Title: Holomorphic functions in the plane and n-dimensional space /
Authors: Gürlebeck, Klaus.; Habetha, Klaus, 1932-; Sprössig, Wolfgang.
Publisher: Birkhäuser ; Springer [distributor],
Year: 2008
Subjects: Holomorphic functions.; Holomorphic functions Problems, exercises, etc.; Functions of complex variables.; Global analysis (Mathematics).; Integral Transforms.; Mathematics.; Potential theory (Mathematics).
Abstract: Complex analysis nowadays has higher-dimensional analoga: the algebra of complex numbers is replaced then by the non-commutative algebra of real quaternions or by Clifford algebras. During the last 30 years the so-called quaternionic and Clifford or hypercomplex analysis successfully developed to a powerful theory with many applications in analysis, engineering and mathematical physics. This textbook introduces both to classical and higher-dimensional results based on a uniform notion of holomorphy. Historical remarks, lots of examples, figures and exercises accompany each chapter.
Contents: Introduction -- I. Numbers -- Complex Numbers - Quaternions - Clifford numbers -- II. Functions -- Topological Aspects - Holomorphic Functions - Power Functions and Möbius Transformations -- III. Integration und Integral Theorems - Integral Theorems and -formulas - Teodorescu Transformation -- IV. Series and Local Properties - Power Series- Orthogonal Series - Elementary Functions -- Local Structure of Holomorphic Functions - Special Functions -- Appendices -- Bibliography -- Index.