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Book review

Author(s) Bell, S.R.
Title The Cauchy transform, potential theory, and conformal mapping
Publisher CRC Press
Year of publication 1992
   
Reviewed by Miroslav Englis

This is a very nice book, which will be of interest both to students and to specialists in complex analysis or related fields. The former will learn beautiful classical results, such as the interplay between the Szegö, Garabedian and Bergman kernels, the Riemann mapping function, Dirichlet and Neumann problems, Ahlfors extremal map, the Hilbert transform, Korenblum's duality between A^() and A-(), etc. The latter will probably appreciate a certain novelty of the approach, which is based on a relatively recent result of Kerzman and Stein on the near selfadjointness of the Cauchy integral transform. Both will be appealed to by a streamlined, elegant, and very understandable exposition, which makes the book a joy to read. There is a short bibliography, an index, and a table of symbols; and both the typesetting and the graphic layout of the book are really excellent. The material is well chosen and arranged, and there are many things which are not readily available elsewhere in the book form.

The only drawback of this book seems to be the price, which is outrageous, at least when compared to the thinness of the volume. Even though, I think that it's a good idea to (have one's library) make the expenditure; and those who do invest it will certainly acquire a really nice and worthy contribution to their library.