Handbook of Complex Variables

300 Pages·1999·13.295 MB·English

by  Steven G. Krantz (auth.)

Handbook of Complex Variables
Steven G. Krantz
Handbook of
Complex Variables
With 102 Figures
Springer Science+Business Media, LLC
Steven G. Krantz
Department of Mathematics
Washington University in St. Louis
St. Louis, MO 63130
USA
Library of Congress Cataloging.in·Publication Data
Krantz, Steven G. (Steven George), 1951-
Handbook of complex variab1es / Steven G. Krantz.
p. cm.
Includes bib1iographica1 references and index.
ISBN 978-1-4612-7206-9 ISBN 978-1-4612-1588-2 (eBook)
DOI 10.1007/978-1-4612-1588-2
1. Functions of complex variab1es. 2. Mathematical analysis.
1. Title.
QA331.7.K744 1999
515'.9-dc21 99.20156
CIP
AMS Subject C1assifications: 30-00, 32-00, 33-00
Printed on acid-free paper.
O»
© 1999 Springer Science+Business Media New York ®
ao»
Originally published by Birkhăuser Boston in 1999
Softcover reprint of the hardcover 1s t edition 1999
AII rights reserved. This work may not be translated or copied in whole or in part without the
written permission ofthe publisher (Springer Science+Business Media, LLC ), except for brief
excerpts in connection with reviews or scholarly analysis. Use in connection with any form of
information storage and retrieval, electronic adaptation, computer software, or by similar or
dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the
former are not especially identified, is not to be taken as a sign that such names, as understood by
the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 978-1-4612-7206-9
SPIN 19954551
J.t.'IEX.
Typeset by the author in
987654321
To the memory ofLars Valerian Ahlfors, 1907-1996.
Contents
Preface xix
List ofFigures xxi
1 The Complex Plane 1
1.1 Complex Arithmetic . . . . . . . 1
1.1.1 The Real Numbers. . . 1
1.1.2 The Complex Numbers 1
1.1.3 Complex Conjugate . . 2
1.1.4 Modulus ofa Complex Number 2
1.1.5 The Topology ofthe Complex Plane. 3
1.1.6 The Complex Numbers as a Field . . 6
1.1.7 TheFundamental Theorem ofAlgebra. 7
1.2 The Exponential and Applications . . . . . . . 7
1.2.1 The Exponential Function 7
1.2.2 The Exponential Using Power Series. 8
1.2.3 Laws ofExponentiation . . . . . . 8
1.2.4 Polar Form ofa Complex Number . . 8
1.2.5 Roots ofComplex Numbers. . . . . . 10
1.2.6 The Argument ofa Complex Number 11
1.2.7 Fundamental Inequalities . . . . . . . 12
1.3 Holomorphic Functions. . . . . . . . . . . . . . 12
1.3.1 Continuously Differentiable and Ck Functions 12
1.3.2 The Cauchy-Riemann Equations. . 13
1.3.3 Derivatives.. . . . . . . . . . . . . 13
1.3.4 Definition ofHolomorphic Function 14
1.3.5 The Complex Derivative 15
vzz
viii Contents
1.3.6 Alternative Terminology for
Holomorphic Functions . . . 16
1.4 The Relationship ofHolomorphic and
Harmonic Functions . . . . . . . . . . 16
1.4.1 Harmonic Functions. . . . . 16
1.4.2 Holomorphic and Harmonic Functions . 17
2 Complex Line Integrals 19
2.1 Real and Complex Line Integrals 19
2.1.1 Curves....... . . 19
2.1.2 Closed Curves . . . . . 19
2.1.3 Differentiable and Ck Curves . 21
2.1.4 Integrals on Curves . . . . . . 21
2.1.5 The Fundamental Theorem ofCalculus
along Curves . . . . . . . . 22
2.1.6 The Complex Line Integral . . . . 22
2.1.7 Properties oflntegrals . . . . . . . 22
2.2 Complex Differentiability and Conformality 23
2.2.1 Limits..... . . . . . . 23
2.2.2 Continuity............. 24
2.2.3 The Complex Derivative ..... 24
2.2.4 Holomorphicity and the Complex Derivative 24
2.2.5 Conformality . . . . . . . . . . . . 25
2.3 The Cauchy Integral Theorem and Formula . . . . . 26
2.3.1 The Cauchy Integral Formula. . . . . . . . 26
2.3.2 The Cauchy Integral Theorem, Basic Form 26
2.3.3 More General Forms ofthe Cauchy Theorems 26
2.3.4 Deformability ofCurves . . . . . 28
2.4 A Codaon the Limitations ofthe Cauchy
Integral Formula . . . . . . . . . . 28
3 Applications ofthe Cauchy Theory 31
3.1 The Derivatives ofa Holomorphic Function 31
3.1.1 A Formula for the Derivative . . . 31
3.1.2 The Cauchy Estimates 31
3.1.3 Entire Functions and Liouville's Theorem. 31
3.1.4 The Fundamental Theorem ofAlgebra. . 32
3.1.5 Sequences ofHolomorphic Functions and
their Derivatives . . . . . . . . . . . . 33
3.1.6 The Power Series Representation ofa
Holomorphic Function . . . . . . . 34
3.1.7 TableofElementary Power Series . . 35
3.2 The Zeros ofa Holomorphic Function ..... 36
3.2.1 The Zero Set ofa Holomorphic Function 36
Contents ix
3.2.2 Discrete Sets and Zero Sets . . . . . . 37
3.2.3 Uniqueness ofAnalytic Continuation 38
4 Isolated Singularities and Laurent Series 41
4.1 The Behavior ofa Holomorphic Function near an
Isolated Singularity. . . . . . . . . . . . 41
4.1.1 Isolated Singularities ..... 41
4.1.2 A Holomorphic Function on a
Punctured Domain . 41
4.1.3 Classification ofSingularities . 41
4.1.4 Removable Singularities, Poles, and
Essential Singularities . . 42
4.1.5 The Riemann Removable
Singularities Theorem . . . . . . . . 42
4.1.6 The Casorati-Weierstrass Theorem. 43
4.2 Expansion around Singular Points . 43
4.2.1 Laurent Series . 43
4.2.2 Convergence ofa Doubly Infinite Series 43
4.2.3 Annulus ofConvergence. . . . . . . . . 44
4.2.4 Uniqueness ofthe Laurent Expansion . 44
4.2.5 The Cauchy Integral Formulafor an Annulus . 45
4.2.6 Existence ofLaurent Expansions ... 45
4.2.7 Holomorphic Functions with Isolated
Singularities .. . . . . . . . . . . . . 45
4.2.8 Classification ofSingularities in Terms of
Laurent Series . . . . . . . . 46
4.3 Examples ofLaurent Expansions . . . . . . . . . . 46
4.3.1 Principal Part ofa Function . 46
4.3.2 Algorithm for Calculating the Coefficients ofthe
Laurent Expansion. . . . . . . . . . . 48
4.4 The Calculus ofResidues . 48
4.4.1 Functions with Multiple Singularities 48
4.4.2 The Residue Theorem . . . . . . . . . 48
4.4.3 Residues........... . . . . . 49
4.4.4 The Index or Winding Number ofa Curve
about a Point . 49
4.4.5 Restatement ofthe Residue Theorem . . . 50
4.4.6 Method for Calculating Residues . . . . . . 50
4.4.7 Summary Charts ofLaurent Series and Residues 51
4.5 Applications to the Calculation ofDefinite
Integrals and Sums . 51
4.5.1 The Evaluation ofDefinite Integrals 51
4.5.2 A Basic Example . 52
4.5.3 Complexification ofthe Integrand . 54
x Contents
4.5.4 An Example with a More Subtle Choice
ofContour . 56
4.5.5 Making the Spurious Part ofthe
Integral Disappear . . . . . . . . 58
4.5.6 The Use ofthe Logarithm. . . . . 60
4.5.7 Summing a Series Using Residues 62
4.5.8 Summary Chart ofSome
Integration Techniques . 63
4.6 Meromorphic Functions and Singularities at
Infinity . 63
4.6.1 Meromorphic Functions . 63
4.6.2 Discrete Sets and Isolated Points. 63
4.6.3 Definition ofMeromorphic Function 64
4.6.4 Examples ofMeromorphic Functions. 64
4.6.5 Meromorphic Functions with Infinitely
Many Poles . 66
4.6.6 Singularities at Infinity . . . . . . . 66
4.6.7 The Laurent Expansion at Infinity . 67
4.6.8 Meromorphic at Infinity. . . . 67
4.6.9 Meromorphic Functions in the
Extended Plane . . . . . . . . 67
5 The Argument Principle 69
5.1 Counting Zeros and Poles . . . . . . . . 69
5.1.1 Local Geometric Behavior ofa
Holomorphic Function . . . . . 69
5.1.2 Locating the Zerosofa Holomorphic Function 69
5.1.3 ZeroofOrder n . . . . . . . . . . . . . . . . . 70
5.1.4 Counting the Zeros ofa Holomorphic Function 70
5.1.5 The Argument Principle .. 71
5.1.6 Location ofPoles 72
5.1.7 The Argument Principle for
Meromorphic Functions 72
5.2 The Local Geometry ofHolomorphic Functions 73
5.2.1 The Open Mapping Theorem 73
5.3 Further Results on the Zeros ofHolomorphic Functions 74
5.3.1 RoucM's Theorem 74
5.3.2 Typical Application ofRoucM's Theorem 74
5.3.3 RoucM's Theorem and the Fundamental
Theorem ofAlgebra 75
5.3.4 Hurwitz's Theorem 76
5.4 The Maximum Principle . . . . . . . . . . . 76
5.4.1 The Maximum Modulus Principle 76
5.4.2 Boundary Maximum Modulus Theorem 76
Contents xi
5.4.3 The Minimum Principle 77
5.4.4 The Maximum Principle on an
Unbounded Domain 77
5.5 The Schwarz Lemma . . . . . . . . . 77
5.5.1 Schwarz's Lemma ..... 78
5.5.2 The Schwarz-Pick Lemma 78
6 The Geometric Theory ofHolomorphic Functions 79
6.1 The Idea ofa Conformal Mapping 79
6.1.1 Conformal Mappings 79
6.1.2 Conformal Self-Maps ofthe Plane 79
6.2 Conformal Mappings ofthe Unit Disc .. 80
6.2.1 Conformal Self-Maps ofthe Disc 80
6.2.2 Mobius Transformations. 81
6.2.3 Self-Maps ofthe Disc . . . . 81
6.3 Linear Fractional Transformations .. 81
6.3.1 Linear Fractional Mappings. 81
6.3.2 The Topology ofthe Extended Plane 83
6.3.3 The Riemann Sphere .. . . . . . . . 83
6.3.4 Conformal Self-Maps ofthe Riemann Sphere 84
6.3.5 The Cayley Transform. . . . . . 85
6.3.6 Generalized Circles and Lines. . 85
6.3.7 The Cayley Transform Revisited 85
6.3.8 Summary Chart ofLinear
Fractional Transformations . . . 85
6.4 The Riemann Mapping Theorem . . . . . 86
6.4.1 The Concept ofHomeomorphism. 86
6.4.2 The Riemann Mapping Theorem . 86
6.4.3 The Riemann Mapping Theorem:
Second Formulation . . . . . . . . 87
6.5 Conformal Mappings ofAnnuli . . . . . . . 87
6.5.1 A Riemann Mapping Theorem for Annuli . 87
6.5.2 Conformal Equivalence ofAnnuli. 87
6.5.3 Classification ofPlanar Domains . . . . . . 88
7 Harmonic Functions 89
7.1 Basic Properties ofHarmonic Functions 89
7.1.1 The Laplace Equation. . . . . 89
7.1.2 Definition ofHarmonic Function 89
7.1.3 Real- and Complex-Valued
Harmonic Functions . . . . . . . 89
7.1.4 Harmonic Functions as the Real Parts of
Holomorphic Functions . . . . . . . 90
7.1.5 Smoothness ofHarmonic Functions ... 90
xii Contents
7.2 The Maximum Principle and the Mean Value
Property . 91
7.2.1 The Maximum Principle for
Harmonic Functions . . . . . 91
7.2.2 The Minimum Principle for
Harmonic Functions . . . . . 91
7.2.3 The Boundary Maximum and
Minimum Principles . . . . . . 91
7.2.4 The Mean Value Property .. 92
7.2.5 Boundary Uniqueness for Harmonic Functions 92
7.3 The Poisson Integral Formula . 92
7.3.1 The Poisson Integral .. 92
7.3.2 The Poisson Kernel .. 93
7.3.3 The Dirichlet Problem. 93
7.3.4 The Solution ofthe Dirichlet Problem
on the Disc . 93
7.3.5 The Dirichlet Problem on a General Disc 94
7.4 Regularity ofHarmonic Functions. . . . . . . . . . 94
7.4.1 The Mean Value Property on Circles .. 94
7.4.2 The Limit ofa SequenceofHarmonic Functions 95
7.5 The Schwarz Reflection Principle . . . . . . 95
7.5.1 Reflection ofHarmonic Functions 95
7.5.2 Schwarz Reflection Principle for
Harmonic Functions . . . . . . . . 95
7.5.3 The Schwarz Reflection Principle for
Holomorphic Functions . . . . . . . . 96
7.5.4 More General Versions ofthe Schwarz
Reflection Principle . . . 96
7.6 Harnack's Principle . . . . . . . . . 97
7.6.1 The Harnack Inequality. 97
7.6.2 Harnack's Principle . . . 97
7.7 The Dirichlet Problem and Subharmonic
Functions . 97
7.7.1 The Dirichlet Problem . 97
7.7.2 Conditions for Solving the Dirichlet Problem 98
7.7.3 Motivation for Subharmonic Functions 98
7.7.4 Definition ofSubharmonic Function 99
7.7.5 Other Characterizations of
Subharmonic Functions . 99
7.7.6 The Maximum Principle . 100
7.7.7 Lack ofA Minimum Principle 100
7.7.8 Basic Properties ofSubharmonic Functions 100
7.7.9 The Concept ofa Barrier . 100
7.8 The General Solution ofthe Dirichlet Problem ... 101