Handbook of Complex Variables, 1999

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.

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 Rouché’s Theorem.- 5.3.2 Typical Application of Rouché’s Theorem.- 5.3.3 Rouché’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 Möbius 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 Köbe 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)®.- 16.1.2 Mathematica®.- 16.1.3 Maple®.- 16.1.4 Mat Lab®.- 16.1.5 Ricci®.- Glossary of Terms from Complex Variable Theory and Analysis.- List of Notation.- Table of Laplace Transforms.- A Guide to the Literature.- References.