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.
General
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