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Commutative quaternions, spectral analysis and boundary value problems

We bring together commutative quaternions, functions of two complex variables and spectral analysis to: (i) introduce some novel formulae for commutative quaternions; (ii) present a new application of this theory, namely the solution of boundary value problems. We first consider functions of two complex variables and derive an analogue of the wellknown Dbar formula appearing in complex analysis. We then focus on the subset of holomorphic functions to prove the fundamental theorem of calculus. Finally, we use this theorem to solve boundary value problems for a quaternionic generalization of the Laplace equation. The relevant domains are Cartesian products of convex polygons and the solution is obtained by spectral analysis in analogy with the analysis of two-dimensional problems, see e.g. [A.S. Fokas and A.A. Kapaev, On a transform approach for the Laplace equation in a polygon, IMA J Appl. Math 68 (2003), p. 355, A.S. Fokas and D.A. Pinotsis, The Dbar formalism for certain non homogeneous linear elliptic equations in two dimensions, Eur. J. Appl. Math. 17(3) (2006), pp. 323–346]. These results could provide a first step towards the construction of nonlinear integrable equations using commutative quaternions.