Book contents
- Frontmatter
- Contents
- Preface to the Second Edition
- Preface to the First Edition
- 0 The Origins of Complex Analysis, and Its Challenge to Intuition
- 1 Algebra of the Complex Plane
- 2 Topology of the Complex Plane
- 3 Power Series
- 4 Differentiation
- 5 The Exponential Function
- 6 Integration
- 7 Angles, Logarithms, and the Winding Number
- 8 Cauchy’s Theorem
- 9 Homotopy Versions of Cauchy’s Theorem
- 10 Taylor Series
- 11 Laurent Series
- 12 Residues
- 13 Conformal Transformations
- 14 Analytic Continuation
- 15 Infinitesimals in Real and Complex Analysis
- 16 Homology Version of Cauchy’s Theorem
- 17 The Road Goes Ever On …
- References
- Index
4 - Differentiation
Published online by Cambridge University Press: 06 September 2018
Book contents
- Frontmatter
- Contents
- Preface to the Second Edition
- Preface to the First Edition
- 0 The Origins of Complex Analysis, and Its Challenge to Intuition
- 1 Algebra of the Complex Plane
- 2 Topology of the Complex Plane
- 3 Power Series
- 4 Differentiation
- 5 The Exponential Function
- 6 Integration
- 7 Angles, Logarithms, and the Winding Number
- 8 Cauchy’s Theorem
- 9 Homotopy Versions of Cauchy’s Theorem
- 10 Taylor Series
- 11 Laurent Series
- 12 Residues
- 13 Conformal Transformations
- 14 Analytic Continuation
- 15 Infinitesimals in Real and Complex Analysis
- 16 Homology Version of Cauchy’s Theorem
- 17 The Road Goes Ever On …
- References
- Index
Summary
Derivative of a complex function defined by analogy with the real case, shown to satisfy corresponding rules for combinations of functions. Differentiable paths. Power series are differentiable an infinite number of times in their circle of convergence unlike real functions which may be differentiable a certain number of times but not more. Examples are given of the 'bad behaviour' of real functions contrasting with the infinite differentiability of complex functions in an open connected domain.
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- Chapter
- Information
- Complex Analysis , pp. 75 - 95Publisher: Cambridge University PressPrint publication year: 2018
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