Application of Holomorphic Functions in Two and Higher Dimensions

ISBN-13: 9783034809627 / Angielski / Twarda / 2016 / 390 str.

Klaus Gurlebeck; Klaus Habetha; Wolfgang Sprossig

Opis

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 Schrodinger 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 Throughout the book the authors endeavor to present historical references and important personalities.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."

Recenzje

"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)

Spis treści

1.Basic Properties of Holomorphic Functions.- 2.Conformal and Quasi-conformal Mappings.- 3.Function Theoretic Function spaces.- 4.Operator Calculus.- 5.Decompositions.- 6.Some First Order Systems of Partial Differential Equations.- 7.Boundary Value Problems of Second Order Partial Differential Equations.- 8.Some Initial-boundary Value Problems.- 9.Riemann-Hilbert Problems.- 10.Initial Boundary Value Problems on the Sphere.- 11.Fourier Transforms.- Bibliography.- Index.

O autorze

Klaus Gürlebeck, born 1954, Dr. rer. nat. 1984 Technische Hochschule Karl-Marx-Stadt (Chemnitz), Habilitation 1988 TU Karl-Marx-Stadt (Chemnitz), since 1999 Full Prof. Bauhaus-Universität Weimar; co-editor of several international mathematical journals; interested in quaternionic analysis, discrete function theories and applications to partial differential equations.

Klaus Habetha, born 1932, Dr. rer. nat. 1959 Freie Universität Berlin, Habilitation 1962 Technische Universität Berlin, Prof. Technische Universität Berlin, Full Prof. Universität Dortmund and since 1975 RWTH Aachen, here Rector 1987 - 1997, vice-president of German Rectors Conference; co-editor of the journal Complex Variables and elliptic equations until 2007; interested in function theory for partial differential equation.

Wolfgang Sprößig, born 1946 , Dr. rer. nat. 1974 Technische Hochschule Karl-Marx-Stadt (Chemnitz) Habilitation 1979 TU Chemnitz, Ass. Prof. TU Chemnitz, since 1986 Full Prof. TU Bergakademie Freiberg, Head of the Institute of Applied Analysis 1993 -2012. Since 1998 Editor-in-Chief of the journal Mathematical Methods in the Applied Sciences, co-editor of several international mathematical journals, interested in hypercomplex analysis and its applications. 

Blurb

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.