A Guide to Complex Variables

A Guide to Complex Variables /

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Bibliographic Details
Online Access: Full text (MCPHS users only)
Main Author: Krantz, Steven G. (Steven George), 1951-
Format: Electronic eBook
Language:English
Published: Cambridge : Cambridge University Press, 2012
Series:Dolciani mathematical expositions.
Subjects:
Functions of complex variables.
Local Note:ProQuest Ebook Central
Table of Contents:
  • A Guide to Complex Variables
  • Preface
  • Contents
  • 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 Numbers1.2.6 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 Holomorphic and Harmonic Functions
  • 1.4.1 Harmonic 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 Curves2.1.3 Differentiable and C^k 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 Differentiabilityand 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 A Coda on the Limitations of The Cauchy Integral Formula3 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 Liouvilleâ€?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 Function3.2.3 Discrete Sets and Zero Sets
  • 3.2.4 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

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