Verfügbarkeit: A guide to complex variables

A guide to complex variables:

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Bibliographische Detailangaben
1. Verfasser:	Krantz, Steven G. 1951- (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Washington, DC Math. Assoc. of America 2008
Schriftenreihe:	MAA guides 1 Dolciani mathematical expositions 32
Schlagworte:	Functions of complex variables Funktionentheorie
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XVIII, 181 S. graph. Darst.
ISBN:	9780883853382

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MARC


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Datensatz im Suchindex

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adam_text	Contents Prefă ce ......................................................... ix 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 of a Complex Number ........... 2 1.1.5 The Topology of the Complex Plane ......... 3 1.1.6 The Complex Numbers as a Field .......... 5 1.1.7 The Fundamental Theorem of Algebra ........ 6 1.2 The Exponential and Applications .............. 6 1.2.1 The Exponential Function .............. 6 1.2.2 The Exponential Using Power Series ........ 7 1.2.3 Laws of Exponentiation ............... 7 1.2.4 Polar Form of a Complex Number .......... 7 1.2.5 Roots of Complex Numbers ............. 9 1.2.6 The Argument of a Complex Number ........ 11 1.2.7 Fundamental Inequalities ............... 11 1.3 Holomorphic Functions ................... 12 1.3.1 Continuously Differentiable and Ck Functions ... 12 1.3.2 The Cauchy-Riemann Equations ........... 12 1.3.3 Derivatives ...................... 13 1.3.4 Definition of Holomorphic Function ......... 14 1.3.5 The Complex Derivative ............... 14 1.3.6 Alternative Terminology for Holomorphic Functions 16 1.4 Holomorphic and Harmonic Functions ........... 16 1.4.1 Harmonic Functions ................. 16 хи А Guide to Complex Variables 1.4.2 How They are Related ................ 17 2 Complex Line Integrals ...................................... 19 2.1 Real and Complex Line Integrals .............. 19 2.1.1 Curves ........................ 19 2.1.2 ClosedCurves .................... 20 2.1.3 Differentiable and Ск Curves ............ 20 2.1.4 Integrals on Curves .................. 21 2.1.5 The Fundamental Theorem of Calculus along Curves 21 2.1.6 The Complex Line Integral .............. 22 2.1.7 Properties of Integrals ................ 23 2.2 Complex Differentiability and Conformality ........ 24 2.2.1 Limits ......................... 24 2.2.2 Holomorphicity and the Complex Derivative .... 24 2.2.3 Conformality ..................... 24 2.3 The Cauchy Integral Formula and Theorem ......... 27 2.3.1 The Cauchy Integral Theorem, Basic Form ..... 27 2.3.2 The Cauchy Integral Formula ............ 28 2.3.3 More General Forms of the Cauchy Theorems ... 28 2.3.4 Deformability of Curves ............... 30 2.4 The Limitations of the Cauchy Formula ........... 30 3 Applications of the Cauchy Theory ........................... 33 3.1 The Derivatives of a Holomorphic Function ......... 33 3.1.1 A Formula for the Derivative ............. 33 3.1.2 The Cauchy Estimates ................ 33 3.1.3 Entire Functions and Liouville s Theorem ...... 34 3.1.4 The Fundamental Theorem of Algebra ........ 35 3.1.5 Sequences of Holomorphic Functions and their Derivatives ...................... 36 3.1.6 The Power Series Representation of a Holomorphic Function ....................... 36 3.2 The Zeros of a Holomorphic Function ............ 39 3.2.1 The Zero Set of a Holomorphic Function ...... 39 3.2.2 Discreteness of the Zeros of a Holomorphic Function 39 3.2.3 Discrete Sets and Zero Sets ............. 39 3.2.4 Uniqueness of Analytic Continuation ........ 40 CONTENTS ХНІ 4 Laurent Series .............................................. 43 4.1 Behavior Near an Isolated Singularity ............ 43 4.1.1 Isolated Singularities ................. 43 4.1.2 A Holomorphic Function on a Punctured Domain . . 44 4.1.3 Classification of Singularities ............ 44 4.1.4 Removable Singularities, Poles, and Essential Singularities ................ 44 4.1.5 The Riemann Removable Singularities Theorem .. 45 4.1.6 The Casorati- Weierstrass Theorem .......... 45 4.2 Expansion around Singular Points .............. 46 4.2.1 Laurent Series .................... 46 4.2.2 Convergence of a Doubly Infinite Series ....... 46 4.2.3 Annulus of Convergence ............... 46 4.2.4 Uniqueness of the Laurent Expansion ........ 47 4.2.5 The Cauchy Integral Formula for an Annulus .... 48 4.2.6 Existence of Laurent Expansions ........... 48 4.2.7 Holomorphic Functions with Isolated Singularities . 49 4.2.8 Classification of Singularities in Terms of Laurent Series ......................... 49 4.3 Examples of Laurent Expansions .............. 50 4.3.1 Principal Part of a Function ............. 50 4.3.2 Algorithm for Calculating the Coefficients of the Laurent Expansion .................. 52 4.4 The Calculus of Residues .................. 52 4.4.1 Functions with Multiple Singularities ........ 52 4.4.2 The Residue Theorem ................ 52 4.4.3 Residues ....................... 53 4.4.4 The Index or Winding Number of a Curve about a Point 53 4.4.5 Restatement of the Residue Theorem ........ 54 4.4.6 Method for Calculating Residues .......... 55 4.4.7 Summary Charts of Laurent Series and Residues . . 55 4.5 Applications to Integrals ................... 55 4.5.1 The Evaluation of Definite Integrals ......... 55 4.5.2 A Basic Example of the Indefinite Integral ..... 56 4.5.3 Complexification of the Integrand .......... 58 4.5.4 An Example with a More Subtle Choice of Contour . 59 4.5.5 Making the Spurious Part of the Integral Disappear . 62 4.5.6 The Use of the Logarithm .............. 64 4.5.7 Summing a Series Using Residues .......... 66 XIV A Guide то complex Variables 4.6 Singularities at Infinity .................... 67 4.6.1 Meromorphic Functions ............... 67 4.6.2 Definition of Meromorphic Function ......... 67 4.6.3 Examples of Meromorphic Functions ........ 68 4.6.4 Meromorphic Functions with Infinitely Many Poles . 68 4.6.5 Singularities at Infinity ................ 68 4.6.6 The Laurent Expansion at Infinity .......... 69 4.6.7 Meromorphic at Infinity ............... 69 4.6.8 Meromorphic Functions in the Extended Plane ... 70 5 The Argument Principle ..................................... 71 5.1 Counting Zeros and Poles .................. 71 5.1.1 Local Geometric Behavior of a Holomorphic Function 71 5.1.2 Locating the Zeros of a Holomorphic Function ... 71 5.1.3 Zero of Order и .................... 72 5.1.4 Counting the Zeros of a Holomorphic Function ... 72 5.1.5 The Argument Principle ............... 73 5.1.6 Location of Poles .................. 74 5.1.7 The Argument Principle for Meromorphic Functions 74 5.2 The Local Geometry of Holomorphic Functions ....... 75 5.2.1 The Open Mapping Theorem ............ 75 5.3 Further Results ........................ 76 5.3.1 Rouché s Theorem .................. 76 5.3.2 Typical Application of Rouché s Theorem ..... 77 5.3.3 Rouché s Theorem and the Fundamental Theorem of Algebra ....................... 77 5.3.4 Hurwitz s Theorem ................. 78 5.4 The Maximum Principle ................... 78 5.4.1 The Maximum Modulus Principle .......... 78 5.4.2 Boundary Maximum Modulus Theorem ...... 79 5.4.3 The Minimum Principle ............... 80 5.4.4 The Maximum Principle on an Unbounded Domain 80 5.5 The Schwarz Lemma ..................... 81 5.5.1 Schwarz s Lemma .................. 81 5.5.2 The Schwarz-Pick Lemma .............. 82 6 The Geometric Theory ....................................... 83 6.1 The Idea of a Conformai Mapping .............. 83 6.1.1 Conformai Mappings ................. 83 6.1.2 Conformai Self-Maps of the Plane .......... 83 Contents xv 6.2 Linear Fractional Transformations .............. 86 6.2.1 Linear Fractional Mappings ............. 86 6.2.2 The Topology of the Extended Plane ......... 87 6.2.3 The Riemann Sphere ................. 87 6.2.4 Conformai Self-Maps of the Riemann Sphere .... 89 6.2.5 The Cayley Transform ................ 89 6.2.6 Generalized Circles and Lines ............ 89 6.2.7 The Cayley Transform Revisited ........... 89 6.2.8 Summary Chart of Linear Fractional Transformations 90 6.3 The Riemann Mapping Theorem ............... 91 6.3.1 The Concept of Homeomorphism .......... 91 6.3.2 The Riemann Mapping Theorem ........... 91 6.3.3 The Riemann Mapping Theorem: Second Formulation 91 6.4 Conformai Mappings of Annuli............... 92 6.4.1 A Riemann Mapping Theorem for Annuli...... 92 6.4.2 Conformai Equivalence of Annuli.......... 93 6.4.3 Classification of Planar Domains ........... 93 7 Harmonic Functions ......................................... 95 7.1 Basic Properties of Harmonic Functions ........... 95 7.1.1 The Laplace Equation ................ 95 7.1.2 Definition of Harmonic Function ........... 95 7.1.3 Real-and Complex-Valued Harmonic Functions . . 96 7.1.4 Harmonic Functions as the Real Parts of Holomorphic Functions ............... 96 7.1.5 Smoothness of Harmonic Functions ......... 97 7.2 The Maximum Principle and the Mean Value Property ... 97 7.2.1 The Maximum Principle for Harmonic Functions . . 97 7.2.2 The Minimum Principle for Harmonic Functions . . 97 7.2.3 The Boundary Maximum and Minimum Principles . 98 7.2.4 The Mean Value Property .............. 98 7.2.5 Boundary Uniqueness for Harmonic Functions ... 99 7.3 The Poisson Integral Formula ................ 99 7.3.1 The Poisson Integral ................. 99 7.3.2 The Poisson Kernel .................. 99 7.3.3 The Dirichlet Problem ................ 100 7.3.4 The Solution of the Dirichlet Problem on the Disc . 100 7.3.5 The Dirichlet Problem on a General Disc ...... 101 xvi A Guide то Complex Variables 7.4 Regularity of Harmonic Functions .............. 101 7.4.1 The Mean Value Property on Circles ......... 101 7.4.2 The Limit of a Sequence of Harmonic Functions . . 102 7.5 The Schwarz Reflection Principle .............. 102 7.5.1 Reflection of Harmonic Functions .......... 102 7.5.2 Schwarz Reflection Principle for Harmonic Functions 102 7.5.3 The Schwarz Reflection Principle for Holomorphic Functions ............... 103 7.5.4 More General Versions of the Schwarz Reflection Principle ................. 103 7.6 Harnack s Principle ...................... 104 7.6.1 The Harnack Inequality ............... 104 7.6.2 Harnack s Principle .................. 104 7.7 The Dirichlet Problem .................... 104 7.7.1 The Dirichlet Problem ................ 104 7.7.2 Conditions for Solving the Dirichlet Problem .... 105 7.7.3 Motivation for Subharmonic Functions ....... 105 7.7.4 Definition of Subharmonic Function ......... 106 7.7.5 Other Characterizations of Subharmonic Functions . 106 7.7.6 The Maximum Principle ............... 107 7.7.7 Lack of A Minimum Principle ............ 107 7.7.8 Basic Properties of Subharmonic Functions ..... 107 7.7.9 The Concept of a Barrier ............... 108 7.8 The General Solution of the Dirichlet Problem ....... 109 7.8.1 Enunciation of the Solution of the Dirichlet Problem 109 8 Infinite Series and Products .................................. Ill 8.1 Basic Concepts ........................ Ill 8.1.1 Uniform Convergence of a Sequence ........ Ill 8.1.2 The Cauchy Condition for a Sequence of Functions . Ill 8.1.3 Normal Convergence of a Sequence ......... 112 8.1.4 Normal Convergence of a Series ........... 112 8.1.5 The Cauchy Condition for a Series .......... 112 8.1.6 The Concept of an Infinite Product .......... 113 8.1.7 Infinite Products of Scalars .............. 113 8.1.8 Partial Products .................... 113 8.1.9 Convergence of an Infinite Product ......... 113 8.1.10 The Value of an Infinite Product ........... 114 8.1.11 Products That Are Disallowed ............ 114 Contents XVII 8.1.12 Condition for Convergence of an Infinite Product . . 114 8.1.13 Infinite Products of Holomorphic Functions ..... 116 8.1.14 Vanishing of an Infinite Product........... 116 8.1.15 Uniform Convergence of an Infinite Productor Functions................. 116 8.1.16 Condition for the Uniform Convergence of an Infinite Product of Functions ............. 117 8.2 The Weierstrass Factorization Theorem ........... 117 8.2.1 Prologue ....................... 117 8.2.2 Weierstrass Factors .................. 117 8.2.3 Convergence of the Weierstrass Product ....... 118 8.2.4 Existence of an Entire Function with Prescribed Zeros 118 8.2.5 The Weierstrass Factorization Theorem ....... 119 8.3 Weierstrass and Mittag-Leffler Theorems .......... 119 8.3.1 The Concept of Weierstrass s Theorem ....... 119 8.3.2 Weierstrass s Theorem ................ 119 8.3.3 Construction of a Discrete Set ............ 119 8.3.4 Domains of Existence for Holomorphic Functions . 120 8.3.5 The Field Generated by the Ring of Holomorphic Functions ............... 120 8.3.6 The Mittag-Leffler Theorem ............. 120 8.3.7 Prescribing Principal Parts .............. 121 8.4 Normal Families ....................... 122 8.4.1 Normal Convergence ................. 122 8.4.2 Normal Families ................... 122 8.4.3 Montel s Theorem, First Version ........... 122 8.4.4 Montel s Theorem, Second Version ......... 122 8.4.5 Examples of Normal Families ............ 123 9 Analytic Continuation .......................................125 9.1 Definitionofan Analytic Function Element ......... 125 9.1.1 Continuation of Holomorphic Functions ....... 125 9.1.2 Examples of Analytic Continuation ......... 125 9.1.3 Function Elements .................. 129 9.1.4 Direct Analytic Continuation ............. 129 9.1.5 Analytic Continuation of a Function ......... 129 9.1.6 Global Analytic Functions .............. 129 9.1.7 An Example of Analytic Continuation ........ 130 xviii A Guide to Complex Variables 9.2 Analytic Continuation along a Curve ............ 131 9.2.1 Continuation on a Curve ............... 131 9.2.2 Uniqueness of Continuation along a Curve ..... 132 9.3 The Monodramy Theorem .................. 132 9.3.1 Unambiguity of Analytic Continuation ....... 132 9.3.2 The Concept of Homotopy .............. 133 9.3.3 Fixed Endpoint Homotopy .............. 134 9.3.4 Unrestricted Continuation .............. 134 9.3.5 The Monodramy Theorem .............. 134 9.3.6 Monodramy and Globally Defined Analytic Functions 134 9.4 The Idea of a Riemann Surface ............... 135 9.4.1 What is a Riemann Surface? ............. 135 9.4.2 Examples of Riemann Surfaces ........... 136 9.4.3 The Riemann Surface for the Square Root Function . 138 9.4.4 Holomorphic Functions on a Riemann Surf ace ... 138 9.4.5 The Riemann Surface for the Logarithm ....... 139 9.4.6 Riemann Surfaces in General ............ 139 9.5 Picard s Theorems ...................... 140 9.5.1 Value Distribution for Entire Functions ....... 140 9.5.2 Picard s Little Theorem ............... 140 9.5.3 Picard s Great Theorem ............... 140 9.5.4 The Little Theorem, the Great Theorem, and the Casorati- Weierstrass Theorem ............ 141 Glossary of Terms from Complex Variable Theory and Analysis .. 143 Bibliography ...................................................175 Index ...........................................................177 About the Author ...............................................182
adam_txt	Contents Prefă ce . ix 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 of a Complex Number . 2 1.1.5 The Topology of the Complex Plane . 3 1.1.6 The Complex Numbers as a Field . 5 1.1.7 The Fundamental Theorem of Algebra . 6 1.2 The Exponential and Applications . 6 1.2.1 The Exponential Function . 6 1.2.2 The Exponential Using Power Series . 7 1.2.3 Laws of Exponentiation . 7 1.2.4 Polar Form of a Complex Number . 7 1.2.5 Roots of Complex Numbers . 9 1.2.6 The Argument of a Complex Number . 11 1.2.7 Fundamental Inequalities . 11 1.3 Holomorphic Functions . 12 1.3.1 Continuously Differentiable and Ck Functions . 12 1.3.2 The Cauchy-Riemann Equations . 12 1.3.3 Derivatives . 13 1.3.4 Definition of Holomorphic Function . 14 1.3.5 The Complex Derivative . 14 1.3.6 Alternative Terminology for Holomorphic Functions 16 1.4 Holomorphic and Harmonic Functions . 16 1.4.1 Harmonic Functions . 16 хи А Guide to Complex Variables 1.4.2 How They are Related . 17 2 Complex Line Integrals . 19 2.1 Real and Complex Line Integrals . 19 2.1.1 Curves . 19 2.1.2 ClosedCurves . 20 2.1.3 Differentiable and Ск Curves . 20 2.1.4 Integrals on Curves . 21 2.1.5 The Fundamental Theorem of Calculus along Curves 21 2.1.6 The Complex Line Integral . 22 2.1.7 Properties of Integrals . 23 2.2 Complex Differentiability and Conformality . 24 2.2.1 Limits . 24 2.2.2 Holomorphicity and the Complex Derivative . 24 2.2.3 Conformality . 24 2.3 The Cauchy Integral Formula and Theorem . 27 2.3.1 The Cauchy Integral Theorem, Basic Form . 27 2.3.2 The Cauchy Integral Formula . 28 2.3.3 More General Forms of the Cauchy Theorems . 28 2.3.4 Deformability of Curves . 30 2.4 The Limitations of the Cauchy Formula . 30 3 Applications of the Cauchy Theory . 33 3.1 The Derivatives of a Holomorphic Function . 33 3.1.1 A Formula for the Derivative . 33 3.1.2 The Cauchy Estimates . 33 3.1.3 Entire Functions and Liouville's Theorem . 34 3.1.4 The Fundamental Theorem of Algebra . 35 3.1.5 Sequences of Holomorphic Functions and their Derivatives . 36 3.1.6 The Power Series Representation of a Holomorphic Function . 36 3.2 The Zeros of a Holomorphic Function . 39 3.2.1 The Zero Set of a Holomorphic Function . 39 3.2.2 Discreteness of the Zeros of a Holomorphic Function 39 3.2.3 Discrete Sets and Zero Sets . 39 3.2.4 Uniqueness of Analytic Continuation . 40 CONTENTS ХНІ 4 Laurent Series . 43 4.1 Behavior Near an Isolated Singularity . 43 4.1.1 Isolated Singularities . 43 4.1.2 A Holomorphic Function on a Punctured Domain . . 44 4.1.3 Classification of Singularities . 44 4.1.4 Removable Singularities, Poles, and Essential Singularities . 44 4.1.5 The Riemann Removable Singularities Theorem . 45 4.1.6 The Casorati- Weierstrass Theorem . 45 4.2 Expansion around Singular Points . 46 4.2.1 Laurent Series . 46 4.2.2 Convergence of a Doubly Infinite Series . 46 4.2.3 Annulus of Convergence . 46 4.2.4 Uniqueness of the Laurent Expansion . 47 4.2.5 The Cauchy Integral Formula for an Annulus . 48 4.2.6 Existence of Laurent Expansions . 48 4.2.7 Holomorphic Functions with Isolated Singularities . 49 4.2.8 Classification of Singularities in Terms of Laurent Series . 49 4.3 Examples of Laurent Expansions . 50 4.3.1 Principal Part of a Function . 50 4.3.2 Algorithm for Calculating the Coefficients of the Laurent Expansion . 52 4.4 The Calculus of Residues . 52 4.4.1 Functions with Multiple Singularities . 52 4.4.2 The Residue Theorem . 52 4.4.3 Residues . 53 4.4.4 The Index or Winding Number of a Curve about a Point 53 4.4.5 Restatement of the Residue Theorem . 54 4.4.6 Method for Calculating Residues . 55 4.4.7 Summary Charts of Laurent Series and Residues . . 55 4.5 Applications to Integrals . 55 4.5.1 The Evaluation of Definite Integrals . 55 4.5.2 A Basic Example of the Indefinite Integral . 56 4.5.3 Complexification of the Integrand . 58 4.5.4 An Example with a More Subtle Choice of Contour . 59 4.5.5 Making the Spurious Part of the Integral Disappear . 62 4.5.6 The Use of the Logarithm . 64 4.5.7 Summing a Series Using Residues . 66 XIV A Guide то complex Variables 4.6 Singularities at Infinity . 67 4.6.1 Meromorphic Functions . 67 4.6.2 Definition of Meromorphic Function . 67 4.6.3 Examples of Meromorphic Functions . 68 4.6.4 Meromorphic Functions with Infinitely Many Poles . 68 4.6.5 Singularities at Infinity . 68 4.6.6 The Laurent Expansion at Infinity . 69 4.6.7 Meromorphic at Infinity . 69 4.6.8 Meromorphic Functions in the Extended Plane . 70 5 The Argument Principle . 71 5.1 Counting Zeros and Poles . 71 5.1.1 Local Geometric Behavior of a Holomorphic Function 71 5.1.2 Locating the Zeros of a Holomorphic Function . 71 5.1.3 Zero of Order и . 72 5.1.4 Counting the Zeros of a Holomorphic Function . 72 5.1.5 The Argument Principle . 73 5.1.6 Location of Poles . 74 5.1.7 The Argument Principle for Meromorphic Functions 74 5.2 The Local Geometry of Holomorphic Functions . 75 5.2.1 The Open Mapping Theorem . 75 5.3 Further Results . 76 5.3.1 Rouché's Theorem . 76 5.3.2 Typical Application of Rouché's Theorem . 77 5.3.3 Rouché's Theorem and the Fundamental Theorem of Algebra . 77 5.3.4 Hurwitz's Theorem . 78 5.4 The Maximum Principle . 78 5.4.1 The Maximum Modulus Principle . 78 5.4.2 Boundary Maximum Modulus Theorem . 79 5.4.3 The Minimum Principle . 80 5.4.4 The Maximum Principle on an Unbounded Domain 80 5.5 The Schwarz Lemma . 81 5.5.1 Schwarz's Lemma . 81 5.5.2 The Schwarz-Pick Lemma . 82 6 The Geometric Theory . 83 6.1 The Idea of a Conformai Mapping . 83 6.1.1 Conformai Mappings . 83 6.1.2 Conformai Self-Maps of the Plane . 83 Contents xv 6.2 Linear Fractional Transformations . 86 6.2.1 Linear Fractional Mappings . 86 6.2.2 The Topology of the Extended Plane . 87 6.2.3 The Riemann Sphere . 87 6.2.4 Conformai Self-Maps of the Riemann Sphere . 89 6.2.5 The Cayley Transform . 89 6.2.6 Generalized Circles and Lines . 89 6.2.7 The Cayley Transform Revisited . 89 6.2.8 Summary Chart of Linear Fractional Transformations 90 6.3 The Riemann Mapping Theorem . 91 6.3.1 The Concept of Homeomorphism . 91 6.3.2 The Riemann Mapping Theorem . 91 6.3.3 The Riemann Mapping Theorem: Second Formulation 91 6.4 Conformai Mappings of Annuli. 92 6.4.1 A Riemann Mapping Theorem for Annuli. 92 6.4.2 Conformai Equivalence of Annuli. 93 6.4.3 Classification of Planar Domains . 93 7 Harmonic Functions . 95 7.1 Basic Properties of Harmonic Functions . 95 7.1.1 The Laplace Equation . 95 7.1.2 Definition of Harmonic Function . 95 7.1.3 Real-and Complex-Valued Harmonic Functions . . 96 7.1.4 Harmonic Functions as the Real Parts of Holomorphic Functions . 96 7.1.5 Smoothness of Harmonic Functions . 97 7.2 The Maximum Principle and the Mean Value Property . 97 7.2.1 The Maximum Principle for Harmonic Functions . . 97 7.2.2 The Minimum Principle for Harmonic Functions . . 97 7.2.3 The Boundary Maximum and Minimum Principles . 98 7.2.4 The Mean Value Property . 98 7.2.5 Boundary Uniqueness for Harmonic Functions . 99 7.3 The Poisson Integral Formula . 99 7.3.1 The Poisson Integral . 99 7.3.2 The Poisson Kernel . 99 7.3.3 The Dirichlet Problem . 100 7.3.4 The Solution of the Dirichlet Problem on the Disc . 100 7.3.5 The Dirichlet Problem on a General Disc . 101 xvi A Guide то Complex Variables 7.4 Regularity of Harmonic Functions . 101 7.4.1 The Mean Value Property on Circles . 101 7.4.2 The Limit of a Sequence of Harmonic Functions . . 102 7.5 The Schwarz Reflection Principle . 102 7.5.1 Reflection of Harmonic Functions . 102 7.5.2 Schwarz Reflection Principle for Harmonic Functions 102 7.5.3 The Schwarz Reflection Principle for Holomorphic Functions . 103 7.5.4 More General Versions of the Schwarz Reflection Principle . 103 7.6 Harnack's Principle . 104 7.6.1 The Harnack Inequality . 104 7.6.2 Harnack's Principle . 104 7.7 The Dirichlet Problem . 104 7.7.1 The Dirichlet Problem . 104 7.7.2 Conditions for Solving the Dirichlet Problem . 105 7.7.3 Motivation for Subharmonic Functions . 105 7.7.4 Definition of Subharmonic Function . 106 7.7.5 Other Characterizations of Subharmonic Functions . 106 7.7.6 The Maximum Principle . 107 7.7.7 Lack of A Minimum Principle . 107 7.7.8 Basic Properties of Subharmonic Functions . 107 7.7.9 The Concept of a Barrier . 108 7.8 The General Solution of the Dirichlet Problem . 109 7.8.1 Enunciation of the Solution of the Dirichlet Problem 109 8 Infinite Series and Products . Ill 8.1 Basic Concepts . Ill 8.1.1 Uniform Convergence of a Sequence . Ill 8.1.2 The Cauchy Condition for a Sequence of Functions . Ill 8.1.3 Normal Convergence of a Sequence . 112 8.1.4 Normal Convergence of a Series . 112 8.1.5 The Cauchy Condition for a Series . 112 8.1.6 The Concept of an Infinite Product . 113 8.1.7 Infinite Products of Scalars . 113 8.1.8 Partial Products . 113 8.1.9 Convergence of an Infinite Product . 113 8.1.10 The Value of an Infinite Product . 114 8.1.11 Products That Are Disallowed . 114 Contents XVII 8.1.12 Condition for Convergence of an Infinite Product . . 114 8.1.13 Infinite Products of Holomorphic Functions . 116 8.1.14 Vanishing of an Infinite Product. 116 8.1.15 Uniform Convergence of an Infinite Productor Functions. 116 8.1.16 Condition for the Uniform Convergence of an Infinite Product of Functions . 117 8.2 The Weierstrass Factorization Theorem . 117 8.2.1 Prologue . 117 8.2.2 Weierstrass Factors . 117 8.2.3 Convergence of the Weierstrass Product . 118 8.2.4 Existence of an Entire Function with Prescribed Zeros 118 8.2.5 The Weierstrass Factorization Theorem . 119 8.3 Weierstrass and Mittag-Leffler Theorems . 119 8.3.1 The Concept of Weierstrass's Theorem . 119 8.3.2 Weierstrass's Theorem . 119 8.3.3 Construction of a Discrete Set . 119 8.3.4 Domains of Existence for Holomorphic Functions . 120 8.3.5 The Field Generated by the Ring of Holomorphic Functions . 120 8.3.6 The Mittag-Leffler Theorem . 120 8.3.7 Prescribing Principal Parts . 121 8.4 Normal Families . 122 8.4.1 Normal Convergence . 122 8.4.2 Normal Families . 122 8.4.3 Montel's Theorem, First Version . 122 8.4.4 Montel's Theorem, Second Version . 122 8.4.5 Examples of Normal Families . 123 9 Analytic Continuation .125 9.1 Definitionofan Analytic Function Element . 125 9.1.1 Continuation of Holomorphic Functions . 125 9.1.2 Examples of Analytic Continuation . 125 9.1.3 Function Elements . 129 9.1.4 Direct Analytic Continuation . 129 9.1.5 Analytic Continuation of a Function . 129 9.1.6 Global Analytic Functions . 129 9.1.7 An Example of Analytic Continuation . 130 xviii A Guide to Complex Variables 9.2 Analytic Continuation along a Curve . 131 9.2.1 Continuation on a Curve . 131 9.2.2 Uniqueness of Continuation along a Curve . 132 9.3 The Monodramy Theorem . 132 9.3.1 Unambiguity of Analytic Continuation . 132 9.3.2 The Concept of Homotopy . 133 9.3.3 Fixed Endpoint Homotopy . 134 9.3.4 Unrestricted Continuation . 134 9.3.5 The Monodramy Theorem . 134 9.3.6 Monodramy and Globally Defined Analytic Functions 134 9.4 The Idea of a Riemann Surface . 135 9.4.1 What is a Riemann Surface? . 135 9.4.2 Examples of Riemann Surfaces . 136 9.4.3 The Riemann Surface for the Square Root Function . 138 9.4.4 Holomorphic Functions on a Riemann Surf ace . 138 9.4.5 The Riemann Surface for the Logarithm . 139 9.4.6 Riemann Surfaces in General . 139 9.5 Picard's Theorems . 140 9.5.1 Value Distribution for Entire Functions . 140 9.5.2 Picard's Little Theorem . 140 9.5.3 Picard's Great Theorem . 140 9.5.4 The Little Theorem, the Great Theorem, and the Casorati- Weierstrass Theorem . 141 Glossary of Terms from Complex Variable Theory and Analysis . 143 Bibliography .175 Index .177 About the Author .182
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spelling	Krantz, Steven G. 1951- Verfasser (DE-588)130535907 aut A guide to complex variables Steven G. Krantz Washington, DC Math. Assoc. of America 2008 XVIII, 181 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier MAA guides 1 Dolciani mathematical expositions 32 Functions of complex variables Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 s DE-604 MAA guides 1 (DE-604)BV035218151 1 Dolciani mathematical expositions 32 (DE-604)BV001900740 32 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016815121&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Krantz, Steven G. 1951- A guide to complex variables MAA guides Dolciani mathematical expositions Functions of complex variables Funktionentheorie (DE-588)4018935-1 gnd
subject_GND	(DE-588)4018935-1
title	A guide to complex variables
title_auth	A guide to complex variables
title_exact_search	A guide to complex variables
title_exact_search_txtP	A guide to complex variables
title_full	A guide to complex variables Steven G. Krantz
title_fullStr	A guide to complex variables Steven G. Krantz
title_full_unstemmed	A guide to complex variables Steven G. Krantz
title_short	A guide to complex variables
title_sort	a guide to complex variables
topic	Functions of complex variables Funktionentheorie (DE-588)4018935-1 gnd
topic_facet	Functions of complex variables Funktionentheorie
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016815121&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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