ebook img

Complex Analysis with Applications PDF

501 Pages·2018·18.14 MB·English
by  Asmar
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Complex Analysis with Applications

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 ©SpringerNatureSwitzerlandAG2018 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 authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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 N.H.AsmarandL.Grafakos,ComplexAnalysiswithApplications, UndergraduateTextsinMathematics,https://doi.org/10.1007/978-3-319-94063-2 1

Description:
This textbook is intended for a one semester course in complex analysis for upper level undergraduates in mathematics. Applications, primary motivations for this text, are presented hand-in-hand with theory enabling this text to serve well in courses for students in engineering or applied sciences.
See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.