Application of Holomorphic Functions in Two and Higher Dimensions
Buch in deiner Nähe kaufen
Beschreibung
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Kolosov-Mushkelishvili formulas in linear elasticity.Throughout the book the authors endeavor to present historical references and important personalities.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.
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 topicsinclude 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.
Presents a unique hypercomplex strategy for the solution of boundary value problems and initial-boundary value problems in higher dimensions Details hypercomplex versions of the Fourier transform and applications Offers new approaches to boundary value problems in elasticity and fluid mechanics from modeling to a solution theory Demonstrates the construction of hyperholomorphic orthogonal polynomial Appell systems in elementary domains in R^3
Autor*in
Klaus Gürlebeck
Themen in »Application of Holomorphic Functions in Two and Higher Dimensions«
hypercomplex numbers hypercomplex versions of Fourier transforms initial-boundary value problems quaternionic anaylsis quaternionic operator calculus partial differential equations
Stimmen zu »Application of Holomorphic Functions in Two and Higher Dimensions«
“The text is well written. It contains a wealth of material previously published in several books and articles. An extended bibliography makes it possible to access the original references. As a final note, the book also contains a number of historical pictures of some of the main mathematicians related to the topics described.” (Alessandro Perotti, Mathematical Reviews, July, 2017)
()