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Huygens’ principle and hyperbolic equations. (English) Zbl 0655.35003

Perspectives in Mathematics, 5. Boston, MA etc.: Academic Press, Inc. lvii, 847 p. $ 69.00 (1988).
This is a rather well written book. As the title of the book suggested, the book is about Huygens’ principle, and this is conveyed very clearly to readers in the preface. Very seldom the preface of a book provides so much information about the book in so little space. Huygens’ principle, stated briefly, is that if the source produces a signal for a time duration \(\Delta\) t, an observer at any distance from the source will receive at a later time a disturbance of the signal for an equal duration of time \(\Delta\) t. Not all differential equations support Huygens’ principle. Second order linear hyperbolic differential equations in even order space obviously do not support Huygens’ principle. A differential equation which supports Huygens’ principle is a Huygens’ operator. One of the objectives of the book is to identify Huygens’ operators and to describe their properties.
There are eight chapters in the book. Chapter I, which has no title, treats wave equations for p-forms over a space of constant sectional curvature. It also presents a spinor calculus in four-dimensional space- times. The titles of the remaining chapters are as follows. II. Riesz distributions. III. The fundamental solutions. IV. Huygens’ operators. V. The Euler-Poisson-Darboux equation. VI. Transformation theory. VII. Some theories on Huygens’ operators over four-dimensional space-times. VIII. Plane wave manifolds and Huygens’ principle.
The book also contains four tables and three appendices. The index section appears to be small considering the size of the book which is over eight hundred pages. The bibliography section is not exhaustive but adequate.
Reviewer: T.C.T.Ting

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q99 Partial differential equations of mathematical physics and other areas of application
35Q05 Euler-Poisson-Darboux equations
35L10 Second-order hyperbolic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35A08 Fundamental solutions to PDEs
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