By Sha Huang

Clifford research, a department of arithmetic that has been built given that approximately 1970, has vital theoretical price and a number of other functions. during this booklet, the authors introduce many houses of normal services and generalized standard features in genuine Clifford research, in addition to harmonic capabilities in complicated 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 price difficulties, and so forth. additionally, the ebook considers harmonic research and boundary price difficulties in 4 sorts of attribute fields proposed by way of Luogeng Hua for advanced research of numerous variables. the nice majority of the contents originate within the authors’ investigations, and this new monograph may be attention-grabbing for researchers learning the idea of capabilities.

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**Example text**

7), we get (a23 − b23 )xj = 0, 1 ≤ j ≤ 3, namely (a23 − b23 ) is a constant. e. (a3 − b3 ) only depends on x3 . 8) δjD lAxj , in which CM = A, CM = A, jD = A. Suppose that D is the A-type index. 8), if the equality jM = jM holds, then D = M is the A-type index. This is a contradiction. 8) disappears. 9) n j=1 δjD lAxj , in which CM = A, CM = A, jD = A. 9) possesses the form ∆ω1 = 3 j=1 (aj + bj )ω1xj + 3 j=1 (aj + bj )xj ω1 + (a1 − b1 )ω2x2 +(a1 − b1 )ω3x3 − (a2 − b2 )ω2x2 − (a2 − b2 )ω23x3 −(a3 − b3 )ω3x3 + (a3 − b3 )ω23x2 − (a23 − b23 )ω23x1 −(a23 − b23 )ω3x2 + (a23 − b23 )ω2x3 + 3 j=1 lj xj .

Then f = f (− z ) is said to n General Regular and Harmonic Functions → be A diﬀerential at the point − a ; if there exists p−1 m−1 l ∈ L(An (R), An (R)) such that →− − →→ − → − → f (→ a + − z ) − f (− a)− l ( lim → − → − z z →0 → − →→ − then l = l (− z ) is called the left diﬀerential, −−→ − → −−→ − − → (f )( a ) or l = (f )(→ a ). 5 Let f : Ap−1 f (→ a ) be given. n (R) → An →− − → → − Then f ( z ) is continuous at a . 6 (see [72]1)) If f : Ap−1 n (R) → An →− − a ) is unique. 7 Let f : Ap−1 n (R) → An →− − m−1 → and g : An (R) → An (R) be An (R) diﬀerentiable at f ( a ).

Then f = f (− z ) is said to n General Regular and Harmonic Functions → be A diﬀerential at the point − a ; if there exists p−1 m−1 l ∈ L(An (R), An (R)) such that →− − →→ − → − → f (→ a + − z ) − f (− a)− l ( lim → − → − z z →0 → − →→ − then l = l (− z ) is called the left diﬀerential, −−→ − → −−→ − − → (f )( a ) or l = (f )(→ a ). 5 Let f : Ap−1 f (→ a ) be given. n (R) → An →− − → → − Then f ( z ) is continuous at a . 6 (see [72]1)) If f : Ap−1 n (R) → An →− − a ) is unique. 7 Let f : Ap−1 n (R) → An →− − m−1 → and g : An (R) → An (R) be An (R) diﬀerentiable at f ( a ).