A Guide to Complex Variables

Dolciani Mathematical Expositions

by Steven G. Krantz

Preface

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 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
       1.4.1Harmonic 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
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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 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 The Limitations of the Cauchy 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
       3.2 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. Laurent Series
       4.1 Behavior 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.l.6 The Casorati-Weierstrass Theorem
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First published June 4, 2008