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Hypercomplex Analysis and Applications

:	Frank Sommen, Irene Sabadini
:	Irene Sabadini, Franciscus Sommen
:	Hypercomplex Analysis and Applications
:	Birkhäuser Basel
:	9783034602464
:	1
:	CHF 85.10
:

:	Analysis
:	English

:	284
:	Wasserzeichen
:	PC/MAC/eReader/Tablet
:	PDF

The purpose of the volume is to bring forward recent trends of research in hypercomplex analysis. The list of contributors includes first rate mathematicians and young researchers working on several different aspects in quaternionic and Clifford analysis. Besides original research papers, there are papers providing the state-of-the-art of a specific topic, sometimes containing interdisciplinary fields. The intended audience includes researchers, PhD students, postgraduate students who are interested in the field and in possible connection between hypercomplex analysis and other disciplines, including mathematical analysis, mathematical physics, algebra.

	Hypercomplex Analysis and Applications	3
	Contents	5
	Preface	7
	On the Geometry of the Quaternionic Unit Disc	9
	1. Introduction	9
	2. Basics of quaternionic invariant geometry	10
	3. Poincaré and Kobayashi distances on the quaternionic unit disc	14
	References	17
	Bounded Perturbations of the Resolvent Operators Associated to the F-Spectrum	20
	1. Introduction	20
	2. Preliminary material	22
	3. Examples of equations for the F-spectrum	25
	4. Bounded perturbations of the SC-resolvent	27
	5. Bounded perturbations of the F-resolvent	30
	References	34
	Harmonic and Monogenic Functions in Superspace	36
	1. Introduction	36
	2. Preliminaries	37
	3. Monogenic functions theory in superspace	40
	4. Basis for the space of symplectic harmonics	43
	References	48
	A Hyperbolic Interpretation of Cauchy-Type Kernels in Hyperbolic Function Theory	49
	1. Introduction	49
	2. Clifford Numbers	52
	3. On the Poincaré Upper-Half Space	53
	4. On Hyperbolic Function Theory	55
	5. Hyperbolic Interpretations of the P- and Q-kernels	58
	6. The Mean-Value Theorem for the P-Part of a Hypermonogenic Function	62
	References	63
	Gyrogroups in Projective Hyperbolic Clifford Analysis	66
	1. Introduction	66
	2. Gyrogroups	69
	3. The projective hyperbolic space model	70
	4. The Möbius gyrogroup (Bn1 ,.M)	74
	5. The Einstein gyrogroup (Bn1 ,.E)	76
	6. The proper velocity gyrogroup (Rn,.U)	77
	7. Relation between different velocities	77
	8. Gyrovector spaces	79
	9. Gyrovector space isomorphims	79
	10. Möbius, Einstein and proper gyrations as spin representation of the group Spin(n)	83
	References	84
	Invariant Operators of First Order Generalizing the Dirac Operator in 2 Variables	86
	1. Introduction	86
	1.1. Invariant differential operators	86
	1.2. Dirac operator in k variables	87
	1.3. Verma modules and invariant operators in parabolic geometry	88
	2. Invariant operators acting between higher spin modules	92
	2.1. Classification of first order operator on G/P in terms of weights	92
	2.2. Explicit realizations in simple cases	93
	References	97
	The Zero Sets of Slice Regular Functions and the Open Mapping Theorem	99
	1. Introduction	99
	2. Preliminary results	103
	3. Algebraic properties of the zero set	105
	4. Topological properties of the zero set	106
	5. The Maximum and Minimum Modulus Principles	106
	6. The Open Mapping Theorem	108
	References	110
	A New Approach to Slice Regularity on Real Algebras	112
	1. Introduction	112
	2. The quadratic cone of a real alternative algebra	114
	3. Slice functions	117
	4. Slice regular functions	119
	5. Product of slice functions	120
	6. Zeros of slice functions	121
	7. Examples	123
	References	124
	On the Incompressible Viscous Stationary MHD Equations and Explicit Solution Formulas for Some Three-dimensional Radially Symmetric Domains	127
	1. Introduction	127
	2. Preliminaries	129
	2.1. The quaternionic operator calculus	129
	3. The incompressible stationary MHD equations revisited in the quaternionic calculus	132
	4. The highly viscous case	133
	5. Outlook for the non-linear case	136
	Acknowledgements	137
	References	137
	The Fischer Decomposition for the H-action and Its Applications	140
	1. Introduction	140
	2. The Fischer Decomposition for the H-action	141
	3. Special Monogenic Polynomials	144
	4. Inframonogenic Polynomials	146
	Acknowledgment	148
	References	148
	Bochner’s Formulae for Dunkl-Harmonics and Dunkl-Monogenics	150
	1. Introduction	150
	2. Clifford Analysis and Dunkl Analysis	151
	3. Bochner’s Formula for Dunkl-Harmonics	153
	4. Bochner’s Formula for Dunkl-Monogenics	157
	References	159
	An Invitation to Split Quaternionic Analysis	161
	1. Introduction	161
	2. The Quaternionic Spaces HC, HR and M	164
	3. Regular Functions on H and HC	168
	4. Regular Functions on HR	169
	5. Fueter Formula for Holomorphic Regular Functions on HR	170
	6. Fueter Formula for Regular Functions on HR	173
	7. Separation of the Series for SL(2,R)	177
	References	179
	On the Hyperderivatives of Moisil–Théodoresco Hyperholomorphic Functions	181
	1. Introduction	181
	2. The left-i-hyperderivative	185
	3. The directional left