionsofacom-
plexvariable(thiswouldrequirefourdimensions),wevisualizethesefunctionsas
mappings from one complex plane, the z-plane, into another plane, the w-plane.
Complex-valued functions of a complex variable and their mapping properties are
exploredinthischapter.
Complex numbers, like many other ideas in mathematics, have significant ap-
plications in the sciences and can be used to solve real-world problems. Some of
theseapplicationsarediscussedinthelasttwochapters.Inthischaptertheapplica-
tionsarelimitedtofindingrootsofcertainpolynomial,algebraic,andtrigonometric
equations.
©SpringerInternationalPublishingAG,partofSpringerNature2018 1
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