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Real quaternion analysis is a multi-faceted subject. Created to describe phenomena in special relativity, electrodynamics, spin etc., it has developed into a body of material that interacts with many branches of mathematics, such as complex analysis, harmonic analysis, differential geometry, and differential equations. It is also a ubiquitous factor in the description and elucidation of problems in mathematical physics. In the meantime real quaternion analysis has become a well established branch in mathematics and has been greatly successful in many different directions. This book is based on concrete examples and exercises rather than general theorems, thus making it suitable for an introductory one- or two-semester undergraduate course on some of the major aspects of real quaternion analysis in exercises. Alternatively, it may be used for beginning graduate level courses and as a reference work. With exercises at the end of each chapter and its straightforward writing style the book addresses readers who have no prior knowledge on this subject but have a basic background in graduate mathematics courses, such as real and complex analysis, ordinary differential equations, partial differential equations, and theory of distributions.
Algebra. --- Combinatorics. --- Functions of complex variables. --- Geometry. --- Mathematics. --- Matrix theory. --- Quaternions --- Functions, Quaternion --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Quaternions. --- Analysis (Mathematics) --- Fluxions (Mathematics) --- Infinitesimal calculus --- Limits (Mathematics) --- Quaternion functions --- Nonassociative rings. --- Rings (Algebra). --- Non-associative Rings and Algebras. --- Functions of a Complex Variable. --- Linear and Multilinear Algebras, Matrix Theory. --- Combinatorics --- Mathematical analysis --- Euclid's Elements --- Complex variables --- Elliptic functions --- Functions of real variables --- Calculus. --- Functions, Quaternion. --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra)
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This book presents applications of hypercomplex analysis to boundary value and initial-boundary value problems from various areas of mathematical physics. Given that quaternion and Clifford analysis offer natural and intelligent ways to enter into higher dimensions, it starts with quaternion and Clifford versions of complex function theory including series expansions with Appell polynomials, as well as Taylor and Laurent series. Several necessary function spaces are introduced, and an operator calculus based on modifications of the Dirac, Cauchy-Fueter, and Teodorescu operators and different decompositions of quaternion Hilbert spaces are proved. Finally, hypercomplex Fourier transforms are studied in detail. All this is then applied to first-order partial differential equations such as the Maxwell equations, the Carleman-Bers-Vekua system, the Schrödinger equation, and the Beltrami equation. The higher-order equations start with Riccati-type equations. Further topics include spatial fluid flow problems, image and multi-channel processing, image diffusion, linear scale invariant filtering, and others. One of the highlights is the derivation of the three-dimensional Kolosov-Mushkelishvili formulas in linear elasticity. Throughout the book the authors endeavor to present historical references and important personalities. The book is intended for a wide audience in the mathematical and engineering sciences and is accessible to readers with a basic grasp of real, complex, and functional analysis.
Mathematics. --- Functional analysis. --- Functions of complex variables. --- Integral transforms. --- Operational calculus. --- Partial differential equations. --- Integral Transforms, Operational Calculus. --- Functions of a Complex Variable. --- Partial Differential Equations. --- Functional Analysis. --- Holomorphic functions. --- Functions, Holomorphic --- Functions of several complex variables --- Integral Transforms. --- Differential equations, partial. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Partial differential equations --- Complex variables --- Elliptic functions --- Functions of real variables --- Transform calculus --- Transformations (Mathematics) --- Operational calculus --- Differential equations --- Electric circuits
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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.
Electronic books. -- local. --- Holomorphic functions -- Problems, exercises, etc. --- Holomorphic functions. --- Holomorphic functions --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Functions, Holomorphic --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Functions of complex variables. --- Integral transforms. --- Operational calculus. --- Potential theory (Mathematics). --- Functions of a Complex Variable. --- Integral Transforms, Operational Calculus. --- Potential Theory. --- Analysis. --- Functions of several complex variables --- Integral Transforms. --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Transform calculus --- Integral equations --- Transformations (Mathematics) --- Complex variables --- Elliptic functions --- Functions of real variables --- Global analysis (Mathematics) --- 517.1 Mathematical analysis --- Operational calculus --- Differential equations --- Electric circuits
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