We proved a theorem about integral of quaternionic-differentiable

functions of spatial variable over the closed surface. It is an

analog of the Cauchy theorem from complex analysis

Герус, О. Ф.

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Герус О. Ф.

Zhytomyr State University Library

On hyperholomorphic functions of spatial variableO. F. GerusIvan Franko Zhytomyr State University, Ukraineolgerus@yahoo.comLet H(C) be the algebra of complex quaternions∑3k=0 akik, where {ak}3k=0 arecomplex numbers, i0 = 1 be the unit, i1, i2, i3 be the quaternion units, i.e. theysatisfy the multiplicative rule i21 = i22 = i23 = i1i2i3 = −1. Let z :=∑3k=1 zkik bea point of Euclidean space R3 with the basic set {ik}3k=1 and let Ω be a domainin R3.Definition. Function f := ∑3k=0 fkik, where fk : Ω → C, is called left-hyperholomorphic or right-hyperholomorphic if its components {fk}3k=0 are R3-differentiable functions and the condition∑3k=1 ik∂f∂zk= 0 or∑3k=1∂f∂zkik = 0respectively holds true in the domain Ω.In known formerly publications (see e. g. [1]) similar definitions included themore strong condition on components of function f to have continuous partialderivatives.By Γz,δ denote the set of points ζ contained in Γ such that |ζ − z| ≤ δ.The next theorem (see [2]) is a quaternion analog of the Cauchy theorem fromcomplex analysis.Theorem. Let Ω be a bounded domain with the piece-wise smooth boundaryΓ such that for all points z from R3 and for all δ > 0 the diameter of the set Γz,δdivided by its square measure is bounded by a positive constant. Let functionf : Ω → H(C) be continuous in Ω and right-hyperholomorphic in Ω and letfunction g : Ω → H(C) be continuous in Ω and left-hyperholomorphic in Ω.Then ∫∫Γf(z) ν(z) g(z) ds = 0,where ν(z) :=∑3k=1 νk(z)ik is the unit normal vector to the surface Γ.References1. Blaya R. A., Reyes J. B., Shapiro M. On the Laplasian vector fields theory in domainswith rectifiable boundary. Mathematical Methods in the Applied Sciences. 2006; 29: 1861 1881.2. Gerus O. F. On hyperholomorphic functions of spatial variable. Ukrain. Mat. Zh. 2011;63(4): 459  465 (Russian).

On hyperholomorphic functions of spatial variable

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