A New Cauchy Type Integral Formula for Quaternionic k-hypermonogenic Functions

Abstract

In complex function theory holomorphic functions are conjugate gradient of real harmonic functions. We may build function theories in higher dimensions based on this idea if we generalize harmonic functions and define the conjugate gradient operator. We study this type of function theory in $\mathbb{R}^3$ connected to harmonic functions with respect to the Laplace–Beltrami operator of the Riemannian metric $ds^{2} = x_2^{-2k} \left(\sum \nolimits ^{2}_{i=0} dx^{2}_{i}\right)$. The domain of the definition of our functions is in $\mathbb{R}^3$ and the image space is the associative algebra of quaternions $\mathbb{H}$ generated by 1, e1, e2 and e12 = e1e2 satisfying the relation e i e j + e j e i = –2δ _ij , i, j = 1, 2. The complex field $\mathbb{C}$ is identified by the set $\{ x0 + x1e1 \, | \, x0,x1 \, \varepsilon \, \mathbb{B} \}$. The conjugate gradient is defined in terms of modified Dirac operator, introduced by $M_{k}f = Df + kx_{2}^{-1}\overline{Qf}$, where Qf is given by the decomposition f (x) = Pf (x) + Qf (x) e2 with Pf (x) and Qf (x) in $\mathbb{C}$ and $\overline{Qf}$ is the usual complex conjugation.

Leutwiler noticed around 1990 that if the usual Euclidean metric is changed to the hyperbolic metric of the Poincaré upper half-space model (k = 1), then the power function (x0 + x1e1 + x2e2)ⁿ, calculated using quaternions, is the conjugate gradient of the a hyperbolic harmonic function. We study functions, called k-hypermonogenic, satisfying M _k f = 0. Monogenic functions are 0-hypermonogenic. Moreover, 1-hypermonogenic functions are hypermonogenic defined by H. Leutwiler and the first author.

We prove a new Cauchy type integral formulas for k-hypermonogenic functions where the kernels are calculated using the hyperbolic distance and are k-hypermonogenic functions. This formula gives the known formulas in case of monogenic and hypermonogenic functions. It also produces new Cauchy and Teodorescu type integral operators investigated in the future research.