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Softcover ISBN:  9780883850466 
eBook: ISBN:  9781614440178 
Product Code:  CAR/17.B 
List Price:  $105.00 $80.00 
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AMS Member Price:  $78.75 $60.00 
Softcover ISBN:  9780883850466 
Product Code:  CAR/17 
List Price:  $55.00 
MAA Member Price:  $41.25 
AMS Member Price:  $41.25 
eBook ISBN:  9781614440178 
Product Code:  CAR/17.E 
List Price:  $50.00 
MAA Member Price:  $37.50 
AMS Member Price:  $37.50 
Softcover ISBN:  9780883850466 
eBook ISBN:  9781614440178 
Product Code:  CAR/17.B 
List Price:  $105.00 $80.00 
MAA Member Price:  $78.75 $60.00 
AMS Member Price:  $78.75 $60.00 

Book DetailsThe Carus Mathematical MonographsVolume: 17; 1974; 228 pp
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 selfcontained. 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.

Table of Contents

Chapters

Chapter 1. Prologue

Chapter 2. Conjugate coordinates in the plane

Chapter 3. Elementary geometric facts

Chapter 4. The ninepoint 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


Additional Material

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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 selfcontained. 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.

Chapters

Chapter 1. Prologue

Chapter 2. Conjugate coordinates in the plane

Chapter 3. Elementary geometric facts

Chapter 4. The ninepoint 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