Cambridge University Press, Jun 20, 2002 - Mathematics - 132 pages
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, as evidenced by the 76 figures based on quantitatively correct illustrative examples. There are over 150 classical and modern reference works cited for readers needing more details. There is also a brief appendix illustrating the use of the Schwarz-Christoffel Toolbox for MATLAB, a package for computation of these maps.
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algorithm analytic annulus applications arcs aspect ratio boundary values bounded solution circular-arc polygons complex computational conformal mapping conformal modulus corners CRDT cross-ratios crowding define derivative boundary conditions determined Dirichlet values disk maps domain Doubly connected regions Elcrat elliptic functions elongated embedding example exterior maps Faber polynomials factor gearlike regions geometry Green's function grid harmonic function infinite infinity integrand interior angles intervals iteration L-shaped Laplace's equation Level curves logarithm MATLAB mesh method Mobius transformation Netlib Oblique derivative boundary oblique derivative problem piecewise piecewise-constant polynomial prevertex prevertices quadrilateral real axis rectangle maps reflection Riemann mapping theorem Riemann surface SC formula SC integral SC map SC Toolbox Schwarz Schwarz-Christoffel formula Schwarz-Christoffel mapping SCPACK section 4.2 segments shown in Figure side lengths simply connected singularities slit standard SC strip map target region Theorem 5.2 Trefethen triangles unbounded unique unit circle unknown upper half-plane vertex vertices