Handbook of Complex Variables

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