The Cauchy Transform * (Mathematical Surveys and Monographs 125) * By Joseph A. Cima, Alec L. Matheson and William T. Ross: * 272 pp., {pound}43.75 (US$75), ISBN 0-8218-3871-7 * (American Mathematical Society, Providence, RI, 2006).

In the border country between complex analysis, harmonic analysisand differential equations, there can be found many populartransforms, of which those of Fourier and Laplace are probablythe most commonly used. This book is concerned with anotherone, which is more rarely sighted. The Cauchy transform of a finite complex measure µ definedon the unit circle in the complex plane is the analytic functionKµ defined by

forz in the unit disc . Its name derives from the observation thatCauchy's integral formula, when applied to an analytic functionf, is equivalent to the statement that the Cauchy transformof the measure is just