Handbook of Complex Variables by Steven G. Krantz , Springer

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Springer Handbook of Complex Variables by Steven G. Krantz

This book is written to be a convenient reference for the working scientist, student, or engineer who needs to know and use basic concepts in complex analysis. It is not a book of mathematical theory. It is instead a book of mathematical practice. All the basic ideas of complex analysis, as well as many typical applica tions, are treated. Since we are not developing theory and proofs, we have not been obliged to conform to a strict logical ordering of topics. Instead, topics have been organized for ease of reference, so that cognate topics appear in one place. Required background for reading the text is minimal: a good ground ing in (real variable) calculus will suffice. However, the reader who gets maximum utility from the book will be that reader who has had a course in complex analysis at some time in his life. This book is a handy com pendium of all basic facts about complex variable theory. But it is not a textbook, and a person would be hard put to endeavor to learn the subject by reading this book._x000D_ Table of contents : - _x000D_ 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 The Relationship of Holomorphic and Harmonic Functions.- 1.4.1 Harmonic Functions.- 1.4.2 Holomorphic and Harmonic Functions.- 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 Ck 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 Differentiability and Conformality.- 2.2.1 Limits.- 2.2.2 Continuity.- 2.2.3 The Complex Derivative.- 2.2.4 Holomorphicity and the Complex Derivative..- 2.2.5 Conformality.- 2.3 The Cauchy Integral Theorem and Formula.- 2.3.1 The Cauchy Integral Formula.- 2.3.2 The Cauchy Integral Theorem, Basic Form.- 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.- 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.- 3.1.7 Table of Elementary Power Series.- 3.2 The Zeros of a Holomorphic Function.- 3.2.1 The Zero Set of a Holomorphic Function.- 3.2.2 Discrete Sets and Zero Sets.- 3.2.3 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.- 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.- 4.5.8 Summary Chart of Some Integration Techniques.- 4.6 Meromorphic Functions and Singularities at Infinity.- 4.6.1 Meromorphic Functions.- 4.6.2 Discrete Sets and Isolated Points.- 4.6.3 Definition of Meromorphic Function.- 4.6.4 Examples of Meromorphic Functions.- 4.6.5 Meromorphic Functions with Infinitely Many Poles.- 4.6.6 Singularities at Infinity.- 4.6.7 The Laurent Expansion at Infinity.- 4.6.8 Meromorphic at Infinity.- 4.6.9 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.- 6.1.2 Conformal Self-Maps of the Plane.- 6.2 Conformal Mappings of the Unit Disc.- 6.2.1 Conformal Self-Maps of the Disc.- 6.2.2 Moebius Transformations.- 6.2.3 Self-Maps of the Disc.- 6.3 Linear Fractional Transformations.- 6.3.1 Linear Fractional Mappings.- 6.3.2 The Topology of the Extended Plane.- 6.3.3 The Riemann Sphere.- 6.3.4 Conformal Self-Maps of the Riemann Sphere.- 6.3.5 The Cayley Transform.- 6.3.6 Generalized Circles and Lines.- 6.3.7 The Cayley Transform Revisited.- 6.3.8 Summary Chart of Linear Fractional Transformations.- 6.4 The Riemann Mapping Theorem.- 6.4.1 The Concept of Homeomorphism.- 6.4.2 The Riemann Mapping Theorem.- 6.4.3 The Riemann Mapping Theorem: Second Formulation.- 6.5 Conformal Mappings of Annuli.- 6.5.1 A Riemann Mapping Theorem for Annuli.- 6.5.2 Conformal Equivalence of Annuli.- 6.5.3 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.- 7.2.4 The Mean Value Property.- 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.- 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.- 7.7.3 Motivation for Subharmonic Functions.- 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 Weierstrass Factorization 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.- 8.3 The Theorems of Weierstrass and Mittag-Leffler.- 8.3.1 The Concept of Weierstrass's Theorem.- 8.3.2 Weierstrass's Theorem.- 8.3.3 Construction of a Discrete Set.- 8.3.4 Domains of Existence for Holomorphic Functions.- 8.3.5 The Field Generated by the Ring of Holomorphic Functions.- 8.3.6 The Mittag-Leffler Theorem.- 8.3.7 Prescribing Principal Parts.- 8.4 Normal Families.- 8.4.1 Normal Convergence.- 8.4.2 Normal Families.- 8.4.3 Montel's Theorem, First Version.- 8.4.4 Montel's Theorem, Second Version.- 8.4.5 Examples of Normal Families.- 9 Applications of Infinite Sums and Products.- 9.1 Jensen's Formula and an Introduction to Blaschke Products.- 9.1.1 Blashke Factors.- 9.1.2 Jensen's Formula.- 9.1.3 Jensen's Inequality.- 9.1.4 Zeros of a Bounded Holomorphic Function.- 9.1.5 The Blaschke Condition.- 9.1.6 Blaschke Products.- 9.1.7 Blaschke Factorization.- 9.2 The Hadamard Gap Theorem.- 9.2.1 The Technique of Ostrowski.- 9.2.2 The Ostrowski-Hadamard Gap Theorem.- 9.3 Entire Functions of Finite Order.- 9.3.1 Rate of Growth and Zero Set.- 9.3.2 Finite Order.- 9.3.3 Finite Order and the Exponential Term of Weierstrass.- 9.3.4 Weierstrass Canonical Products.- 9.3.5 The Hadamard Factorization Theorem.- 9.3.6 Value Distribution Theory.- 10 Analytic Continuation.- 10.1 Definition of an Analytic Function Element.- 10.1.1 Continuation of Holomorphic Functions.- 10.1.2 Examples of Analytic Continuation.- 10.1.3 Function Elements.- 10.1.4 Direct Analytic Continuation.- 10.1.5 Analytic Continuation of a Function.- 10.1.6 Global Analytic Functions.- 10.1.7 An Example of Analytic Continuation.- 10.2 Analytic Continuation along a Curve.- 10.2.1 Continuation on a Curve.- 10.2.2 Uniqueness of Continuation along a Curve.- 10.3 The Monodromy Theorem.- 10.3.1 Unambiguity of Analytic Continuation.- 10.3.2 The Concept of Homotopy.- 10.3.3 Fixed Endpoint Homotopy.- 10.3.4 Unrestricted Continuation.- 10.3.5 The Monodromy Theorem 134 10.3.6 Monodromy and Globally Defined Analytic Functions.- 10.3.6 Monodromy and Globally Defined Analytic Functions.- 10.4 The Idea of a Riemann Surface.- 10.4.1 What is a Riemann Surface?.- 10.4.2 Examples of Riemann Surfaces.- 10.4.3 The Riemann Surface for the Square Root Function.- 10.4.4 Holomorphic Functions on a Riemann Surface.- 10.4.5 The Riemann Surface for the Logarithm.- 10.4.6 Riemann Surfaces in General.- 10.5 Picard's Theorems.- 10.5.1 Value Distribution for Entire Functions.- 10.5.2 Picard's Little Theorem.- 10.5.3 Picard's Great Theorem.- 10.5.4 The Little Theorem, the Great Theorem, and the Casorati-Weierstrass Theorem.- 11 Rational Approximation Theory.- 11.1 Runge's Theorem.- 11.1.1 Approximation by Rational Functions.- 11.1.2 Runge's Theorem.- 11.1.3 Approximation by Polynomials.- 11.1.4 Applications of Runge's Theorem.- 11.2 Mergelyan's Theorem.- 11.2.1 An Improvement of Runge's Theorem.- 11.2.2 A Special Case of Mergelyan's Theorem.- 11.2.3 Generalized Mergelyan Theorem.- 12 Special Classes of Holomorphic Functions.- 12.1 Schlicht Functions and the Bieberbach Conjecture.- 12.1.1 Schlicht Functions.- 12.1.2 The Bieberbach Conjecture.- 12.1.3 The Lusin Area Integral.- 12.1.4 The Area Principle.- 12.1.5 The Koebe 1/4 Theorem.- 12.2 Extension to the Boundary of Conformal Mappings.- 12.2.1 Boundary Continuation.- 12.2.2 Some Examples Concerning Boundary Continuation.- 12.3 Hardy Spaces.- 12.3.1 The Definition of Hardy Spaces.- 12.3.2 The Blaschke Factorization for H?.- 12.3.3 Monotonicity of the Hardy Space Norm.- 12.3.4 Containment Relations among Hardy Spaces.- 12.3.5 The Zeros of Hardy Functions.- 12.3.6 The Blaschke Factorization for HP Functions.- 13 Special Functions.- 13.0 Introduction.- 13.1 The Gamma and Beta Functions.- 13.1.1 Definition of the Gamma Function.- 13.1.2 Recursive Identity for the Gamma Function.- 13.1.3 Holomorphicity of the Gamma Function.- 13.1.4 Analytic Continuation of the Gamma Function.- 13.1.5 Product Formula for the Gamma Function.- 13.1.6 Non-Vanishing of the Gamma Function.- 13.1.7 The Euler-Mascheroni Constant.- 13.1.8 Formula for the Reciprocal of the Gamma Function.- 13.1.9 Convexity of the Gamma Function.- 13.1.10 The Bohr-Mollerup Theorem.- 13.1.11 The Beta Function.- 13.1.12 Symmetry of the Beta Function.- 13.1.13 Relation of the Beta Function to the Gamma Function.- 13.1.14 Integral Representation of the Beta Function.- 13.2 Riemann's Zeta Function.- 13.2.1 Definition of the Zeta Function.- 13.2.2 The Euler Product Formula.- 13.2.3 Relation of the Zeta Function to the Gamma Function.- 13.2.4 The Hankel Contour and Hankel Functions.- 13.2.5 Expression of the Zeta Function as a Hankel Integral.- 13.2.6 Location of the Pole of the Zeta Function.- 13.2.7 The Functional Equation.- 13.2.8 Zeros of the Zeta Function.- 13.2.9 The Riemann Hypothesis.- 13.2.10 The Lambda Function.- 13.2.11 Relation of the Zeta Function to the Lambda Function.- 13.2.12 More on the Zeros of the Zeta Function.- 13.2.13 Zeros of the Zeta Function and the Boundary of the Critical Strip.- 13.3 Some Counting Functions and a Few Technical Lemmas.- 13.3.1 The Counting Functions of Classical Number Theory.- 13.3.2 The Function ?.- 13.3.3 The Prime Number Theorem.- 14 Applications that Depend on Conformal Mapping.- 14.1 Conformal Mapping.- 14.1.1 A List of Useful Conformal Mappings.- 14.2 Application of Conformal Mapping to the Dirichlet Problem.- 14.2.1 The Dirichlet Problem.- 14.2.2 Physical Motivation for the Dirichlet Problem.- 14.3 Physical Examples Solved by Means of Conformal Mapping.- 14.3.1 Steady State Heat Distribution on a Lens-Shaped Region.- 14.3.2 Electrostatics on a Disc.- 14.3.3 Incompressible Fluid Flow around a Post.- 14.4 Numerical Techniques of Conformal Mapping.- 14.4.1 Numerical Approximation of the Schwarz-Christoffel Mapping.- 14.4.2 Numerical Approximation to a Mapping onto a Smooth Domain.- Appendix to Chapter 14: A Pictorial Catalog of Conformal Maps.- 15 Transform Theory.- 15.0 Introductory Remarks.- 15.1 Fourier Series.- 15.1.1 Basic Definitions.- 15.1.2 A Remark on Intervals of Arbitrary Length..- 15.1.3 Calculating Fourier Coefficients.- 15.1.4 Calculating Fourier Coefficients Using Complex Analysis.- 15.1.5 Steady State Heat Distribution.- 15.1.6 The Derivative and Fourier Series.- 15.2 The Fourier Transform.- 15.2.1 Basic Definitions.- 15.2.2 Some Fourier Transform Examples that Use Complex Variables.- 15.2.3 Solving a Differential Equation Using the Fourier Transform.- 15.3 The Laplace Transform.- 15.3.1 Prologue.- 15.3.2 Solving a Differential Equation Using the Laplace Transform.- 15.4 The z-Transform.- 15.4.1 Basic Definitions.- 15.4.2 Population Growth by Means of the z-Transform.- 16 Computer Packages for Studying Complex Variables.- 16.0 Introductory Remarks.- 16.1 The Software Packages.- 16.1.1 The Software f (z) (R).- 16.1.2 Mathematica (R).- 16.1.3 Maple (R).- 16.1.4 Mat Lab (R).- 16.1.5 Ricci (R).- Glossary of Terms from Complex Variable Theory and Analysis.- List of Notation.- Table of Laplace Transforms.- A Guide to the Literature.- References._x000D_ show more