Monogenic Functions in a Finite-Dimensional Semi-Simple Commutative Algebra

Abstract

We obtain a constructive description of monogenic functions taking values in a finite-dimensional semi-simple commutative algebra by means of holomorphic functions of the complex variable. We prove that the mentioned monogenic functions have the Gateaux derivatives of all orders. For monogenic functions we prove also analogues of classical integral theorems of the holomorphic function theory: the Cauchy integral theorems for surface and curvilinear integrals, the Morera theorem and the Cauchy integral formula.Introduction. William Hamilton (1843) constructed an algebra of noncommutative quaternions over the field of real numbers R, and developing the hypercomplex analysis began. C. Segre [1] constructed an algebra of commutative quaternions {x + iy + jz + kt : i 2 = j 2 = −1, ij = k, x, y, z, t ∈ R} over the field R that can be considered as a two-dimensional commutative semi-simple algebra of bicomplex numbers {z 1 + jz 2 : j 2 = −1, z 1 , z 2 ∈ C} over the field of complex numbers C.A theory of functions of a bicomplex variable was developed in papers of many authors (see, e.g., [2,3,4,5,6]). In particular, in the papers of F. Ringleb [2] and J. Riley [3], it is proved that any analytic function of a bicomplex variable can be constructed with an use of two holomorphic functions of complex variables. In addition, G. Price [4] considered multicomplex

Integral theorems for the quaternionic G-monogenic mappings

“…We note that the method of this proof is similarly to the proof of Theorem 6 of the paper [8], where Cauchy integral formula is obtained in a finitedimensional semi-simple commutative algebra. If a domain Ω ⊂ R 3 has a closed piece-smooth boundary ∂Ω and a mapping Ψ : Ω ζ → H(C) is continuous together with partial derivatives of the first order up to the boundary ∂Ω ζ , then the following analogues of the Gauss -Ostrogradsky formula are true:…”

“…Now, the next theorem is a result of the formulae (7), (8) and the equalities (2), (3), respectively. Theorem 1.…”

Constructive description of monogenic functions in finite-dimensional commutative algebras

“…. , e k in A m n , where 2 ≤ k ≤ 2n , and these vectors are linearly independent over the field of real numbers R (see [22]). It means that the equality…”

“…Furthermore, constructive descriptions of monogenic functions taking values in special n -dimensional commutative algebras by means n holomorphic functions of complex variables are obtained in the papers [21,22].…”

“…The formula (32) generalizes representations of monogenic functions in both three-dimensional harmonic algebras (see [16,17,18]) and specific n -dimensional algebras (see [21,22]) to the case of algebras more general form and to a variable of more general form.…”