Softcover ISBN: | 978-0-88385-046-6 |
Product Code: | CAR/17 |
List Price: | $55.00 |
MAA Member Price: | $41.25 |
AMS Member Price: | $41.25 |
eBook ISBN: | 978-1-61444-017-8 |
Product Code: | CAR/17.E |
List Price: | $50.00 |
MAA Member Price: | $37.50 |
AMS Member Price: | $37.50 |
Softcover ISBN: | 978-0-88385-046-6 |
eBook: ISBN: | 978-1-61444-017-8 |
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 |
Softcover ISBN: | 978-0-88385-046-6 |
Product Code: | CAR/17 |
List Price: | $55.00 |
MAA Member Price: | $41.25 |
AMS Member Price: | $41.25 |
eBook ISBN: | 978-1-61444-017-8 |
Product Code: | CAR/17.E |
List Price: | $50.00 |
MAA Member Price: | $37.50 |
AMS Member Price: | $37.50 |
Softcover ISBN: | 978-0-88385-046-6 |
eBook ISBN: | 978-1-61444-017-8 |
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 |
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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 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.
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Table of Contents
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Chapters
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Chapter 1. Prologue
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Chapter 2. Conjugate coordinates in the plane
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Chapter 3. Elementary geometric facts
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Chapter 4. The nine-point circle
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Chapter 5. The Schwarz function for an analytic arc
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Chapter 6. Geometrical Interpretation Of The schwarz Function; Schwarzian reflection
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Chapter 7. The Schwarz function and differential geometry
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Chapter 8. Conformal maps, reflections, and their algebra
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Chapter 9. What figure is the $\sqrt {-1}$ power of a circle?
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Chapter 10. Properties in the large of the Schwarz function
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Chapter 11. Derivatives and integrals
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Chapter 12. Application to elementary fluid mechanics
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Chapter 13. The Schwarz function and the Dirichlet problem
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Chapter 14. Schwarz functions of specified type
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Chapter 15. Schwarz functions and iteration
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Chapter 16. Dictionary of functional relationships
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Chapter 17. Bibliographical and supplementary notes
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Chapter 18. Bibliography
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Additional Material
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RequestsReview Copy – for publishers of book reviewsAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- 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 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.
-
Chapters
-
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
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Chapter 18. Bibliography