An introduction to commutative quaternions
Since the important articles by \textit{A. Scheffers} [Ber. Verh. Königl. Sächs. Gesellsch. Wiss. Leipzig, Math.-Phys. Kl. 45, 828--848 (1893); 46, 120--134 (1894; JFM 25.0667.03)] many authors have dealt with differentiable functions in commutative algebras leading to generalized Cauchy-Riemann equations. More success and more applications seem to be reached via non-commutative algebras, e.g. Clifford algebras. The authors of this paper deal with algebras they call commutative quaternions (a name enclosing a contradiction), which is an algebra with base elements \(1, i, j, k\) and the multiplication rules \(i^2=k^2=\alpha, j^2=1, ij=ji=k, ik=ki=\alpha j\) with \(\alpha = -1\) (elliptic case), \(\alpha=0\) (parabolic case), \(\alpha = 1\) (hyperbolic case). The generalized Cauchy-Riemann equations and some elementary functions are given. The mappings given by such functions are claimed to be conformal, so it would be very interesting to see which functions are in this class of differentiable functions (because of the theorem of Liouville).
scientific article; zbMATH DE number 5117978