Curvilinear integral theorem for $G$-monogenic mappings in the algebra of complex quaternion

For $G$-monogenic mappings taking values in the algebra of complex quaternion we prove a curvilinear analogue of the Cauchy integral theorem in the case where a curve of integration lies on the boundary of a domain.


Introduction
Let H(C) be the quaternion algebra over the field of complex numbers C, whose basis consists of the unit 1 of the algebra and of the elements I, J, K satisfying the multiplication rules: The unit of the algebra can be decomposed as 1 = e 1 + e 2 .
Let us consider the vectors i 1 = e 1 + e 2 , i 2 = a 1 e 1 + a 2 e 2 , i 3 = b 1 e 1 + b 2 e 2 , a k , b k ∈ C, k = 1, 2, which are linearly independent over the field of real numbers R. It means that the equality α 1 i 1 + α 2 i 2 + α 3 i 3 = 0 for α 1 , α 2 , α 3 ∈ R holds if and only if In the algebra H(C) we consider the linear span E 3 := {ζ = xi 1 +yi 2 +zi 3 : x, y, z ∈ R} generated by the vectors i 1 , i 2 , i 3 over the field R. A set S ⊂ R 3 is associated with the set S ζ := {ζ = xi 1 + yi 2 + zi 3 : (x, y, z) ∈ S} in E 3 . We also note that a topological property of a set S ζ in E 3 understand as the same topological property of the set S in R 3 . For example, we will say that a curve γ ζ ⊂ E 3 is homotopic to a point if γ ⊂ R 3 is homotopic to a point, etc.
We say (see [2]) that a continuous mapping Φ : Ω ζ → H(C) or Φ : Ω ζ → H(C) is right-G-monogenic or left-G-monogenic in a domain Ω ζ ⊂ E 3 , if Φ or Φ is differentiable in the sense of the Gâteaux at every point of Ω ζ , i. e. for every ζ ∈ Ω ζ there exists an The Cauchy integral theorems for holomorphic functions of the complex variable are fundamental results of the classical complex analysis. Analogues of these results are also important tools in the quaternionic analysis.
In the paper [3] were established some analogues of classical integral theorems of the theory of analytic functions of the complex variable: the surface and curvilinear Cauchy integral theorems and the Cauchy integral formula. The Morera theorem was proved in the paper [4]. Taylor's and Laurent's expansions of G-monogenic mappings are obtained in [5].
Namely, in the paper [3] was proved a curvilinear analogue of the Cauchy integral theorem in the case where a curve of integration lies in a domain of G-monogeneity.
In the present paper we prove the curvilinear Cauchy integral theorem for G-monogenic mappings in the case where a curve of integration lies on the boundary of a domain of G-monogeneity.

The main result
Let γ be a Jordan rectifiable curve in R 3 . For a continuous mapping Ψ : γ ζ → H(C) of the form where (x, y, z) ∈ γ and U k : γ → R, V k : γ → R, we define integrals along a Jordan rectifiable curve γ ζ by the equalities where dζ := dxi 1 + dyi 2 + dzi 3 . In the paper [3] for right-G-monogenic mappings was obtained the following analogue of the Cauchy integral theorem.
Theorem 1 [3]. Let Φ : Ω ζ → H(C) be a right-G-monogenic mapping in a domain Ω ζ . Then for every closed Jordan rectifiable curve γ ζ homotopic to a point in Ω ζ , the following equality is true: Below we establish sufficient conditions for the curve γ ζ lying on the boundary ∂Ω ζ of a domain Ω ζ such that the equality (2) holds. For this goal we apply a scheme of the paper [6] for G-monogenic mappings.
Let on a boundary ∂Ω ζ of the domain Ω ζ given closed Jordan rectifiable curve γ ζ ≡ γ ζ (t), where 0 ≤ t ≤ 1, homotopic to an interior point ζ 0 ∈ Ω ζ . It means that there exists the mapping H(s, t) continuous on the square As in the paper [4], for the element ζ = xi 1 + yi 2 + zi 3 we define the Euclidian norm Using the Theorem of equivalents of norms, for the element a := 4 k=1 (a 1k + ia 2k )e k , a 1k , a 2k ∈ R, we have the following inequalities where c is a positive constant does not dependent on a.
Now passing to the limit in the last inequality as ε → 0, we obtain the equality (2). The Theorem is proved.
The similar statement is true for the left-G-monogenic mappings.
Theorem 3. Suppose that Φ : Ω ζ → H(C) is a continuous mapping in the closure Ω ζ of a domain Ω ζ and left-G-monogenic in Ω ζ . Suppose also that γ ζ ⊂ ∂Ω ζ is a closed Jordan rectifiable curve homotopic to an interior point ζ 0 ∈ Ω ζ , the curves of the family {Γ t ζ : 0 ≤ t ≤ 1} are rectifiable and the set {mes γ s ζ : 0 ≤ s ≤ 1} is bounded, then the following equality is true: