The Schwarz function and its applications / by Philip J. Davis.

Format:

Book

Author/Creator:

Davis, Philip J., 1923- author.

Series:

Carus mathematical monographs ; Number 17.

Carus Mathematical Monographs ; Number 17

Language:

English

Subjects (All):

Analytic functions.

Geometry, Analytic--Plane.

Geometry, Analytic.

Physical Description:

1 online resource (xi, 228 pages) : digital, PDF file(s).

Edition:

1st ed.

Other Title:

The Schwarz Function & its Applications

Place of Publication:

Washington : Mathematical Association of America, 1974.

Language Note:

English

Summary:

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.

Contents:

Prologue

Conjugate coordinates in the plane

Elementary geometrical facts

The none-point circle

The Schwarz function for an analytic arc

Geometrical interpretation of the Schwarz function; Schwarzian reflection

The Schwarz function and differential geometry

Conformal maps, reflections, and their algebra

What figure is the [square root]-1 power of a circle?

Properties in the large of the Schwarz function

Derivatives and integrals

Application to elementary fluid mechanics

The Schwarz function and the Dirichlet problem

Schwarz functions of specified type

Schwarz functions and iterations

Dictionary of functional relationships

Bibliographical and supplementary notes

Bibliography.

Notes:

Title from publisher's bibliographic system (viewed on 02 Oct 2015).

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-61444-017-4