By Runge C.
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April 7, 2008 17:13 World Scientific Review Volume - 9in x 6in 22 H. Begehr The last rule identifies z/|z|m as the fundamental solution to the Dirac equation. For more detailed information see [10, 11]. Basic for representation formulas for functions in Clifford algebra is a version of the Gauss theorem. Because of the anticommutativity of multiplication two functions are involved here. Gauss Theorem. Let D ⊂ Rm be a regular domain and f, g ∈ C 1 (D; Cm ) ∩ C(D; Cm ). Then [(f ∂) g + f (∂g)] dv = D f dσ g , ∂G [(f ∂) g + f (∂g)] dv = f dσ g .
181 (2002), 55–71. [8] H. N. Hile, A hierarchy of integral operators. Rocky Mountain J. Math. 27 (1997), 669–706. [9] H. C. Wen, Some second order systems in the complex plane. Revue Roumaine Math. Pure Appl. 44 (1999), 521–554. [10] F. Bracks, R. Delanghe and F. Sommen, Clifford analysis. Pitman, London, 1982. [11] E. Obolashvili, Partial differential equations in Clifford analysis. Addison Wesley Longman, Harlow, 1998. [12] W. Rudin, Function theory in the unit ball of Cn . , 1980. N. Vekua, Generalized analytic functions.
Upper gradient and ACCp functions A C 1 –function u in a domain Ω of Rn satisfies l(γ) u(x) − u(y) = γ ∇u · ds = ∇u(˜ γ (s)) · γ˜ ′ (s)ds 0 where γ is any rectifiable path in Ω with endpoints x and y. This leads to |u(x) − u(y)| ≤ γ |∇u|dx. We will now see that this inequality is almost as useful as the previous equality. We extend the latter inequality to a metric space (X, d). 11) γ for each rectifiable path γ joining x and y in X. Every function has an upper gradient, namely g ≡ ∞, and upper gradients are never unique.