By Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig
This publication provides purposes of hypercomplex research to boundary worth and initial-boundary price difficulties from quite a few components of mathematical physics. on condition that quaternion and Clifford research supply normal and clever how one can input into larger dimensions, it begins with quaternion and Clifford types of complicated 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 diverse decompositions of quaternion Hilbert areas are proved. eventually, hypercomplex Fourier transforms are studied in detail.
All this can be then utilized to first-order partial differential equations comparable to the Maxwell equations, the Carleman-Bers-Vekua method, the Schrödinger equation, and the Beltrami equation. The higher-order equations commence with Riccati-type equations. extra themes comprise 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 3-dimensional Kolosov-Mushkelishvili formulation in linear elasticity.
Throughout the e-book the authors recreation to give historic references and critical personalities. The ebook is meant for a large viewers within the mathematical and engineering sciences and is out there to readers with a simple seize of genuine, complicated, and useful analysis.
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Extra resources for Application of Holomorphic Functions in Two and Higher Dimensions
2 c) If we are in H and if we choose n = 3, we have formally the same polynomials with the variables z1 , z2 , z3 as in C (3), but the values of the polynomials lie always in H, which is not true in C (3). We then have P(1,0,0) (x) = z1 , P(0,1,0) (x) = z2 , P(0,0,1) (x) = z3 and for k = 2 polynomials similar to those in b) P(2,0,0) (x) = z12 , P(1,1,0) (x) = 1 (z1 z2 + z2 z1 ), . .. 2 The polynomials are much more complicated for larger k. We need not only polynomials for series expansions but we need also singular holomorphic functions to be able to express the coeﬃcients of an expansion by means of an integral formula.
5 (Nabla). We call the vector operator ∇ := (∂0 , ∂1 , . . , ∂n )T Nabla, it corresponds to our Cauchy-Riemann operator ∂ in the algebra. For a multiindex k = (k0 , . . , kn ) one deﬁnes the symbol ∇k := ∂0k0 ∂1k1 . . ∂nkn . 32 Chapter 1. Basic properties of holomorphic functions If we take the scalar product of the vector ∇ with itself we get n Δ = ∇ · ∇ = ∂∂ = ∂i2 . i=0 A solution of the equation Δf = 0 is called a harmonic function, so the coordinate functions of a holomorphic function are harmonic, just as in the plane.
The symmetrization compensates in some sense for the non-commutativity. 1 (Fueter polynomials). Let x be in C, H, or Rn+1 . (i) We call k := (k0 , . . , kn ) with integers ki a multiindex; for multiindices with non-negative coordinates deﬁne n k := |k| := n ki , k! := i=0 ki !. i=0 We call k = |k| the degree of the multiindex k. (ii) For a multiindex k with at least one negative coordinate we deﬁne Pk (x) := 0. If all coordinates ki = 0 we abbreviate k = (0, . . , 0) = 0 and deﬁne P0 (x) := 1.