A Guide To Complex Variables
A Guide to Complex Variables gives the reader a quick and accessible introduction to the key topics. While the coverage is not comprehensive, it certainly gives the reader a solid grounding in this fundamental area. There are many figures and examples to illustrate the principal ideas, and the expos...
Saved in:
| Main Author: | |
|---|---|
| Format: | eBook Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
2008
Mathematical Association of America The Mathematical Association of America |
| Edition: | 1 |
| Series: | Dolciani Mathematical Expositions |
| Subjects: | |
| ISBN: | 9780883853382, 0883853388 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | A Guide to Complex Variables gives the reader a quick and accessible introduction to the key topics. While the coverage is not comprehensive, it certainly gives the reader a solid grounding in this fundamental area. There are many figures and examples to illustrate the principal ideas, and the exposition is lively and inviting. An undergraduate wanting to have a first look at this subject or a graduate student preparing for the qualifying exams, will find this book to be a useful resource. In addition to important ideas from the Cauchy theory, the book also includes the Riemann mapping theorem, harmonic functions, the argument principle, general conformal mapping and dozens of other central topics. Readers will find this book to be a useful companion to more exhaustive texts in the field. It is a valuable resource for mathematicians and non-mathematicians alike. Steven Krantz is well-known for his skill in expository writing and this volume confirms it. He is the author of more than 50 books, and more than 150 scholarly papers. The MAA has awarded him both the Beckenbach Book Prize and the Chauvenet Prize. |
|---|---|
| AbstractList | A Guide to Complex Variables gives the reader a quick and accessible introduction to the key topics. While the coverage is not comprehensive, it certainly gives the reader a solid grounding in this fundamental area. There are many figures and examples to illustrate the principal ideas, and the exposition is lively and inviting. An undergraduate wanting to have a first look at this subject or a graduate student preparing for the qualifying exams, will find this book to be a useful resource. In addition to important ideas from the Cauchy theory, the book also includes the Riemann mapping theorem, harmonic functions, the argument principle, general conformal mapping and dozens of other central topics. Readers will find this book to be a useful companion to more exhaustive texts in the field. It is a valuable resource for mathematicians and non-mathematicians alike. Steven Krantz is well-known for his skill in expository writing and this volume confirms it. He is the author of more than 50 books, and more than 150 scholarly papers. The MAA has awarded him both the Beckenbach Book Prize and the Chauvenet Prize. |
| Author | Krantz, Steven G. |
| Author_xml | – sequence: 1 givenname: Steven G. surname: Krantz fullname: Krantz, Steven G. organization: Washington University in St. Louis |
| BackLink | https://cir.nii.ac.jp/crid/1130000797270712704$$DView record in CiNii |
| BookMark | eNpdkM1LAzEQxSNasa29exDZgyAI1SSTz2NdahUK9VB7DeluAlu3m7rZUv98t6wH9fKGBz_ezLwBOqtC5RC6IviBa6Ye398WWiqsFCiuCVMnaPTHn_7yAIr20IBirDSVnONz1NdUE4oZURdoFOMGY0wJ6Bbuo-tJMtsXuUuWIUnDdle6r2Rl68KuSxcvUc_bMrrRzxyi1fN0mb6M54vZazqZj60kjMPYcZEDc1oxLyznDmfCS5BMgCbKc26lh1x5yZnN1-2ljnJKM89FphnzVsMQ3XXBuzp87l1sjFuH8JG5qqltaaZPKQBgELwlbzuyKgqTFUclBNqPsNSSSixJK6zFbjpsE5tQm2NYNBuTNY047A4ZtMB9B9ht7JZFQ7A51m3-1w3fwUVoMA |
| ContentType | eBook Book |
| Copyright | Copyright 2008 American Mathematical Society 2008 The Mathematical Association of America (Incorporated) |
| Copyright_xml | – notice: Copyright 2008 American Mathematical Society – notice: 2008 The Mathematical Association of America (Incorporated) |
| DBID | RYH |
| DEWEY | 515/.9 |
| DOI | 10.5948/UPO9780883859148 |
| DatabaseName | CiNii Complete |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISBN | 9780883859148 0883859149 |
| Edition | 1 |
| ExternalDocumentID | EBC3330365 BA8815521X 10.4169/j.ctt6wpwc3 10_5948_UPO9780883859148 |
| GroupedDBID | AABBV AAWPO ABARN ABIAV ABQPQ ACLGV ACNAM ADVEM AEKGI AERYV AFOJC AGXRL AHWGJ AJFER ALMA_UNASSIGNED_HOLDINGS AMYDA AMYDF AZZ BBABE BPBUR BQKUL FEAQG GDRSO GEOUK PQQKQ -G2 089 A4J AAAKM ACFYN AGLEC AIXPE AKHYG ANGWU BFATZ CDLIL HF4 IWG JJU NK1 NK2 RW~ RYH SN- |
| ID | FETCH-LOGICAL-a71453-e56d34e984f6a55e0c6f737463918f55a7f3d8f754adb978e2522cf56c944fa93 |
| ISBN | 9780883853382 0883853388 |
| ISICitedReferencesCount | 0 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000086694&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| IngestDate | Wed Nov 26 03:58:12 EST 2025 Thu Jun 26 23:16:47 EDT 2025 Sun Jun 29 12:53:29 EDT 2025 Thu Aug 14 15:25:19 EDT 2025 |
| IsPeerReviewed | false |
| IsScholarly | false |
| LCCN | 2008927550 |
| LCCallNum_Ident | QA331.7 .K743 2008 |
| Language | English |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-a71453-e56d34e984f6a55e0c6f737463918f55a7f3d8f754adb978e2522cf56c944fa93 |
| Notes | Includes bibliography (p. 175-176) and index |
| OCLC | 929120418 |
| PQID | EBC3330365 |
| PageCount | 202 |
| ParticipantIDs | proquest_ebookcentral_EBC3330365 nii_cinii_1130000797270712704 jstor_books_j_ctt6wpwc3 ams_ebooks_10_5948_UPO9780883859148 |
| PublicationCentury | 2000 |
| PublicationDate | 20080000 20080101 c2008 2008 |
| PublicationDateYYYYMMDD | 2008-01-01 |
| PublicationDate_xml | – year: 2008 text: 20080000 |
| PublicationDecade | 2000 |
| PublicationPlace | Providence, Rhode Island |
| PublicationPlace_xml | – name: Providence, Rhode Island – name: Washington, D.C – name: Washington |
| PublicationSeriesTitle | Dolciani Mathematical Expositions |
| PublicationYear | 2008 |
| Publisher | American Mathematical Society Mathematical Association of America The Mathematical Association of America |
| Publisher_xml | – name: American Mathematical Society – name: Mathematical Association of America – name: The Mathematical Association of America |
| SSID | ssj0002139385 ssj0000577638 |
| Score | 1.9667101 |
| Snippet | A Guide to Complex Variables gives the reader a quick and accessible introduction to the key topics. While the coverage is not comprehensive, it certainly... |
| SourceID | proquest nii jstor ams |
| SourceType | Publisher |
| SubjectTerms | Functions of complex variables Mathematics |
| TableOfContents | Preface
--
The Complex Plane
--
Complex Line Integrals
--
Applications of the Cauchy Theory
--
Laurent Series
--
The Argument Principle
--
The Geometric Theory
--
Harmonic Functions
--
Infinite Series and Products
--
Analytic Continuation
--
Glossary of Terms from Complex Variable Theory and Analysis
--
Bibliography Front Matter Preface Table of Contents CHAPTER 1: The Complex Plane CHAPTER 2: Complex Line Integrals CHAPTER 3: Applications of the Cauchy Theory CHAPTER 4: Isolated Singularities and Laurent Series CHAPTER 5: The Argument Principle CHAPTER 6: The Geometric Theory of Holomorphic Functions CHAPTER 7: Harmonic Functions CHAPTER 8: Infinite Series and Products CHAPTER 9: Analytic Continuation Glossary of Terms from Complex Variable Theory and Analysis Bibliography Index About the Author 8.3 The Theorems of Weierstrass and Mittag-Leffler 7.5.3 The Schwarz Reflection Principle for Holomorphic Functions -- 7.5.4 More General Versions of the Schwarz Reflection Principle -- 7.6 Harnack's Principle -- 7.6.1 The Harnack Inequality -- 7.6.2 Harnack's Principle -- 7.7 The Dirichlet Problem and Subharmonic Functions -- 7.7.1 The Dirichlet Problem -- 7.7.2 Conditions for Solving the Dirichlet Problem -- 7.7.4 Definition of Subharmonic Function -- 7.7.5 Other Characterizations of Subharmonic Functions -- 7.7.6 The Maximum Principle -- 7.7.7 Lack of A Minimum Principle -- 7.7.8 Basic Properties of Subharmonic Functions -- 7.7.9 The Concept of a Barrier -- 7.8 The General Solution of the Dirichlet Problem -- 7.8.1 Enunciation of the Solution of the Dirichlet Problem -- 8 Infinite Series and Products -- 8.1 Basic Concepts Concerning Infinite Sums and Products -- 8.1.1 Uniform Convergence of a Sequence -- 8.1.2 The Cauchy Condition for a Sequence of Functions -- 8.1.3 Normal Convergence of a Sequence -- 8.1.4 Normal Convergence of a Series -- 8.1.5 The Cauchy Condition for a Series -- 8.1.6 The Concept of an Infinite Product -- 8.1.7 Infinite Products of Scalars -- 8.1.8 Partial Products -- 8.1.9 Convergence of an Infinite Product -- 8.1.10 The Value of an Infinite Product -- 8.1.11 Products That Are Disallowed -- 8.1.12 Condition for Convergence of an Infinite Product -- 8.1.13 Infinite Products of Holomorphic Functions -- 8.1.14 Vanishing of an Infinite Product -- 8.1.15 Uniform Convergence of an Infinite Product of Functions -- 8.1.16 Condition for the Uniform Convergence of an Infinite Product of Functions -- 8.2 The WeierstrassFactorization Theorem -- 8.2.1 Prologue -- 8.2.2 Weierstrass Factors -- 8.2.3 Convergence of the Weierstrass Product -- 8.2.4 Existence of an Entire Function with Prescribed Zeros -- 8.2.5 The Weierstrass Factorization Theorem 3.1.1 A Formula for the Derivative -- 3.1.2 The Cauchy Estimates -- 3.1.3 Entire Functions and Liouville's Theorem -- 3.1.4 The Fundamental Theorem of Algebra -- 3.1.5 Sequences of Holomorphic Functions and their Derivatives -- 3.1.6 The Power Series Representation of a Holomorphic Function -- 3.2 The Zeros of a Holomorphic Function -- 3.2.1 The Zero Set of a Holomorphic Function -- 3.2.2 Discreteness of the Zeros of a Holomorphic Function -- 3.2.3 Discrete Sets and Zero Sets -- 3.2.4 Uniqueness of Analytic Continuation -- 4 Isolated Singularities and Laurent Series -- 4.1 The Behavior of a Holomorphic Function near an Isolated Singularity -- 4.1.1 Isolated Singularities -- 4.1.2 A Holomorphic Function on a Punctured Domain -- 4.1.3 Classification of Singularities -- 4.1.4 Removable Singularities, Poles, and Essential Singularities -- 4.1.5 The Riemann Removable Singularities Theorem -- 4.1.6 The Casorati-Weierstrass Theorem -- 4.2 Expansion around Singular Points -- 4.2.1 Laurent Series -- 4.2.2 Convergence of a Doubly Infinite Series -- 4.2.3 Annulus of Convergence -- 4.2.4 Uniqueness of the Laurent Expansion -- 4.2.5 The Cauchy Integral Formula for an Annulus -- 4.2.6 Existence of Laurent Expansions -- 4.2.7 Holomorphic Functions with Isolated Singularities -- 4.2.8 Classification of Singularities in Terms of Laurent Series -- 4.3 Examples of Laurent Expansions -- 4.3.1 Principal Part of a Function -- 4.3.2 Algorithm for Calculating the Coefficients of the Laurent Expansion -- 4.4 The Calculus of Residues -- 4.4.1 Functions with Multiple Singularities -- 4.4.2 The Residue Theorem -- 4.4.3 Residues -- 4.4.4 The Index or Winding Number of a Curve about a Point -- 4.4.5 Restatement of the Residue Theorem -- 4.4.6 Method for Calculating Residues -- 4.4.7 Summary Charts of Laurent Series and Residues 4.5 Applications to the Calculation of Definite Integrals and Sums -- 4.5.1 The Evaluation of Definite Integrals -- 4.5.2 A Basic Example of the Indefinite Integral -- 4.5.3 Complexification of the Integrand -- 4.5.4 An Example with a More Subtle Choice of Contour -- 4.5.5 Making the Spurious Part of the Integral Disappear -- 4.5.6 The Use of the Logarithm -- 4.5.7 Summing a Series Using Residues -- 4.6 Singularities at Infinity -- 4.6.1 Meromorphic Functions -- 4.6.2 Definition of Meromorphic Function -- 4.6.3 Examples of Meromorphic Functions -- 4.6.4 Meromorphic Functions with Infinitely Many Poles -- 4.6.5 Singularities at Infinity -- 4.6.6 The Laurent Expansion at Infinity -- 4.6.7 Meromorphic at Infinity -- 4.6.8 Meromorphic Functions in the Extended Plane -- 5 The Argument Principle -- 5.1 Counting Zeros and Poles -- 5.1.1 Local Geometric Behavior of a Holomorphic Function -- 5.1.2 Locating the Zeros of a Holomorphic Function -- 5.1.3 Zero of Order n -- 5.1.4 Counting the Zeros of a Holomorphic Function -- 5.1.5 The Argument Principle -- 5.1.6 Location of Poles -- 5.1.7 The Argument Principle for Meromorphic Functions -- 5.2 The Local Geometry of Holomorphic Functions -- 5.2.1 The Open Mapping Theorem -- 5.3 Further Results on the Zeros of Holomorphic Functions -- 5.3.1 Rouche's Theorem -- 5.3.2 Typical Application of Rouche's Theorem -- 5.3.3 Rouche's Theorem and the Fundamental Theorem of Algebra -- 5.3.4 Hurwitz's Theorem -- 5.4 The Maximum Principle -- 5.4.1 The Maximum Modulus Principle -- 5.4.2 Boundary Maximum Modulus Theorem -- 5.4.3 The Minimum Principle -- 5.4.4 The Maximum Principle on an Unbounded Domain -- 5.5 The Schwarz Lemma -- 5.5.1 Schwarz's Lemma -- 5.5.2 The Schwarz-Pick Lemma -- 6 The Geometric Theory of Holomorphic Functions -- 6.1 The Idea of a Conformal Mapping -- 6.1.1 Conformal Mappings Intro -- A Guide to Complex Variables -- Preface -- Contents -- 1 The Complex Plane -- 1.1 Complex Arithmetic -- 1.1.1 The Real Numbers -- 1.1.2 The Complex Numbers -- 1.1.3 Complex Conjugate -- 1.1.4 Modulus of a Complex Number -- 1.1.5 The Topology of the Complex Plane -- 1.1.6 The Complex Numbers as a Field -- 1.1.7 The Fundamental Theorem of Algebra -- 1.2 The Exponential and Applications -- 1.2.1 The Exponential Function -- 1.2.2 The Exponential Using Power Series -- 1.2.3 Laws of Exponentiation -- 1.2.4 Polar Form of a Complex Number -- 1.2.5 Roots of Complex Numbers -- 1.2.6 The Argument of a Complex Number -- 1.2.7 Fundamental Inequalities -- 1.3 Holomorphic Functions -- 1.3.1 Continuously Differentiable and Ck Functions -- 1.3.2 The Cauchy-Riemann Equations -- 1.3.3 Derivatives -- 1.3.4 Definition of Holomorphic Function -- 1.3.5 The Complex Derivative -- 1.3.6 Alternative Terminology for Holomorphic Functions -- 1.4 Holomorphic and Harmonic Functions -- 1.4.1 Harmonic Functions -- 1.4.2 How They are Related -- 2 Complex Line Integrals -- 2.1 Real and Complex Line Integrals -- 2.1.1 Curves -- 2.1.2 Closed Curves -- 2.1.3 Differentiable and C^k Curves -- 2.1.4 Integrals on Curves -- 2.1.5 The Fundamental Theorem of Calculus along Curves -- 2.1.6 The Complex Line Integral -- 2.1.7 Properties of Integrals -- 2.2 Complex Differentiabilityand Conformality -- 2.2.1 Limits -- 2.2.2 Holomorphicity and the Complex Derivative -- 2.2.3 Conformality -- 2.3 The Cauchy Integral Formula and Theorem -- 2.3.1 The Cauchy Integral Theorem, Basic Form -- 2.3.2 The Cauchy Integral Formula -- 2.3.3 More General Forms of the Cauchy Theorems -- 2.3.4 Deformability of Curves -- 2.4 A Coda on the Limitations of The Cauchy Integral Formula -- 3 Applications of the Cauchy Theory -- 3.1 The Derivatives of a Holomorphic Function 6.1.2 Conformal Self-Maps of the Plane -- 6.2 Linear Fractional Transformations -- 6.2.1 Linear Fractional Mappings -- 6.2.2 The Topology of the Extended Plane -- 6.2.3 The Riemann Sphere -- 6.2.4 Conformal Self-Maps of the Riemann Sphere -- 6.2.5 The Cayley Transform -- 6.2.6 Generalized Circles and Lines -- 6.2.7 The Cayley Transform Revisited -- 6.2.8 Summary Chart of Linear Fractional Transformations -- 6.3 The Riemann Mapping Theorem -- 6.3.1 The Concept of Homeomorphism -- 6.3.2 The Riemann Mapping Theorem -- 6.3.3 The Riemann Mapping Theorem: Second Formulation -- 6.4 Conformal Mappings of Annuli -- 6.4.1 A Riemann Mapping Theorem for Annuli -- 6.4.2 Conformal Equivalence of Annuli -- 6.4.3 Classification of Planar Domains -- 7 Harmonic Functions -- 7.1 Basic Properties of Harmonic Functions -- 7.1.1 The Laplace Equation -- 7.1.2 Definition of Harmonic Function -- 7.1.3 Real- and Complex-Valued Harmonic Functions -- 7.1.4 Harmonic Functions as the Real Parts of Holomorphic Functions -- 7.1.5 Smoothness of Harmonic Functions -- 7.2 The Maximum Principle and the Mean Value Property -- 7.2.1 The Maximum Principle for Harmonic Functions -- 7.2.2 The Minimum Principle for Harmonic Functions -- 7.2.3 The Boundary Maximum and Minimum Principles -- 7.2.5 Boundary Uniqueness for Harmonic Functions -- 7.3 The Poisson Integral Formula -- 7.3.1 The Poisson Integral -- 7.3.2 The Poisson Kernel -- 7.3.3 The Dirichlet Problem -- 7.3.4 The Solution of the Dirichlet Problem on the Disc -- 7.3.5 The Dirichlet Problem on a General Disc -- 7.4 Regularity of Harmonic Functions -- 7.4.1 The Mean Value Property on Circles -- 7.4.2 The Limit of a Sequence of Harmonic Functions -- 7.5 The Schwarz Reflection Principle -- 7.5.1 Reflection of Harmonic Functions -- 7.5.2 Schwarz Reflection Principle for Harmonic Functions |
| Title | A Guide To Complex Variables |
| URI | https://www.ams.org/dol/032 https://www.jstor.org/stable/10.4169/j.ctt6wpwc3 https://cir.nii.ac.jp/crid/1130000797270712704 https://ebookcentral.proquest.com/lib/[SITE_ID]/detail.action?docID=3330365 |
| Volume | 32 |
| WOSCitedRecordID | wos000086694&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1LT9wwELZg6YGeWtrCtoAitbdV2g3j5xHQlkoVjwNUe4u8fqBwCIgE2J_fsfNYkR4qDlysOIrsZD7H_mbGniHkGxhtLIBLuaE6pbDAfy4zMuV8YTPPqXfgY7IJcXYm53N10cZwrWI6AVGWcrlUd68KNd5DsMPR2RfA3TeKN_AaQccSYcdywIj7arvfeHL9UNiYCiNuFHfLySNqwuFs1Mpjg0tTzOA6adKZTU6-B47ZVaKBvDMOdLYAObAFhJF12kd7fQ5xJLaNE-iZComzDOCSDU0KoOGEGoK54AdfXZz3j6qsCY05CFN9dChlCOeWzdfJuuCoB2-czM6vfvcGLySGOJfJGIOz7VK2EZD6V2j8yKHTH8MuYyTcqts1irWyKP5ZPyMpuHxHRuGgyHuy5sot8nYlkeoD2T1MIhZJfZu0WCQ9Fh_Jn5-zy-NfaZuHItUiowxSx7gF6pSknmvG3NRwL0BQZHeZ9Ixp4cFKLxjVdoFv7Q6Q1RrPuFGUeq3gExmVt6XbIYkGr5hjU62Vo1PLpPMKtOCWW4-qth-Tr_iZefSUVzlqaEEY-VAYY7IdxZA3j93kpq75092TgTHZQ8HkpghlFnyVSAIVklSkkljQMUk6kTWdtHuA89nRMUDgM-zzf5r4QjZXY2-XjOr7B7dH3pjHuqju91vY_wLrKS2- |
| linkProvider | ProQuest Ebooks |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.title=A+guide+to+complex+variables&rft.au=Krantz%2C+Steven+G.+%28Steven+George%29&rft.date=2008-01-01&rft.pub=The+Mathematical+Association+of+America&rft.isbn=9780883853382&rft_id=info:doi/10.5948%2FUPO9780883859148&rft.externalDocID=BA8815521X |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=9780883853382/lc.gif&client=summon&freeimage=true |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=9780883853382/mc.gif&client=summon&freeimage=true |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=9780883853382/sc.gif&client=summon&freeimage=true |