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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 SL(2, ℝ) and SL(2, ℂ)/SL(2, ℝ)″ joint with Igor Frenkel we naturally apply the methods and formulas of quaternionic analysis to solve the problems of harmonic analysis on SL(2,ℝ) and the imaginary Lobachevski space SL(2, ℂ)/SL(2, ℝ). [ABSTRACT FROM AUTHOR]