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This book provides a very readable description of a technique, developed by the author years ago but as current as ever, for proving that solutions to certain (nonelliptic) partial differential equations only have real analytic solutions when the data are real analytic (locally). The technique is completely elementary but relies on a construction, a kind of a noncommutative power series, to localize the analysis of high powers of derivatives in the socalled bad direction. It is hoped that this work will permit a far greater audience of researchers to come to a deep understanding of this technique and its power and flexibility.
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