Book contents
- Frontmatter
- Contents
- Preface
- On the Structure of Mathematics
- Brief Summaries of Topics
- 1 Linear Algebra
- 2 ∈ and δ Real Analysis
- 3 Calculus for Vector-Valued Functions
- 4 Point Set Topology
- 5 Classical Stokes' Theorems
- 6 Differential Forms and Stokes' Thm.
- 7 Curvature for Curves and Surfaces
- 8 Geometry
- 9 Complex Analysis
- 10 Countability and the Axiom of Choice
- 11 Algebra
- 12 Lebesgue Integration
- 13 Fourier Analysis
- 14 Differential Equations
- 15 Combinatorics and Probability
- 16 Algorithms
- A Equivalence Relations
- Bibliography
- Index
9 - Complex Analysis
Published online by Cambridge University Press: 11 April 2011
- Frontmatter
- Contents
- Preface
- On the Structure of Mathematics
- Brief Summaries of Topics
- 1 Linear Algebra
- 2 ∈ and δ Real Analysis
- 3 Calculus for Vector-Valued Functions
- 4 Point Set Topology
- 5 Classical Stokes' Theorems
- 6 Differential Forms and Stokes' Thm.
- 7 Curvature for Curves and Surfaces
- 8 Geometry
- 9 Complex Analysis
- 10 Countability and the Axiom of Choice
- 11 Algebra
- 12 Lebesgue Integration
- 13 Fourier Analysis
- 14 Differential Equations
- 15 Combinatorics and Probability
- 16 Algorithms
- A Equivalence Relations
- Bibliography
- Index
Summary
Basic Object: The complex numbers
Basic Map: Analytic functions
Basic Goal: Equivalences of analytic functions
Complex analysis in one variable studies a special type of function (called analytic or holomorphic) mapping complex numbers to themselves. There are a number of seemingly unrelated but equivalent ways for defining an analytic function. Each has its advantages; all should be known.
We will first define analyticity in terms of a limit (in direct analogy with the definition of a derivative for a real-valued function). We will then see that this limit definition can also be captured by the Cauchy-Riemann equations, an amazing set of partial differential equations. Analyticity will then be described in terms of relating the function with a particular path integral (the Cauchy Integral Formula). Even further, we will see that a function is analytic if and only if it can be locally written in terms of a convergent power series. We will then see that an analytic function, viewed as a map from R2 to R2, must preserve angles (which is what the term conformal means), provided that the function has a nonzero derivative.
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- Information
- All the Mathematics You MissedBut Need to Know for Graduate School, pp. 171 - 200Publisher: Cambridge University PressPrint publication year: 2001