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E-raamat: Application of Holomorphic Functions in Two and Higher Dimensions

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  • Ilmumisaeg: 20-Jun-2016
  • Kirjastus: Birkhauser Verlag AG
  • Keel: eng
  • ISBN-13: 9783034809641
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Application of Holomorphic Functions in Two and Higher DimensionsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 20-Jun-2016
  • Kirjastus: Birkhauser Verlag AG
  • Keel: eng
  • ISBN-13: 9783034809641
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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 spa

tial 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Kolosov-Mushkelishvili formulas in linear elasticity.Throughout the book the authors endeavor to present historical references and important personalities.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.

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.

Arvustused

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)

Preface xi
1 Basic properties of holomorphic functions 1(42)
1.1 Number systems
1(15)
1.1.1 Real Clifford numbers
1(3)
1.1.2 Quaternion algebra
4(1)
1.1.3 On rotations
5(2)
1.1.4 Complex quaternions
7(1)
1.1.5 Clifford's geometric algebra
7(4)
1.1.6 The ± split with respect to two square roots of -1
11(4)
1.1.7 Bicomplex numbers
15(1)
1.2 Classical function spaces in quaternions
16(2)
1.3 New types of holomorphic functions
18(5)
1.3.1 Definitions
18(3)
1.3.2 Construction of holomorphic functions
21(2)
1.4 Integral theorems
23(5)
1.4.1 General integral theorems
23(2)
1.4.2 Integral theorems for holomorphic functions
25(3)
1.5 Polynomial systems
28(15)
1.5.1 Fueter polynomials
29(4)
1.5.2 Holomorphic Appell polynomials
33(1)
1.5.3 Holomorphic polynomials for the Riesz system
34(5)
1.5.4 Orthogonal polynomials in H
39(1)
1.5.5 Series expansions
39(4)
2 Conformal and quasi-conformal mappings 43(32)
2.1 Mobius transformations
43(2)
2.1.1 Schwarzian derivative
44(1)
2.2 Conformal mappings
45(8)
2.2.1 Conformal mappings in the plane
46(1)
2.2.2 Conformal mappings in space
47(5)
2.2.3 Mercator projection
52(1)
2.3 Quasi-conformal mappings
53(6)
2.3.1 Basic definitions
53(3)
2.3.2 Quaternionic quasi-conformal mappings
56(3)
2.4 M-conformal mappings
59(16)
2.4.1 Characterization of M-conformal mappings
60(10)
2.4.2 M-conformal mappings in a plane
70(2)
2.4.3 M-conformal mappings of curves on the unit sphere
72(3)
3 Function theoretic function spaces 75(20)
3.1 Qp-spaces
75(3)
3.2 Properties of Qp-spaces
78(4)
3.3 Another characterization of Qp-spaces
82(7)
3.4 Bergman and Hardy spaces
89(4)
3.4.1 Bergman space
89(2)
3.4.2 Hardy space
91(2)
3.5 Riesz potentials
93(2)
4 Operator calculus 95(56)
4.1 Teodorescu transform and its left inverse
95(5)
4.1.1 Historical prologue
95(1)
4.1.2 Borel-Pompeiu formula
96(4)
4.2 On generalized II-operators
100(19)
4.2.1 The complex II-operator
100(2)
4.2.2 Shevchenko's generalization
102(1)
4.2.3 A generalization via the Teodorescu transform
103(8)
4.2.4 The second generalization of the II-operator
111(3)
4.2.5 The third generalization of the II-operator
114(3)
4.2.6 The special case of quaternions
117(2)
4.3 A general operator approach to holomorphy
119(12)
4.3.1 A general holomorphy
120(2)
4.3.2 Types of L-holomorphy
122(6)
4.3.3 Taylor type formula
128(3)
4.3.4 Taylor-Gontcharov formula for generalized Dirac operators of higher order
131(1)
4.4 A modified operator calculus in the plane
131(8)
4.4.1 Modified Borel-Pompeiu type formulas
132(2)
4.4.2 Modified Plemelj-Sokhotski formulas
134(1)
4.4.3 A modified Dirichlet problem
135(2)
4.4.4 A norm estimate for the modified Teodorescu transform
137(2)
4.5 Modified operator calculus in space
139(5)
4.5.1 Modified fundamental solutions
139(3)
4.5.2 A modified Borel-Pompeiu formula
142(2)
4.6 Operator calculus on the sphere
144(7)
4.6.1 Gegenbauer functions
144(1)
4.6.2 Spherical harmonics
145(4)
4.6.3 Borel-Pompeiu formula
149(2)
5 Decompositions 151(18)
5.1 Vector fields in Euclidean space
151(5)
5.1.1 Helmholtz decomposition
151(3)
5.1.2 Associated boundary value problems
154(1)
5.1.3 Original Hodge decomposition theorem
155(1)
5.2 Bergman-Hodge decompositions
156(8)
5.2.1 Suitable fundamental solutions
157(2)
5.2.2 An orthogonal decomposition formula with complex potential
159(3)
5.2.3 Generalized Bergman-Hodge decomposition
162(1)
5.2.4 Decompositions in domains on the unit sphere
162(2)
5.3 Representations of functions by holomorphic generators
164(5)
5.3.1 Almansi decomposition
164(2)
5.3.2 Fischer decomposition
166(3)
6 Some first-order systems of partial differential equations 169(34)
6.1 Maxwell equations
169(6)
6.1.1 A brief historical review
169(3)
6.1.2 Stationary Maxwell equations
172(1)
6.1.3 Stationary Maxwell equations with variable permitivities
173(2)
6.2 Bers-Vekua systems
175(20)
6.2.1 History of the Vekua equation
175(2)
6.2.2 Pseudoanalytic functions
177(2)
6.2.3 Generating sequences and formal powers
179(2)
6.2.4 An important special case
181(1)
6.2.5 Orthogonal coordinates and explicit generating sequences
182(2)
6.2.6 Completeness of the systems of formal powers
184(1)
6.2.7 Factorization of second-order operators in the plane
185(3)
6.2.8 Complete systems of solutions for the stationary Schrodinger equation and their applications
188(2)
6.2.9 The Riccati equation in two dimensions
190(1)
6.2.10 On the solution of the Riccati equation
191(3)
6.2.11 Factorization in the hyperbolic case
194(1)
6.3 Biquaternions and factorization of spatial models
195(8)
6.3.1 Biquaternionic Vekua-type equations from physics
195(2)
6.3.2 Factorization of the 3D-Schrodinger operator and the main biquaternionic Vekua equation
197(6)
7 Boundary value problems for second-order partial differential equations 203(62)
7.1 p-harmonicity
203(5)
7.1.1 Poisson equation
203(4)
7.1.2 p-harmonic functions
207(1)
7.2 A class of non-linear boundary value problems
208(2)
7.3 Helmholtz equation
210(9)
7.3.1 Motivation and historical note
210(2)
7.3.2 Square roots of the Helmholtz operator
212(7)
7.4 Yukawa's equation
219(3)
7.4.1 An operator theory
219(3)
7.5 Equations of linear elasticity
222(25)
7.5.1 Stress
222(4)
7.5.2 Deformation
226(4)
7.5.3 Solution theory for the stationary problem
230(2)
7.5.4 Kolosov-Muskhelishvili formulas
232(2)
7.5.5 Fundamentals of the linear theory of elasticity
234(2)
7.5.6 General solution of Papkovic-Neuber
236(1)
7.5.7 The representation theorem of Goursat in H
237(2)
7.5.8 Spatial Kolosov-Muskhelishvili formulas in H
239(2)
7.5.9 Generalized Kolosov-Muskhelishvili formulas for stresses
241(6)
7.6 Transmission problems in linear elasticity
247(7)
7.6.1 Boundary value problems in multiply connected domains
248(2)
7.6.2 Solution of the transmission problem
250(4)
7.6.3 Transmission problems for the Lame system
254(1)
7.7 Stationary fluid flow problems
254(11)
7.7.1 A brief history of fluid dynamics
254(1)
7.7.2 Stationary linear Stokes problem
255(1)
7.7.3 Non-linear Stokes equations
256(1)
7.7.4 Stationary Navier-Stokes problem
257(1)
7.7.5 Stationary equations of thermo-fluid dynamics
258(1)
7.7.6 Stationary magneto-hydromechanics
259(6)
8 Some initial-boundary value problems 265(38)
8.1 Rothe's method
266(1)
8.2 Stokes equation
267(2)
8.3 Galpern-Sobolev equations
269(7)
8.3.1 Description of the problem
270(3)
8.3.2 Quaternionic integral operators
273(1)
8.3.3 A representation formula
274(2)
8.4 The Poisson-Stokes problem
276(6)
8.4.1 Semi-discretization
278(2)
8.4.2 Operator decomposition
280(1)
8.4.3 Representation formulas
280(2)
8.5 Higher dimensional versions of Korteweg-de Vries' and Burgers' equation
282(6)
8.5.1 Multidimensional version of Burgers equation
282(1)
8.5.2 Airy's equation
283(4)
8.5.3 A quaternionic Korteweg-de Vries-Burgers equation
287(1)
8.6 Solving the Maxwell equations
288(2)
8.7 Alternative treatment of parabolic problems
290(4)
8.7.1 The Witt basis approach
290(2)
8.7.2 Harmonic extension method
292(2)
8.8 Fluid flow through porous media
294(9)
8.8.1 Governing equations
294(1)
8.8.2 Representation in a quaternionic operator calculus
295(5)
8.8.3 Error analysis
300(3)
9 Riemann-Hilbert problems 303(16)
9.1 Riemann-Hilbert problem in the plane
303(4)
9.2 Riemann-Hilbert problems in Cl(3, 0)
307(12)
9.2.1 Plemelj formula for functions with a parameter
308(8)
9.2.2 Riemann boundary value problem for harmonic functions
316(1)
9.2.3 Riemann boundary value problem for biharmonic functions
317(2)
10 Initial-boundary value problems on the sphere 319(10)
10.1 Forecasting equations
319(7)
10.1.1 Forecasting equations - a physical description
319(2)
10.1.2 Toroidal flows on the sphere
321(1)
10.1.3 Tangential derivatives
322(1)
10.1.4 Oseen's problem on the sphere
323(2)
10.1.5 Forecasting equations in a ball shell
325(1)
10.2 Viscous shallow water equations
326(3)
11 Fourier transforms 329(32)
11.1 Hypercomplex Fourier transforms
329(17)
11.1.1 Introduction
329(1)
11.1.2 General two-sided Clifford Fourier transforms
330(1)
11.1.3 Properties of the general two-sided CFT
331(3)
11.1.4 Fourier transforms in quaternions
334(6)
11.1.5 Clifford Fourier-Mellin transform
340(1)
11.1.6 Clifford-Fourier transforms with pseudoscalar square roots of -1
341(2)
11.1.7 Spacetime Fourier transform
343(2)
11.1.8 Summary
345(1)
11.2 Fractional Fourier transform
346(5)
11.2.1 Exponentials of the Dirac operator
346(2)
11.2.2 Fourier transform of fractional order
348(3)
11.3 Radon transforms
351(10)
11.3.1 A basic problem
351(1)
11.3.2 At the very beginning
351(1)
11.3.3 Passing to higher dimensions
352(1)
11.3.4 Derivatives
352(1)
11.3.5 Relation to the Fourier transform
353(1)
11.3.6 Radon transform and spherical means
354(1)
11.3.7 Inversion formula for radial functions
355(1)
11.3.8 Relation to the Hilbert transform
356(1)
11.3.9 Radon transform on SO(3)
356(5)
Bibliography 361(22)
Index 383
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. 
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