Holomorphic Functions in the Plane and n-dimensional Space + CD-Rom
This textbook introduces both classical complex analysis and its higher dimensional generalization, Clifford analysis. In higher dimensions, algebra of complex numbers is replaced by non commutative algebra of (real) quaternions or with a Clifford algebra. After developing standard results and notions from classical complex analysis, a discussion of its higher dimensional counterpart follows. In the first chapter complex numbers, quaternions and Clifford numbers are introduced and their algebraic and geometric properties are studied. In chapter 2, holomorphic functions in the plane are defined. After a discussion of possible higher dimensional generalizations, the definition of holomorphic functions is extended to functions in n-dimensional space with values in the corresponding Clifford algebra. Then, ‘simple’ holomorphic functions in the plane and n-dimensional space are dealt with, including powers and Möbius transformations.
The next chapter presents integral formulae for holomorphic functions in the plane and their higher dimensional analogues, namely the Cauchy integral formula, the Borel-Pompeiu formula and the Plemelj-Sokhotski formula. The last chapter deals with Taylor and Laurent series, isolated singularities of holomorphic functions and the residue theorem, and elementary functions and special functions, including the Gamma function, the Zeta function and automorphic functions. The book is very well written. Each chapter contains historical remarks, examples and exercises illustrating the topics treated. Moreover, the enclosed CD contains an up-to-date literature database and a Maple package solving some of the problems discussed in the text. This book can be recommended as a text for a basic course on complex analysis, as well as a readable introduction to Clifford analysis accessible to students of mathematics and physics.