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Methods of complex analysis in partial differential equations with applications. (English) Zbl 0644.35005
Canadian Mathematical Society Series of Monographs and Advanced Texts; Wiley-Interscience Publication. New York etc: Wiley. xiv, 394 p. £47.50 (1988).

This book is a self-contained presentation of the function theoretical method in partial differential equations. It describes some methods for constructing integral operators which transform analytic functions into solutions to certain partial differential equations.

Such operators were introduced by S. Bergman (1937) and by I. N. Vekua (1937). The plan of the book is explained in the introduction. In Chapters 2-7 the authors develop various methods for constructing the integral operators and study integral representations of solutions to various PDE. Construction of Riemann’s function is discussed in detail.

In Chapters 8-10, the authors study: 1) singularities of solutions to some PDE in terms of the properties of the coefficients of the series which represent the solutions; 2) approximation of solutions to PDE by some special solutions (Runge’s, Mergelyan’s and Walsh’s type theorems); and 3) value distribution theory for solutions to some PDE.

In Chapters 11-13, various applications are given. In particular, the Bauer-Peschl equation, Chaplygin’s equation, and Tricomi’s equation are treated.

The book has an extensive bibliography and author, subject and symbol indices. The authors are active contributors to the field.

Reviewer: A.G.Ramm

MSC:
35-02Research monographs (partial differential equations)
35C15Integral representations of solutions of PDE