This book provides a comprehensive look at the Schwarz-Christoffel transformation, including its history and foundations, practical computation, common and less common variations, and many applications in fields such as electromagnetism, fluid flow, design and inverse problems, and the solution of linear systems of equations. It is an accessible resource for engineers, scientists, and applied mathematicians who seek more experience with theoretical or computational conformal mapping techniques. The most important theoretical results are stated and proved, but the emphasis throughout remains on concrete understanding and implementation. There is a brief appendix illustrating the use of the Schwarz-Christoffel Toolbox for MATLAB, the state-of-the-art package for computation of these maps.
Author(s): Tobin A. Driscoll, Lloyd N. Trefethen
Edition: 1
Year: 2002
Language: English
Pages: 149
Tags: Математика;Комплексное исчисление;
0521807263......Page 1
Contents......Page 10
Figures......Page 12
Preface......Page 16
1.1 The Schwarz–Christoffel idea......Page 18
1.2 History......Page 21
2.1 Polygons......Page 26
2.2 The Schwarz–Christoffel formula......Page 27
2.3 Polygons with one or two vertices......Page 29
2.4 Triangles......Page 33
2.5 Rectangles and elliptic functions......Page 35
2.6 Crowding......Page 37
3.1 Side-length parameter problem......Page 40
3.2 Quadrature......Page 44
3.3 Inverting the map......Page 46
3.4 Cross-ratio parameter problem......Page 47
3.5 Mapping using cross-ratios......Page 53
3.6 Software......Page 56
4 Variations......Page 58
4.1 Mapping from the disk......Page 59
4.2 Mapping from a strip......Page 61
4.3 Mapping from a rectangle......Page 64
4.4 Exterior maps......Page 68
4.5 Periodic regions and fractals......Page 72
4.6 Reflections and other transformations......Page 74
4.7 Riemann surfaces......Page 75
4.8 Gearlike regions......Page 77
4.9 Doubly connected regions......Page 81
4.10 Circular-arc polygons......Page 87
4.11 Curved boundaries......Page 90
5.1 Why use Schwarz–Christoffel maps?......Page 92
5.2 Piecewise-constant boundary conditions......Page 94
5.3 Alternating Dirichlet and Neumann conditions......Page 100
5.4 Oblique derivative boundary conditions......Page 104
5.5 Generalized parameter problems......Page 116
5.6 Free-streamline flows......Page 118
5.7 Mesh generation......Page 122
5.8 Polynomial approximation and matrix iterations......Page 125
5.9 Symmetric multiply connected domains......Page 128
Appendix: Using the SC Toolbox......Page 132
Bibliography......Page 138
J......Page 148
W......Page 149
