By Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig

ISBN-10: 303480962X

ISBN-13: 9783034809627

ISBN-10: 3034809646

ISBN-13: 9783034809641

This e-book offers purposes of hypercomplex research to boundary price and initial-boundary worth difficulties from a variety of parts of mathematical physics. provided that quaternion and Clifford research provide usual and clever how one can input into larger dimensions, it begins with quaternion and Clifford types of complicated functionality concept together with sequence expansions with Appell polynomials, in addition to Taylor and Laurent sequence. a number of helpful functionality areas are brought, and an operator calculus in keeping with adjustments of the Dirac, Cauchy-Fueter, and Teodorescu operators and diverse decompositions of quaternion Hilbert areas are proved. ultimately, hypercomplex Fourier transforms are studied in detail.

All this can be then utilized to first-order partial differential equations equivalent to the Maxwell equations, the Carleman-Bers-Vekua process, the Schrödinger equation, and the Beltrami equation. The higher-order equations commence with Riccati-type equations. additional subject matters comprise spatial fluid move difficulties, picture and multi-channel processing, snapshot diffusion, linear scale invariant filtering, and others. one of many highlights is the derivation of the three-d Kolosov-Mushkelishvili formulation in linear elasticity.

Throughout the publication the authors exercise to give ancient references and critical personalities. The e-book is meant for a large viewers within the mathematical and engineering sciences and is available to readers with a simple seize of actual, advanced, and practical analysis.

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**Additional resources for Application of Holomorphic Functions in Two and Higher Dimensions**

**Example text**

Having in mind the Cliﬀord algebras or the quaternions, a disadvantage is that this structure is not closed under the quaternionic resp. Cliﬀord multiplication. 5. Polynomial systems 35 square integrable functions over R3 . , Lam´e system [130], Stokes system [131]). Looking for homogeneous holomorphic polynomial solutions of the Riesz system as a basis for the approximation of all square integrable solutions and staying in the real vector space of functions f : G ⊂ R3 → A := spanR {1, e1 , e2 }, it makes sense to start with well-known homogeneous harmonic polynomials.

Here in contrast to the Sobolev spaces the second upper index may be a real number, not only a natural number. The norm is introduced by ( u p,k ) p =( u p,k ) p Iλ (∇s u). + |s|=[k] The space W0p,k (G) is just the closure of D(G) (space of quaternion-valued Schwartz distributions, componentwise deﬁned) using the topology of W p,k (G). The space W p,0 (G) coincides with the quaternion version of the space Lp (G). We may also deal with functions having trace zero on the boundary of the domain; ◦ the corresponding spaces are denoted by W p,k (G).

1. In exceptional cases we need Banach spaces of Cliﬀord algebravalued functions. These functions have to belong componentwise to the corresponding Banach space. Next, a bounded domain G ⊂ R3 is called a Lipschitz domain if ∂G = Γ can be locally decribed by means of graphs of Lipschitz functions. Let 1 ≤ p < ∞ and 0 < s < 1. Then the Besov space Bsp,p is deﬁned to be the set of all measurable functions u on Γ with ⎛ ⎞1/p p |u(x) − u(y)| |dσ(x)||dσ(y)|⎠ < ∞. u Bsp,p (Γ) := u Lp (Γ) + ⎝ |x − y|2+sp Γ Γ It is easy to prove that the Besov space Bs∞,∞ (Γ), 0 < s < 1, coincides with the q,q (Γ) is the classical Banach space of H¨older continuous functions.

### Application of Holomorphic Functions in Two and Higher Dimensions by Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig

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