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 Integrat
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.
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