Real and Complex Analysis: Volume 2

688 Pages·2018·5.228 MB·English

by  Rajnikant Sinha

Rajnikant Sinha
Real and
Complex
Analysis
Volume 2
Real and Complex Analysis
Rajnikant Sinha
Real and Complex Analysis
Volume 2
123
Rajnikant Sinha
Varanasi, Uttar Pradesh, India
ISBN978-981-13-2885-5 ISBN978-981-13-2886-2 (eBook)
https://doi.org/10.1007/978-981-13-2886-2
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Preface
The book is an introduction to real and complex analysis that will be useful to
undergraduatestudentsofmathematicsandengineering.Itisdesignedtoequipthe
readerwithtoolsthatwillhelpthemtounderstandtheconceptsofrealanalysisand
complex analysis. In addition, it contains the essential topics of analysis that are
needed for the study offunctional analysis. Its guiding principles help develop the
necessary concepts rigorously with enough detail and with the minimum prereq-
uisites. Further, I have developed the necessary tools to enhance the readability.
Thisbookcontainscompletesolutionstoalmostalltheproblemsdiscussedwithin.
This will be beneficial to readers only if used correctly: readers are encouraged to
look at the solution to a problem only after trying to solve the problem.
Certainly,attimes,thereadermayfindtheproofsexcruciatinglydetailed,butit
is better to be detailed than concise. Furthermore, eliding over detailed calculation
can sometimes be perplexing for the beginners. I have tried to make it a readable
text that caters to a broad audience. This approach should certainly benefit begin-
ners who have not yet tussled with the subject in a serious way.
Thisbookcontainsseveralusefultheorems andtheirproofs intherealm ofreal
and complex analysis. Most of these theorems are the works of some the great
mathematiciansofthe19thand20thcenturies.Inalphabeticalorder,theseinclude:
Arzela, Ascoli, Baire, Banach, Carathéodory, Cauchy, Dirichlet, Egoroff,
Fatou, Fourier, Fubini, Hadamard, Jordan, Lebesgue, Liouville, Minkowski,
Mittag-Leffler,Morera,Nikodym,Ostrowski,Parseval,Picard,Plancherel,Poisson,
Radon, Riemann, Riesz, Runge, Schwarz, Taylor, Tietze, Urysohn, Weierstrass,
and Young. I have spent several years providing their proofs in unprecedented
detail.
There are plenty of superb texts on real and complex analysis, but there is a
dearth of books that blend real analysis with complex analysis. Libraries already
contain several excellent reference books on real and complex analysis, which
interested students can consult for a deeper understanding. It was notmy intention
to replace such books. This book is written under the assumption that students
alreadyknowthefundamentalsofadvancedcalculus.Theproofsofvariousnamed
v
vi Preface
theorems should be considered to be at the core of the book by any reader who is
serious about learning the subject.
This is the second volume of the two-volume book. Volume 2 contains four
chapters: Holomorphic and Harmonic Functions, Conformal Mapping, Analytic
Continuation, and Special Functions. In Chap. 1, we study the holomorphic func-
tions and harmonic functions. Here, we have proved the fundamental theorem of
algebra and global Cauchy theorem. The chapter ends with a discussion on the
Schwarzreflectionprinciple.ThetopicsofdiscussioninChap.2areinfiniteproduct
and the Riemann mapping theorem. In Chap. 3, we have introduced analytic
continuation and proved the monodromy theorem. A branch of logarithm function
is also discussed in this chapter. In Chap. 4, we prove the prime number theorem.
Forthispurpose,alltheneededtoolsofanalyticnumbertheoryaredevelopedfrom
scratches. Finally, we have supplied a beautiful proof of Picard’s little theorem.
I am particularly indebted to Walter Rudin and Paul Richard Halmos for their
letters discussing academic questions. By great good fortune, some colleagues of
minewereabletojoininwiththisenterpriseafewyearsago,someofwhomhave
provideda meticulous reading of themanuscript from a user’sviewpoint. Iextend
my great thanks to all of them for their expert services.
Whilestudyingthisbook,Ihopethatreaderswillexperiencethethrillofcreative
effort and the joy of achievement.
Varanasi, India Rajnikant Sinha
Contents
1 Holomorphic and Harmonic Functions. . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Locally Path-Connected Function . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Representable by Power Series . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Winding Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4 Cauchy’s Theorem in a Triangle . . . . . . . . . . . . . . . . . . . . . . . . 54
1.5 Cauchy’s Theorem in a Convex Set. . . . . . . . . . . . . . . . . . . . . . 59
1.6 Cauchy Integral Formula in a Convex Set . . . . . . . . . . . . . . . . . 63
1.7 Morera’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
1.8 Parseval’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
1.9 Casorati–Weierstrass Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 78
1.10 Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
1.11 Maximum Modulus Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 86
1.12 Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . . . 90
1.13 Cauchy’s Estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
1.14 Open Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
1.15 Global Cauchy Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
1.16 Meromorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
1.17 Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
1.18 Rouché’s Theorem and Hurwitz’s Theorem . . . . . . . . . . . . . . . . 134
1.19 Cauchy–Riemann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
1.20 Dual Space of a Banach Space . . . . . . . . . . . . . . . . . . . . . . . . . 145
1.21 Poisson’s Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
1.22 Mean Value Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
1.23 Schwarz Reflection Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . 181
2 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
2.1 Schwarz Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
2.2 Radial Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
2.3 Infinite Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
2.4 Elementary Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
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viii Contents
2.5 Weierstrass Factorization Theorem. . . . . . . . . . . . . . . . . . . . . . . 217
2.6 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
2.7 Linear Fractional Transformations . . . . . . . . . . . . . . . . . . . . . . . 232
2.8 Arzela–Ascoli Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
2.9 Montel’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
2.10 Riesz Representation Theorem for Bounded Functionals . . . . . . . 261
2.11 Rung’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
2.12 Mittag-Leffler Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
2.13 Simply Connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
2.14 Riemann Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
2.15 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
3 Analytic Continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
3.1 Analytic Continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
3.2 Ostrowski’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
3.3 Hadamard’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
3.4 Modular Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
3.5 Harnack’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
3.6 Nontangential and Radial Maximal Functions. . . . . . . . . . . . . . . 346
3.7 Lebesgue Point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
3.8 Representation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
3.9 Lindelöf’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
3.10 Monodromy Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
3.11 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
3.12 Branch of the Logarithm Function. . . . . . . . . . . . . . . . . . . . . . . 410
3.13 Riemann Surface of the Logarithm Function . . . . . . . . . . . . . . . 419
3.14 Infinite Product Converges Absolutely . . . . . . . . . . . . . . . . . . . . 421
4 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
4.1 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
4.2 Laurent’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
4.3 Evaluation of Improper Riemann Integrals . . . . . . . . . . . . . . . . . 453
4.4 Meromorphic Function as a Quotient of Holomorphic
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
4.5 Euler’s Gamma Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
4.6 Beta Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
4.7 Euler’s Constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
4.8 Riemann Zeta Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
4.9 Gauss’s Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
4.10 Preparation for the Prime Number Theorem . . . . . . . . . . . . . . . . 548
4.11 Abscissa of Convergence of a Dirichlet Series . . . . . . . . . . . . . . 560
4.12 Riemann Zeta Function Has a Simple Pole at 1 . . . . . . . . . . . . . 580
Contents ix
4.13 Riemann Hypothesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
4.14 Chebyshev’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
4.15 Prime Number Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
4.16 Picard’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663
Bibliography .. .... .... .... ..... .... .... .... .... .... ..... .... 679
About the Author
RajnikantSinha isFormerProfessorofMathematicsatMagadhUniversity,Bodh
Gaya,India.Asapassionatemathematician,hehaspublishednumerousinteresting
research findings in international journals and books, including Smooth Manifolds
(Springer)andthecontributedbookSolutionstoWeatherburn’sElementaryVector
Analysis. His research focuses on topological vector spaces, differential geometry
and manifolds.
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