A New Cauchy Type Integral Formula for Quaternionic k-hypermonogenic Functions
Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Scientific › peer-review
|Title of host publication||Modern Trends in Hypercomplex Analysis|
|Editors||Swanhild Bernstein, Uwe Kähler, Irene Sabadini, Franciscus Sommen|
|Publisher||Springer International Publishing|
|Number of pages||15|
|Publication status||Published - 23 Nov 2016|
|Publication type||A4 Article in a conference publication|
|Event||ISAAC Congress - |
Duration: 1 Jan 1900 → …
|Name||Trends in Mathematics|
|Period||1/01/00 → …|
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.