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By Sha Huang

Clifford research, a department of arithmetic that has been built due to the fact that approximately 1970, has vital theoretical worth and several other purposes. during this publication, the authors introduce many houses of normal features and generalized general capabilities in genuine Clifford research, in addition to harmonic capabilities in advanced Clifford research. It covers very important advancements in dealing with the incommutativity of multiplication in Clifford algebra, the definitions and computations of high-order singular integrals, boundary worth difficulties, etc. furthermore, the ebook considers harmonic research and boundary worth difficulties in 4 sorts of attribute fields proposed by way of Luogeng Hua for complicated research of a number of variables. the nice majority of the contents originate within the authors’ investigations, and this new monograph could be fascinating for researchers learning the idea of capabilities.

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In order to discuss the uniqueness of its solution, we first consider the sectional domains of Ω. Cutting Ω by “the planes”: ⎧ ⎪ x3 = a3 , ⎪ ⎪ ⎪ ⎪ ⎨ x4 = a4 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ... xn = an , we obtain a sectional domain Ga in the x1 x2 plane: xa xTa = x21 + x22 + n a2m < 1. , an ). For given continuous functions B uB (ξ)eB (ξ ∈ ∂Ω), φB (ξa ) (ξa ∈ Γa ) and the complex constants dBa , we find a regular function f (x) = IB eB + IB eB in Ω, which is continuous in Ω with the following B B pseudo-modified conditions: ⎧ ⎪ J1 f (ξ) ⎪ ⎪ ⎪ ⎨ ReI | B Γa ⎪ ⎪ ⎪ ⎪ ⎩ I (O ) a B = B uB (ξ)eB , ξ ∈ ∂Ω, = φB (ξa ) + hB (ξa ), ξa ∈ Γa , = dBa , where hB (ξa ) = hBa (ξa ∈ Γa ) are all unknown real constants to be determined appropriately, and ReIB = Re(FB + F2B e2 ) = FB .

1 Denote by L(Am−1 n n (R)) the set of linear → − m−1 p−1 mappings from An (R) to An (R). A bounded mappings l ∈ L(Am−1 (R), Ap−1 n n (R)) is said to be A linear (or linear for short), if → − → − u , v ∈ Am−1 (R), λ, µ ∈ An (R); then we have n → − − →→ − →→ − → l (λ→ u + µ− v ) = λ l (− u ) + µ l (− v ). , m. Proof have Let l ∈ L(H m−1 , An (R)) be an A linear mapping. , m, and then → l(− z ) = z2 A2 + · · · + zm Am . , m. 2 (see [48]) Let − a ∈ H m−1 and f be a continuous → → z) mapping from some domain of − a to An (R).

Z1 j=2 ∂zj Noting that V ec(A) ⎫ ⎬ ⎞ ⎤ n ∂ ∂ ⎠ ∂uA (sz) ⎦ sn−3 ⎣⎝e1 − ej zeA ds ∂z1 j=2 ∂zj ∂zl 38 Chapter I is 1 0 n sn−3 zk k=1 ∂ 2 uA (sz) ds, ∂zk ∂zl we have V ec(A) 1 = 0 ⎩ 0 ⎡⎛ ⎡⎛ ⎞ ⎫ ⎤ n ⎬ ∂ ∂ ⎠ ∂uA (sz) ⎦ sn−3 ⎣⎝e1 − ej zeA ds ⎭ ∂z1 j=2 ∂zj ∂zl ⎞ ⎤ n ∂ ∂ ⎠ ∂uA (sz) ⎦ sn−3 ⎣⎝e1 − ej zeA ds ∂z1 j=2 ∂zj ∂zl 1 − ⎧ ⎨ 1 0 n sn−3 zk k=1 Because ∂ 2 uA (sz) eA ds. ∂zk zl ⎛ ⎞ n ∂ ∂ ⎠ ⎝e1 − ej uA (sz) ∂z1 j=2 ∂zj is a complex left regular function, we get n 1 − el = − 1 0 0 A l=1 ⎡⎛ sn−3 ⎣⎝e1 ⎧ ⎨ n sn−3 A ⎩ ⎡ el ⎣ l=1 ⎞ n ⎤ ∂ ∂ ⎠ ∂uA (sz) ⎦ − ej zeA ds ∂z1 j=2 ∂zj ∂zl ⎛ n ∂ ⎝ ∂ e1 − ∂zl ∂z1 j=2 ⎤⎫ ⎬ ∂ ⎠ ej uA (sz)⎦ zeA ds = 0, ⎭ ∂zj ⎞ and n el l=1 ∂F = ∂zl n − 1 = 0 1 el A l=1 V ec(A) el l=1 n A 0 ⎧ ⎨ 1 ⎩ 0 n sn−3 el ek A l,k=1 n sn−3 zk k=1 ∂ 2 uA (sz) eA ds ∂zk zl ⎡⎛ ⎞ ⎤ ⎫ n ⎬ ∂ ∂ ⎠ sn−3 ⎣⎝e1 − ej uA (sz)⎦ el eA ds ⎭ ∂z1 j=2 ∂zj ∂ 2 uA (sz) eA ds ∂zk zl General Regular and Harmonic Functions 39 ⎧⎡⎛ ⎫ ⎞ ⎤ n ⎨ ⎬ ∂ ∂ ⎠ uA (sz)⎦ el eA ds sn−3 el V ec(A) ⎣⎝e1 − ej − ⎩ ⎭ ∂z1 j=2 ∂zj 0 A l=1 n 1 1 = 0 n ∂ ∂uA (sz) zl ej eA ds ∂zj ∂zl j,l=1 sn−3 A ⎧⎡⎛ ⎫ ⎞ ⎤ n ⎨ ⎬ ∂ ∂ ⎠ uA (sz)⎦ el eA ds.

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