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, ...

Author: Tobin A. Driscoll

Publisher: Cambridge University Press

ISBN: 113943392X

Category: Mathematics

Page: 132

View: 561

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.

This is a unique monograph on numerical conformal mapping that gives a comprehensive account of the theoretical, computational and application aspects of the problems of determining conformal modules of quadrilaterals and of mapping ...

Author: Nicolas Papamichael

Publisher: World Scientific

ISBN: 9789814289535

Category: Mathematics

Page: 229

View: 618

This is a unique monograph on numerical conformal mapping that gives a comprehensive account of the theoretical, computational and application aspects of the problems of determining conformal modules of quadrilaterals and of mapping conformally onto a rectangle. It contains a detailed study of the theory and application of a domain decomposition method for computing the modules and associated conformal mappings of elongated quadrilaterals, of the type that occur in engineering applications. The reader will find a highly useful and up-to-date survey of available numerical methods and associated computer software for conformal mapping. The book also highlights the crucial role that function theory plays in the development of numerical conformal mapping methods, and illustrates the theoretical insight that can be gained from the results of numerical experiments.This is a valuable resource for mathematicians, who are interested in numerical conformal mapping and wish to study some of the recent developments in the subject, and for engineers and scientists who use, or would like to use, conformal transformations and wish to find out more about the capabilities of modern numerical conformal mapping.

Let tpp(w) be the Riemann map of a simply connected domain D onto D.
Suppose WQ is a corner of D with interior angle AR. Set w(£) = wo + A£a, ...
Express the limit as an integral, and relate it to the Schwarz- Christoffel map. 5.
For fixed 0 < k ...

Author: THEODORE GAMELIN

Publisher: Springer Science & Business Media

ISBN: 0387950699

Category: Mathematics

Page: 478

View: 414

An introduction to complex analysis for students with some knowledge of complex numbers from high school. It contains sixteen chapters, the first eleven of which are aimed at an upper division undergraduate audience. The remaining five chapters are designed to complete the coverage of all background necessary for passing PhD qualifying exams in complex analysis. Topics studied include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces, with emphasis placed on the three geometries: spherical, euclidean, and hyperbolic. Throughout, exercises range from the very simple to the challenging. The book is based on lectures given by the author at several universities, including UCLA, Brown University, La Plata, Buenos Aires, and the Universidad Autonomo de Valencia, Spain.

Author: Royal Eugene CollinsPublish On: 2012-06-11

Set (7-4) (1) Use the Schwartz–Christoffel mapping to derive Mapping Number (4
) of our Table 7-1. (2) Use the Schwartz–Christoffel mapping to derive Mapping
Number (8) of our Table 7-1. [Hint: treat the plane strip there as the initial z plane,
...

Author: Royal Eugene Collins

Publisher: Courier Corporation

ISBN: 9780486150123

Category: Science

Page: 400

View: 116

Practical text focuses on fundamental applied math needed to deal with physics and engineering problems: elementary vector calculus, special functions of mathematical physics, calculus of variations, much more. 1968 edition.

7 , 12 . 2 . 2 ) , it follows that , in specifying the Schwarz - Christoffel map , we may
select three of the x ; ' s arbitrarily . We will take x1 = - 1 , x2 = 0 , and In = too . It
remains to determine the other n - 3 vertices ( which may not be freely chosen ) .

Author: Steven G. Krantz

Publisher: Springer Science & Business Media

ISBN: 0817640118

Category: Mathematics

Page: 290

View: 528

Handbook of Complex Variables is a reference work for scientists and engineers who need to know and use essential information and methods involving complex variables and analysis. Its focus is on basic concepts and informational tools for mathematical "practice": solving problems in applied mathematics, science, and engineering. This handbook is a reference and authoritative resource for all professionals, practitioners, and researchers in mathematics, physical science, and engineering.

This book is a guide on conformal mappings, their applications in physics and technology, and their computer-aided visualization.

Author: Valentin I. Ivanov

Publisher: CRC Press

ISBN: 0849389364

Category: Mathematics

Page: 368

View: 774

This book is a guide on conformal mappings, their applications in physics and technology, and their computer-aided visualization. Conformal mapping (CM) is a classical part of complex analysis having numerous applications to mathematical physics. This modern handbook on CM includes recent results such as the classification of all triangles and quadrangles that can be mapped by elementary functions, mappings realized by elliptic integrals and Jacobian elliptic functions, and mappings of doubly connected domains. This handbook considers a wide array of applications, among which are the construction of a Green function for various boundary-value problems, streaming around airfoils, the impact of a cylinder on the surface of a liquid, and filtration under a dam. With more than 160 domains included in the catalog of mapping, Handbook of Conformal Mapping with Computer-Aided Visualization is more complete and useful than any previous volume covering this important topic. The authors have developed an interactive ready-to-use software program for constructing conformal mappings and visualizing plane harmonic vector fields. The book includes a floppy disk for IBM-compatible computers that contains the CONFORM program.

This result is called the transformation function of Schwarz–Christoffel. ...
adjusted to match those of P by introducing the appropriate constants cl and c2,
as multiplication of a mapping function by a complex number causes a rotation
combined ...

Author: G.A. Bruggeman

Publisher: Elsevier

ISBN: 0080527132

Category: Science

Page: 956

View: 851

This book has been divided into two parts, A and B. Part A comprises analytical solutions of about 1100 geohydrological problems in the saturated zone. Classification of the problems according to certain characteristics. Part B consists of three chapters, describing the basic principles for saturated ground water flow, analytical solution methods and mathematical functions respectively.

4.4.3 Applications of Conformal Mapping Conformal transformation has been
employed frequently to determine the ... The Schwarz-Christoffel transformations
help solve an important class of boundary value problems that involve regions
with ...

Author: Ramesh Garg

Publisher: Artech House

ISBN: 9781596933866

Category: Electromagnetic waves

Page: 528

View: 576

Achieve optimal microwave system performance by mastering the principles and methods underlying today's powerful computational tools and commercial software in electromagnetics. This authoritative resource offers you clear and complete explanation of this essential electromagnetics knowledge, providing you with the analytical background you need to understand such key approaches as MoM (method of moments), FDTD (Finite Difference Time Domain) and FEM (Finite Element Method), and Green's functions. This comprehensive book includes all math necessary to master the material. Moreover, it features numerous solved problems that help ensure your understanding of key concepts throughout the book.

10.4.6 Schwarz - Christoffel Transformation The properties of conformal mappings at singular points mentioned in connection with the mapping function 5
= z2 can also be used to map the x axis onto a polygon . We shall consider the mapping ...

Author: Joseph H. Spurk

Publisher: Springer Science & Business Media

ISBN: 3540616519

Category: Technology & Engineering

Page: 514

View: 982

This textbook emphasizes the unified nature of all the disciplines of Fluid Mechanics as they emerge from the general principles of continuum mechaniccs. The different branches of Fluid Mechanics, always originating from simplifying assumptions, are developed according to the basic rule: from the general to the specific. The first part of the book contains a concise but readable introduction into kinematics and the formulation of the laws of mechanics and thermodynamics. The second part consists of the methodical application of these principles to technology. This book is offered to engineers, physicists and applied mathematicians; it can be used for self study, as well as in conjunction with a lecture course.

This book evolved out of a graduate course given at the University of New Orleans in 1997.

Author: Prem Kythe

Publisher: Springer Science & Business Media

ISBN: 9781461220022

Category: Mathematics

Page: 462

View: 656

This book evolved out of a graduate course given at the University of New Orleans in 1997. The class consisted of students from applied mathematics andengineering. Theyhadthebackgroundofatleastafirstcourseincomplex analysiswithemphasisonconformalmappingandSchwarz-Christoffeltrans formation, a firstcourse in numerical analysis, and good to excellent working knowledgeofMathematica* withadditionalknowledgeofsomeprogramming languages. Sincetheclasshad nobackground inIntegralEquations, thechap tersinvolvingintegralequationformulations werenotcoveredindetail, except for Symm's integral equation which appealed to a subsetofstudents who had some training in boundary element methods. Mathematica was mostly used for computations. In fact, it simplified numerical integration and other oper ations very significantly, which would have otherwise involved programming inFortran, C, orotherlanguageofchoice, ifclassical numericalmethods were attempted. Overview Exact solutions of boundary value problems for simple regions, such as cir cles, squares or annuli, can be determined with relative ease even where the boundaryconditionsarerathercomplicated. Green'sfunctionsforsuchsimple regions are known. However, for regions with complex structure the solution ofa boundary value problem often becomes more difficult, even for a simple problemsuchastheDirichletproblem. Oneapproachtosolvingthesedifficult problems is to conformally transform a given multiply connected region onto *Mathematica is a registered trade mark of Wolfram Research, Inc. ix x PREFACE simpler canonical regions. This will, however, result in change not only in the region and the associated boundary conditions but also in the governing differential equation. As compared to the simply connected regions, confor mal mapping ofmultiply connected regions suffers from severe limitations, one of which is the fact that equal connectivity ofregions is not a sufficient condition to effect a reciprocally connected map ofone region onto another.

Suppose f1 (z) map D onto the disk |w| < 1 so that z0 maps into the origin and a
given direction at z0 maps into the ... Also, the Schwarz-Christoffel method, which
we study in the next section, gives a more or less general scheme for mapping ...

Author: John W. Dettman

Publisher: Courier Corporation

ISBN: 9780486158280

Category: Mathematics

Page: 512

View: 668

Fundamentals of analytic function theory — plus lucid exposition of 5 important applications: potential theory, ordinary differential equations, Fourier transforms, Laplace transforms, and asymptotic expansions. Includes 66 figures.

A and B are found by the requirements that 71: —ia =f(—1)= A{—5} + B 0 = f(0) =
B. Thus, the mapping is f(2)=igg{2,/1—z2 + arcsin 2}. El 71: Example 9 Find the Schwarz—Christoffel mapping of the upper half-plane U onto the region in Figure
...

Author: Stephen D. Fisher

Publisher: Courier Corporation

ISBN: 9780486134840

Category: Mathematics

Page: 448

View: 875

Topics include the complex plane, basic properties of analytic functions, analytic functions as mappings, analytic and harmonic functions in applications, transform methods. Hundreds of solved examples, exercises, applications. 1990 edition. Appendices.

552 3.2. The Schwarz–Christoffel Mapping Theorem . . . . . . . 556 3.3. An Integral
Equation for Mapping on the Upper-Half Plane 561 3.4. Generalizations of the
S-C Mapping Theorem . . . . . . 563 3.5. The Mapping of Nearly Circular Domains .

Author: A. Cemal Eringen

Publisher: Academic Press

ISBN: 9781483277165

Category: Science

Page: 694

View: 884

Continuum Physics: Volume 1 — Mathematics is a collection of papers that discusses certain selected mathematical methods used in the study of continuum physics. Papers in this collection deal with developments in mathematics in continuum physics and its applications such as, group theory functional analysis, theory of invariants, and stochastic processes. Part I explains tensor analysis, including the geometry of subspaces and the geometry of Finsler. Part II discusses group theory, which also covers lattices, morphisms, and crystallographic groups. Part III reviews the theory of invariants that includes isotrophy, transverse isotrophy, and nonpolynomial invariants. Part IV explains functional analysis that also includes set theory, vector spaces, topological spaces, and topological vector spaces. Part V deals with analytic function theory and covers topics, such as Cauchy's theorem, the residue theorem, and the Plemelj formulas. Part VI reviews the elements of stochastic processes and cites some examples where stochastic theory is applied. This book can be valuable for researchers and scientists involved in nuclear physicists, students, and academicians in the field of advanced physics.

... by successive application of the Riemann-Schwarz symmetry principle, a
function Z = Z(z) giving the required mapping ... amount to a distinctive
generalization of the Schwarz-Christoffel formulas used to implement a conformal mapping of a ...

Author: Naum Ilʹich Akhiezer

Publisher: American Mathematical Soc.

ISBN: 0821886770

Category: Mathematics

Page: 237

View: 567

This book contains a systematic presentation of the theory of elliptic functions and some of its applications. A translation from the Russian, this book is intended primarily for engineers who work with elliptic functions. It should be accessible to those with background in the elements of mathematical analysis and the theory of functions contained in approximately the first two years of mathematics and physics courses at the college level.

This volume introduces the basic mathematical tools behind conformal mapping, describes advances in technique, and illustrates a broad range of applications. 1991 edition. Includes 247 figures and 38 tables.

Author: Roland Schinzinger

Publisher: Courier Corporation

ISBN: 9780486150741

Category: Mathematics

Page: 608

View: 416

This volume introduces the basic mathematical tools behind conformal mapping, describes advances in technique, and illustrates a broad range of applications. 1991 edition. Includes 247 figures and 38 tables.

Chapter 1 Conformal Mapping and the Riemann Mapping Theorem 1.1
Introduction Without question, the basic theorem in the theory of ... Finally the Schwarz-Christoffel Theorem giving explicit mappings of polygonal regions is
treated.

Author: Sanford L. Segal

Publisher: Elsevier

ISBN: 0080550762

Category: Mathematics

Page: 500

View: 820

The book addresses many topics not usually in "second course in complex analysis" texts. It also contains multiple proofs of several central results, and it has a minor historical perspective. - Proof of Bieberbach conjecture (after DeBranges) - Material on asymptotic values - Material on Natural Boundaries - First four chapters are comprehensive introduction to entire and metomorphic functions - First chapter (Riemann Mapping Theorem) takes up where "first courses" usually leave off

Computing the map parameters ( setting up and solving the least - squares
system ) took approximately 3 s on a 1 . 4 GHz PentiumM laptop . References
Crowdwains . Computhall , J . 20 Crowdy , D . 2005 The Schwarz - Christoffel mapping to ...

They parameterized D by a conformal map from the exterior of the unit circle onto
the exterior of D. The dynamics of ... the growing region D is a tree graph in the
complex to-plane parameterized by a singular Schwarz-Christoffel map w = H(z)
...