By Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig
This e-book provides purposes of hypercomplex research to boundary worth and initial-boundary worth difficulties from quite a few parts of mathematical physics. on condition that quaternion and Clifford research supply usual and clever how one can input into better dimensions, it starts off with quaternion and Clifford types of advanced functionality thought together with sequence expansions with Appell polynomials, in addition to Taylor and Laurent sequence. a number of important functionality areas are brought, and an operator calculus in response to differences 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 similar to the Maxwell equations, the Carleman-Bers-Vekua procedure, the Schrödinger equation, and the Beltrami equation. The higher-order equations commence with Riccati-type equations. additional themes comprise spatial fluid stream difficulties, picture and multi-channel processing, snapshot 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 e-book the authors pastime to provide historic references and significant personalities. The publication is meant for a large viewers within the mathematical and engineering sciences and is out there to readers with a easy seize of actual, complicated, and sensible analysis.
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Additional resources for Application of Holomorphic Functions in Two and Higher Dimensions
13 (Schwarz integral formula). Let f = u + iv be a function holomorphic in Br ⊂ C and continuous in Br . Then for z ∈ Br we have f (z) = 1 2πi ∂Br dζ ζ +z u(ζ) + i Imf (0). ζ −z ζ An analogous formula in terms of the imaginary part of f holds as well: f (z) = 1 2π ∂Br dζ ζ +z v(ζ) + Re f (0). ζ −z ζ For the proof we refer again to . 14 (Poisson integral formula). Let u be harmonic in Bρ ⊂ C and continuous in Bρ . With z = r(cos ϕ + i sin ϕ) and ζ = ρ(cos θ + i sin θ) we then have 2π 1 u(z) = 2π 0 ρ2 − r2 u(ζ)dθ, ρ2 + r2 − 2rρ cos (ϕ − θ) z ∈ Bρ .
X|→∞ f1 is called the Taylor part and f2 is called the principal part of the Laurent series. 5. 18. Let f ∈ L2 (B3 ; H; H) ∩ ker ∂. ∞ n Aln tn,l , f := with tn,l = n=0 l=0 1 ¯ l n−l ∂ ∂ f (x) n! C 0 x=0 , ¯ The operators is called the generalized Taylor-type series in L2 (B3 ; H; H) ∩ ker ∂. 0 0 ¯ ∂0 and ∂C are identiﬁed with the identity operators. There is also a formal argument for calling the obtained series expansion a Taylor type series. By deﬁnition, the coeﬃcients are determined by the derivatives ∂¯Cl ∂0n−l f (x) x=0 .
We therefore obtain the following result, which in particular is true for quaternions [145, 155]. 14 (Orthogonality of two ± split parts). Let G(p, q) ∼ = M (2d, C) or M (d, H) or M (d, H2 ), or let both f, g be blades in G(p, q) ∼ = M (2d, R) or M (2d, R2 ). 13) with respect to two square roots f, g of −1 we get zero for the scalar part of the mixed products Sc(x+ y− ) = 0, Sc(x− y+ ) = 0. 1. Number systems 15 We also have for all real Cliﬀord algebras G(p, q) the following general identity for exponential factors : eαf x± eβg = x± e(β∓α)g = e(α∓β)f x± for all α, β ∈ R, where f, g ∈ G(p, q), f 2 = g 2 = −1.