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
23 - Physics in three and four dimensions I
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
When we come to explore real dimensions greater than two, matters become considerably more interesting. Indeed, in my own undergraduate studies, the question as to how to solve Laplace's equation in three or more dimensions using methods analogous to those presented here went unanswered, and remains unanswered in most undergraduate curricula the world over. I did not see the answer until my postgraduate studies, studying twistor methods in Oxford, and did not fully understand many of the geometrical aspects until my own post-doctoral work on twistor descriptions of minimal surfaces and strings. However, this author at least is convinced that many of the concepts are easily understood using only the elementary complex analysis already presented here, and this chapter is in part an attempt to get the message across in such a fashion. Another goal of this chapter and the subsequent one is to persuade some of you that, as well as being a basis for research in fundamental theoretical physics, there are also some interesting problems in basic and very applied mathematics that might well be solved with such methods, if only more people worked on it!
In some ways the presentation is easier if we make the jump straight to four dimensions, and treat the relativistic case. Results for three dimensions can then be obtained by constraining matters to a hyperplane t = 0.
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- Information
- Complex Analysis with MATHEMATICA® , pp. 513 - 539Publisher: Cambridge University PressPrint publication year: 2006