The authors study various kinds of generalised analytic continuations (GAC) of meromorphic functions. They present a broad scope of methods used to solve very interesting problems of GAC. The methods presented are closely related to methods of summability of divergent series. It is worth noting that the methods of GAC included here do not provide, in general, the same results.

The chapters cover the following topics: the Poincaré example, non-tangential boundary values, Borel’s ideas, superconvergence, Borel series, Gončar continuation, pseudocontinuation, a problem of Walsh and Tumarkinn, the Darlington synthesis problem, gap theorems, non-continuability, formal multiplication of series, spectral properties, uniqueness and an axiomatic approach. Another approach to GAC developed in potential theory (Fuglede’s finely holomorphic functions) is not mentioned.

The book is very clearly written, and contains a large number of results (including proofs) and historical comments. The presented solutions of GAC are far from the final exhausting answer, and several open problems are presented. The book will be useful for working mathematicians in GAC and complex function theory. The authors will place updates, corrections and additions at www.richmond.edu/~wross.

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