Overview

H. A. Schwarz showed us how to extend the notion of reflection in straight lines and circles to reflection in an arbitrary analytic arc. Notable applications were made to the symmetry principle and to problems of analytic continuation. Reflection, in the hands of Schwarz, is an antianalytic mapping. By taking its complex conjugate, we arrive at an analytic function that we have called here the Schwarz Function of the analytic arc. This function is worthy of study in its own right and this essay presents such a study. In dealing with certain familiar topics, the use of the Schwarz Function lends a point of view, a clarity and elegance, and a degree of generality which might otherwise be missing. It opens up a line of inquiry which has yielded numerous interesting things in complex variables; it illuminates some functional equations and a variety of iterations which interest the numerical analyst. The perceptive reader will certainly find here some old wine in relabelled bottles. But one of the principles of mathematical growth is that the relabelling process often suggests a new generation of problems. Means become ends; the medium rapidly becomes the message. This book is not wholly self-contained. Readers will find that they should be familiar with the elementary portions of linear algebra and of the theory of functions of a complex variable.

Full Product Details

Author:   Philip J. Davis
Publisher:   American Mathematical Society
Imprint:   American Mathematical Society
Weight:   0.228kg
ISBN:  

9781470485559

ISBN 10:   1470485559
Pages:   228
Publication Date:   30 April 1974
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

Chapter 1. Prologue Chapter 2. Conjugate coordinates in the plane Chapter 3. Elementary geometric facts Chapter 4. The nine-point circle Chapter 5. The Schwarz function for an analytic arc Chapter 6. Geometrical Interpretation Of The schwarz Function Schwarzian reflection Chapter 7. The Schwarz function and differential geometry Chapter 8. Conformal maps, reflections, and their algebra Chapter 9. What figure is the $\sqrt {-1}$ power of a circle? Chapter 10. Properties in the large of the Schwarz function Chapter 11. Derivatives and integrals Chapter 12. Application to elementary fluid mechanics Chapter 13. The Schwarz function and the Dirichlet problem Chapter 14. Schwarz functions of specified type Chapter 15. Schwarz functions and iteration Chapter 16. Dictionary of functional relationships Chapter 17. Bibliographical and supplementary notes Chapter 18. Bibliography

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