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By Sha Huang
Clifford research, a department of arithmetic that has been constructed given that approximately 1970, has very important theoretical price and several other purposes. during this e-book, the authors introduce many houses of normal features and generalized standard services in actual Clifford research, in addition to harmonic features in advanced Clifford research. It covers vital advancements in dealing with the incommutativity of multiplication in Clifford algebra, the definitions and computations of high-order singular integrals, boundary worth difficulties, etc. additionally, the booklet considers harmonic research and boundary worth difficulties in 4 different types of attribute fields proposed by means of Luogeng Hua for complicated research of numerous variables. the nice majority of the contents originate within the authors’ investigations, and this new monograph might be attention-grabbing for researchers learning the idea of services.
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Additional resources for Real and Complex Clifford Analysis (Advances in Complex Analysis and Its Applications)
1. Hypercomplex Diﬀerential Let m λk ek , (m ≤ n, λk are real numbers) λ= k=1 General Regular and Harmonic Functions 25 be a hypercomplex number and the set of all λ make up a space Am (R). , zm ). Denote by H m−1 the set of all − z , and dem−1 note the topological product of m − 1 spaces An (R) by An (R), then → → (R). For any − z ∈ H m−1 deﬁne the norm of − z by H m−1 ⊂ Am−1 n 1 − → → → z = (− z ,− z )2 = ( m 1 z k zk ) 2 = (mx21 + k=2 → − → → − − where the inner product (→ z , ξ ) = ( ξ ,− z)= m 1 x2k ) 2 , k=2 m k=2 z k ξk .
It is clear that F (x) is regular in Ω and is a solution of the corresponding homogeneous equation of Problem D∗ and F (1) |∂Ω = uB (ξ)eB − uB (ξ)eB = 0. For convenience, we shall adopt the same B B symbols for f (x) as before, namely denote F (x) = B IB eB + B IB eB . Since F (x) is regular in Ω, thus it is harmonic in Ω, therefore for all B, IB (x) are all harmonic in Ω. Since IB eB |∂Ω = F (1) |∂Ω = 0, IB |∂Ω = 0, B again by using the uniqueness of the solution of the Dirichlet problem for harmonic functions in a ball (see 1)), we get IB ≡ 0 in Ω, thus J1 F ≡ 0, ImB ≡ 0 in Ω.
1, we denote A by A and the A is called the A-type index. e. (−1) 2 = −1, we denote A by A and the A is called the A-type index. 6 Let Ω ⊂ Rn be a bounded domain, and w(x) = (r) wA (x)eA ∈ FΩ (r ≥ 1). 1) then w(x) is called a generalized regular function in Ω, where a(x) = aC (x)eC , b(x) = C bC (x)eC , l(x) = C lA (x)eA . 2, at the same time letting w = wM eM + M wM eM , M we have aw+bw+l = ( aC eC )( C bC eC )( +( C = C,A − wM eM + M M wM eM − aC wM δCM eA + C,A + C,A wM eM ) M C,A bC wM δCM eA + wM eM ) + M aC wM δCM eA + lA eA = A (aC − bC )wM δCM eA + lA eA A C,A bC wM δCM eA (aC +bC )wM δCM eA lA eA , A C,A 22 Chapter I where CM = A, CM = A.
Real and Complex Clifford Analysis (Advances in Complex Analysis and Its Applications) by Sha Huang