Summary: Six years after William Rowan Hamilton’s discovery of quaternions, in 1849, James Cockle introduced the algebra of split quaternions. (He called them “coquaternions.”) In this paper, we define regular functions on split quaternions and prove two different analogues of the Cauchy-Fueter formula for these functions. In the paper [Split quaternionic analysis and the separation of the series for \(\text{SL}(2, \mathbb R)\) and \(\text{SL}(2, \mathbb C)/ \text{SL}(2, \mathbb R)\)] joint with I. Frenkel we naturally apply the methods and formulas of quaternion analysis to solve the problems of harmonic analysis on \(\text{SL}(2,\mathbb R)\) and the imaginary Lobachevskyi space \(\text{SL}(2, \mathbb C) / \text{SL}(2, \mathbb R)\).
For the entire collection see [Zbl 1205.30003].