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Function theory in plane and space. (Funktionentheorie in der Ebene und im Raum.) (German) Zbl 1104.30001
Grundstudium Mathematik. Basel: Birkhäuser (ISBN 3-7643-7369-5/pbk). xiii, 406 p. with CD-ROM. (2006).
In one of his courses on complex analysis some 40 years ago at Freie Universität Berlin Alexander Dinghas has mentioned quaternions as generalization of complex numbers. He also has remarked that there once before World War II even has been a society to promote the use of quaternions. It seems that there exist no traces of this society. But nowadays there is no need for such an institution. Quaternionic and more general Clifford analysis have become quite fashionable in the meantime. A text book presenting the fundamentals of complex, quaternionic and Clifford analysis as a unit stressing the similarities and explaining as well their peculiarities as their advantages is a natural consequence or the altered situation. The present book is very well and enthusiastically written and in many details marvellously elaborated. This is in particular the case for the typographically exposed historical remarks including photos of some of the main initiators of the number systems etc. involved. Only one of W. R. Hamilton is missing. In the past decades one of the authors has systematically collected references on generalized complex analysis. These about 9000 references – too many for being printed – are provided on a CD-Rom attached to the book. The reference list shows 159 publications. The CD-Rom also provides a Maple package for applying the tools developed in the book. The only disadvantage is that the book is written in German. It demands to be translated into English. Chapter 1 is devoted to number systems: complex, quaternionic, and Clifford. Besides discussing algebraic consequences, special attention is paid to geometrical aspects as e.g. rotations in plane and space and also some basics for spherical trigonometry. Functions are the subjects of Chapter 2 covering continuity, series, differentiability, holomorphy (avoiding the terminology monogeneity), power functions and Möbius transformations. Cauchy Pompeiu representations and the Pompeiu Integral transform, in connection with Clifford analysis also denoted as Borel Pompeiu representation and Teodorescu transform, respectively and their consequences are the main subjects of Chapter 3. The final chapter 4 is remarkably larger than the others. It includes consequences from the Cauchy representation such as power and Laurent series representations, representations through orthogonal systems as e.g. through spherical harmonics, useful for numerical treatments – a speciality of one of the authors -, Fourier series, local behaviour of holomorphic functions, residue theory and argument principle, a subject not yet completely well understood in higher dimensions, evaluation of real determined integrals, some special functions as e.g. gamma and zeta functions and automorphic forms and functions. Each chapter includes a few exercises. An Appendix is devoted to differential forms, integration on manifolds, Stokes (without proof), Gauss and Green integral theorems, some basics on function spaces as far as needed for the Hodge decomposition of the Hilbert function space and on spherical functions. Also this part concludes some exercises. This book is recommended as a text book for basic courses in complex analysis and also for self studies.

30-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable
30G30 Other generalizations of analytic functions (including abstract-valued functions)