Quaternion-valued integral representations of the harmonic electromagnetic and spinor fields (Q1363938)

Let \(\mathbb{Q}\) be a set of complex quaternions, \(a\in\mathbb{Q}\) and \(D_a: =D+M^a\), where \(D\) is the Dirac operator and \(M^af=fa\). Then the equation \(D_a f=0\) for \(\mathbb{Q}\)-valued function \(f\in C^1(G)\), \(G\subset\mathbb{R}^3\) is (an equivalent notation) the generalized Moisil-Theodoresku system, considered by \textit{V. V. Kravchenko} and \textit{M. V. Shapiro} [Dokl. Akad. Nauk Ross. Akad. Nauk, 329, No. 5, 547-549 (1993)]. The author proves that the solutions of \(D_af=0\) are connected with the harmonic electromagnetic and spinor fields. For them he obtains the Cauchy integral formula and the Sokhotskij formulas.