By Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig

ISBN-10: 303480962X

ISBN-13: 9783034809627

ISBN-10: 3034809646

ISBN-13: 9783034809641

This ebook provides 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 supply ordinary and clever how you can input into greater dimensions, it begins with quaternion and Clifford types of complicated functionality conception together with sequence expansions with Appell polynomials, in addition to Taylor and Laurent sequence. a number of priceless functionality areas are brought, and an operator calculus in response to changes of the Dirac, Cauchy-Fueter, and Teodorescu operators and various decompositions of quaternion Hilbert areas are proved. ultimately, hypercomplex Fourier transforms are studied in detail.

All this is often then utilized to first-order partial differential equations reminiscent of the Maxwell equations, the Carleman-Bers-Vekua approach, the Schrödinger equation, and the Beltrami equation. The higher-order equations commence with Riccati-type equations. extra subject matters contain spatial fluid movement difficulties, picture and multi-channel processing, picture diffusion, linear scale invariant filtering, and others. one of many highlights is the derivation of the third-dimensional Kolosov-Mushkelishvili formulation in linear elasticity.

Throughout the publication the authors activity to offer historic references and critical personalities. The booklet is meant for a large viewers within the mathematical and engineering sciences and is obtainable to readers with a easy seize of actual, advanced, and sensible analysis.

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

**Example text**

As the dual space to the Hardy space H 1 (with the operator norm), the space BMO is complete. It is possible to introduce a semi-norm u := sup E⊂Sn−1 1 m(E) |u − uE ||dσx |. E Obviously, from u = 0 does only follow u = const. , BMO is a Banach space of function classes. Furthermore, we need also quaternion-valued spaces of Sobolev type, which are introduced in the following way: Let be k = 0, 1, . . and p ≥ 1, then W p,k (G) = {u ∈ Lp (G) : ∇s u ∈ Lp (G), |s| ≤ k}, equipped with the norm u p,k ∇s u p .

N + 1. An important property is that the application of ∂ to the orthogonal system of solid spherical harmonics preserves the orthogonality. This was proved in [52]. 10. 1). The scalar parts of the (R)-holomorphic spherical polynomials are again harmonic and must have a representation in terms of the original solid spherical harmonics. Surprisingly, this representation is very simple and convenient: (n + l + 1) l,† Un , 2 (n + m + 1) m,† Vn . , their holomorphic derivative vanishes. It should be emphasized that the set of constants in the case of the Riesz system is richer than in complex analysis.

We cite another theorem of this type, in which we use the Dirac operator in H, 3 D= ej ∂ j , j=1 which operates only on the variables x1 , x2 , x3 . 6. Let G ⊂ R3 . A function f ∈ C 1 (G) is left-monogenic in G if and only if 1 d(dx ∧ dxf ) = dx∗ (Df ). 2 Analogously, a function f is right-monogenic in G if and only if 1 d(f dx ∧ dx) = (f D)dx∗ . 2 Again the proof may be found in [118]. The nice properties of this type of holomorphic functions have to be studied now. 2 Construction of holomorphic functions A harmonic function is a solution of the Laplace equation Δu = ∂∂u = 0.