Book contents
- Frontmatter
- Contents
- Preface
- 1 Why you need complex numbers
- 2 Complex algebra and geometry
- 3 Cubics, quartics and visualization of complex roots
- 4 Newton—Raphson iteration and complex fractals
- 5 A complex view of the real logistic map
- 6 The Mandelbrot set
- 7 Symmetric chaos in the complex plane
- 8 Complex functions
- 9 Sequences, series and power series
- 10 Complex differentiation
- 11 Paths and complex integration
- 12 Cauchy's theorem
- 13 Cauchy's integral formula and its remarkable consequences
- 14 Laurent series, zeroes, singularities and residues
- 15 Residue calculus: integration, summation and the argument principle
- 16 Conformal mapping I: simple mappings and Möbius transforms
- 17 Fourier transforms
- 18 Laplace transforms
- 19 Elementary applications to two-dimensional physics
- 20 Numerical transform techniques
- 21 Conformal mapping II: the Schwarz—Christoffel mapping
- 22 Tiling the Euclidean and hyperbolic planes
- 23 Physics in three and four dimensions I
- 24 Physics in three and four dimensions II
- Bibliograpy
- Index
16 - Conformal mapping I: simple mappings and Möbius transforms
Published online by Cambridge University Press: 05 August 2014
- Frontmatter
- Contents
- Preface
- 1 Why you need complex numbers
- 2 Complex algebra and geometry
- 3 Cubics, quartics and visualization of complex roots
- 4 Newton—Raphson iteration and complex fractals
- 5 A complex view of the real logistic map
- 6 The Mandelbrot set
- 7 Symmetric chaos in the complex plane
- 8 Complex functions
- 9 Sequences, series and power series
- 10 Complex differentiation
- 11 Paths and complex integration
- 12 Cauchy's theorem
- 13 Cauchy's integral formula and its remarkable consequences
- 14 Laurent series, zeroes, singularities and residues
- 15 Residue calculus: integration, summation and the argument principle
- 16 Conformal mapping I: simple mappings and Möbius transforms
- 17 Fourier transforms
- 18 Laplace transforms
- 19 Elementary applications to two-dimensional physics
- 20 Numerical transform techniques
- 21 Conformal mapping II: the Schwarz—Christoffel mapping
- 22 Tiling the Euclidean and hyperbolic planes
- 23 Physics in three and four dimensions I
- 24 Physics in three and four dimensions II
- Bibliograpy
- Index
Summary
Introduction
Complex functions have an elegant interpretation in terms of mappings of the complex plane into itself. We explored this briefly in Chapter 8. Now we wish to study the geometrical aspects in rather more detail. Our plan is as follows. First, we shall literally play with Mathematica to get a feel for what some simple mappings do to simple regions. Next we shall look at the property of ‘conformality’ – that holormorphic functions, when interpreted as mappings, preserve angles between curves at most points. Then we shall explore the relationship between the geometry of circles and lines and a special class of mappings called Möbius transforms.
This chapter is the foundation for several that follow. In particular, in Chapter 19 we shall explore the application of conformal mapping to problems in physics in 2-dimensional regions. Chapter 23 will explore how some of this material may be generalized to higher dimensions. Chapter 21 will look at how conformal maps, and the Schwarz—Christ-offel transformation in particular, can be managed numerically. Chapter 23 will also reveal the real physics underlying the Möbius transform when it is seen in terms of Einstein's theory of special relativity.
Recall of visualization tools
Our first goal is to use Mathematica to explore some simple mappings. We shall do so by loading the ComplexMap Package and making a pair of additional functions, CartesianMap and PolarMap.
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- Chapter
- Information
- Complex Analysis with MATHEMATICA® , pp. 338 - 356Publisher: Cambridge University PressPrint publication year: 2006