Motor Variable - La Plata Lessons

La Plata Lessons

At the National University of La Plata in 1935, J.C. Vignaux, an expert in convergence of infinite series, contributed four articles on the motor variable to the university’s annual periodical. He is the sole author of the introductory one, and consulted with his department head A. Durañona y Vedia on the others. In "Sobre las series de numeros complejos hiperbolicos" he says (p. 123):

This system of hyperbolic complex numbers is the direct sum of two fields isomorphic to the field of real numbers; this property permits explication of the theory of series and of functions of the hyperbolic complex variable through the use of properties of the field of real numbers.

He then proceeds, for example, to generalize theorems due to Cauchy, Abel, Mertens, and Hardy to the domain of the motor variable.

In the primary article, cited below, he considers D-holomorphic functions, and the satisfaction of d’Alembert’s equation by their components. He calls a rectangle with sides parallel to the diagonals y = x and y = − x, an isotropic rectangle.He concludes his abstract with these words:

Isotropic rectangles play a fundamental role in this theory since they form the domains of existence for holomorphic functions, domains of convergence of power series, and domains of convergence of functional series.

Vignaux completed his series with a six page note on the approximation of D-holomorphic functions in a unit isotropic rectangle by Bernstein polynomials.While there are some typographical errors as well as a couple of technical stumbles in this series, Vignaux succeeded in laying out the main lines of the theory that lies between real and ordinary complex analysis. The text is especially impressive as an instructive document for students and teachers due to its exemplary development from elements. Furthermore, the entire excursion is rooted in “its relation to Émile Borel’s geometry” so as to underwrite its motivation.