[Paper Review] Integral theorems for the quaternionic G-monogenic mappings

Vitalii Shpakivskyi, T. S. Kuzmenko|arXiv (Cornell University)|Dec 17, 2014

Algebraic and Geometric Analysis5 references3 citations

TL;DR

This paper establishes quaternionic analogues of classical Cauchy integral theorems for a new class of mappings called $G$-monogenic functions in the quaternion algebra over the complex numbers. Using Gateaux differentiability and a specialized basis, the authors prove a Cauchy integral formula via curvilinear integrals and surface integral theorems, generalizing Fueter-type results to a broader class of functions with applications in hypercomplex analysis.

ABSTRACT

In the paper [1] considered a new class of quaternionic mappings, so-called $G$-monogenic mappings. In this paper we prove analogues of classical integral theorems of the holomorphic function theory: the Cauchy integral theorems for surface and curvilinear integrals, and the Cauchy integral formula for $G$-monogenic mappings.

Motivation & Objective

• To extend classical integral theorems of complex analysis to the quaternionic setting for a new class of functions.
• To define and analyze $G$-monogenic mappings in the quaternion algebra over $\mathbb{C}$ using a specific basis and linear span.
• To establish analogues of the Cauchy integral theorem and formula for surface and curvilinear integrals in this generalized framework.
• To generalize prior results by Fueter and others by relaxing differentiability assumptions and introducing a new algebraic structure.

Proposed method

• Introduces a complexified quaternion algebra $\mathbb{H}(\mathbb{C})$ with basis $\{e_1, e_2, e_3, e_4\}$ and defines $G$-monogenic mappings via Gateaux differentiability.
• Represents quaternionic mappings as $\Phi(\zeta) = \xi_1 e_1 + \xi_2 e_2$ with $\xi_1, \xi_2 \in \mathbb{C}$, enabling complex-variable techniques.
• Applies generalized Gauss–Ostrogradsky formulas to relate volume and surface integrals, using differential forms $\sigma = dydz + dzdx\,i_2 + dxdy\,i_3$.
• Derives surface integral theorems by combining Gateaux derivatives with the condition $1 + i_2^2 + i_3^2 = 0$, linking to the three-dimensional Laplace equation.
• Proves the Cauchy integral formula via contour integrals in $\xi_1$ and $\xi_2$ variables, using complex residues and basis decomposition.
• Uses the assumption $f_1(E_3) = f_2(E_3) = \mathbb{C}$ to ensure full coverage of complex parameters in the representation.

Experimental results

Research questions

• RQ1 How can the classical Cauchy integral theorems be generalized to quaternionic $G$-monogenic mappings in $\mathbb{H}(\mathbb{C})$?
• RQ2 What are the necessary conditions on the differential structure and basis for extending Cauchy-type integral formulas to this class of functions?
• RQ3 Can the Cauchy integral formula be derived using curvilinear integrals in a complexified quaternion algebra framework?
• RQ4 How do the surface and volume integral theorems for $G$-monogenic mappings relate to the three-dimensional Laplace equation?
• RQ5 What role does the algebraic structure $1 + i_2^2 + i_3^2 = 0$ play in simplifying the integral identities?

Key findings

• The Cauchy integral formula for $G$-monogenic mappings is established as $\Phi(\zeta_0) = \sum_{k=1}^4 e_k \cdot \frac{1}{2\pi i} \int_{C_k} \frac{F_k(\xi_k)}{\xi_k - \xi_k^{(0)}} d\xi_k$, with $\zeta_0 = \xi_1^{(0)}e_1 + \xi_2^{(0)}e_2$, proving the formula via complex contour integration.
• The surface integral theorem is proven via the identity $\int_{\partial\Omega_\zeta} \sigma \Phi(\zeta) = \int_{\Omega_\zeta} (1 + i_2^2 + i_3^2) \Phi'(\zeta) \, dxdydz$, linking the boundary integral to the derivative.
• When $1 + i_2^2 + i_3^2 = 0$, the surface integrals vanish: $\int_{\partial\Omega_\zeta} \sigma \Phi(\zeta) = \int_{\partial\Omega_\zeta} \widehat{\Phi}(\zeta) \sigma = 0$, implying solutions to the 3D Laplace equation satisfy a vanishing integral condition.
• The generalized Gauss–Ostrogradsky formulas (15) and (16) are derived for surface integrals involving $\sigma \Psi(\zeta)$ and $\Psi(\zeta) \sigma$, using differential forms and partial derivatives.
• The proof method mirrors that of Theorem 6 in [8], adapting finite-dimensional semi-simple commutative algebra techniques to the quaternionic setting.
• The results extend classical quaternionic analysis by relaxing differentiability assumptions and introducing a new class of functions with richer algebraic structure.