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Quarternionic calculus covers a branch of mathematics which uses computational techniques to help solve problems from a wide variety of physical systems which are mathematically modelled in 3, 4 or more dimensions. Examples of the application areas include thermodynamics, hydrodynamics, geophysics and structural mechanics. Focusing on the Clifford algebra approach the authors have drawn together the research into quarternionic calculus to provide the non-expert or research student with an accessible introduction to the subject. This book fills the gap between the theoretical representations and the requirements of the user.

differentiaalvergelijkingen --- Harmonic analysis. Fourier analysis --- Partial differential equations --- complexe analyse (wiskunde) --- Mathematical physics --- 512.5 --- Boundary value problems --- Clifford algebras --- Differential equations, Partial --- Quaternions --- Algebra, Universal --- Algebraic fields --- Curves --- Surfaces --- Numbers, Complex --- Vector analysis --- Geometric algebras --- Algebras, Linear --- Boundary conditions (Differential equations) --- Differential equations --- Functions of complex variables --- Initial value problems --- General algebra --- Boundary value problems. --- Clifford algebras. --- Differential equations, Partial. --- Quaternions. --- 512.5 General algebra

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Die Funktionentheorie einer komplexen Variablen hat heute höher-dimensionale Analoga: dabei wird die Algebra der komplexen Zahlen durch die nicht-kommutative Algebra der reellen Quaternionen bzw. Clifford-Algebren ersetzt. In den letzten 30 Jahren hat sich die so genannte Quaternionen- und die reelle Clifford-Analysis erfolgreich entwickelt. Eine Vielzahl von Anwendungen haben diese Funktionentheorie höher-dimensionaler Variablen zu einem wichtigen Instrument der Analysis und deren Anwendungen in der mathematischen Physik werden lassen. Das Buch reflektiert den neuesten Stand der Forschung und entwickelt sowohl die höher-dimensionalen Ergebnisse als auch die klassischen komplexen Resultate aus einem einheitlichen Begriff der Holomorphie. Der fundamentale Begriff der holomorphen Funktion als Lösung des Cauchy-Riemann-Systems wird im Höher-dimensionalen unter Beibehaltung der Bezeichnung als Lösung eines entsprechenden Systems partieller Differentialgleichungen 1. Ordnung verstanden. Historische Bemerkungen, zahlreiche Beispiele, viele Abbildungen sowie eine angemessene Auswahl von Übungsaufgaben festigen und erweitern die erworbenen Kenntnisse. Das vorliegende Buch ist für Studenten der Mathematik, Physik und mathematisch orientierten Ingenieurstudenten im Grund- und Fachstudium geeignet. Es kann auch als Grundlage von Proseminaren oder Seminaren dienen. Die beiliegende CD enthält eine umfangreiche Literaturdatenbank sowie ein Maple-Package, das die im Buch eingeführten Werkzeuge und Methoden als Kommandos bzw. vorgefertigte Prozeduren enthält. Einige Beispiel-Worksheets unterstützen die Einarbeitung in das Package.

Mathematics. --- Functions of a Complex Variable. --- Analysis. --- Special Functions. --- Global analysis (Mathematics). --- Functions of complex variables. --- Functions, special. --- Mathématiques --- Analyse globale (Mathématiques) --- Fonctions d'une variable complexe --- Functions, Special. --- Mathematical analysis. --- Analysis (Mathematics). --- Special functions.

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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 --- Calculus, Operational. --- Differential equations, Partial.

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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.

Algebra --- Functional analysis --- Differential equations --- Mathematical analysis --- differentiaalvergelijkingen --- algebra --- analyse (wiskunde) --- Laplacetransformatie --- functies (wiskunde)

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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.

Algebra --- Algebraic geometry --- Functional analysis --- Partial differential equations --- Mathematical analysis --- Mathematics --- differentiaalvergelijkingen --- algebra --- analyse (wiskunde) --- complexe veranderlijken --- functies (wiskunde) --- wiskunde