Motor variable

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The function of a motor variable is a concept developed in Germany, Argentina, and Russia (see references).It presumes knowledge of the plane of split-complex numbers which form the motor plane D, the term initiated by William Kingdon Clifford.Let

f(z) = u(z) + j v(z) , z = x + j y , x, y ∈ R , j² = +1 , u(z), v(z) ∈ R.

The function f is called D-holomorphic when

0 = (∂/∂x − j ∂/∂y) ( u + j v)

= u_x − j² v_y + j (v_x − u_y ).

The consequent partial differential equations are called "Scheffer's conditions" by Yaglom who credits George Scheffer's work of 1893. See Duren's text for the use of similar differential operators to establish the relation of harmonic function theory to analytic functions on the ordinary complex plane C .It is apparent that the components u and v of a D-holomorphic function f satisfy the wave equation, associated with D'Alembert, whereas components of C-holomorphic functions satisfy Laplace's equation.

[edit] Elementary functions of a motor variable

• f(z) = uz + c , u = exp(aj) = cosh a + j sinh a . This affine transformation of D

amounts to an hyperbolic rotation combined with a translation by c.

• Let U₁ = {z ∈ D : |y| < x }. f(z) = z² , f: D → U₁ since f ({1, -1, j, -j }) = {1} .
• f(z) = 1/z = z*/|z|² where |z|² = z z* . Note that z z* = 1 forms the hyperbola x² − y² = 1.Thus this

reciprocation involves the hyperbola as reference unit as opposed to the circle in C.

• f(z) = (az + b) / (cz + d) These analogues of Mobius transformations are encountered in inversive ring geometry.
• Rectangle T = {z = x + j y : |y| < x < 1 or |y| < x − 1 when 1<x<2}. Then

f(z) = 1/ (z + 1/2), f: U₁ → T . Since T is bounded this mapping can compactify the open and unbounded U₁.(This service is similar to the Cayley transform that takes the half-plane in C to the unit disk.)

[edit] La Plata lessons

At La Plata in 1935, J.C. Vignaux, an expert in convergence of infinite series, contributed four articles on the motor variable to his 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 [motor variables] 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 Emile Borel’s geometry” so as to underwrite its motivation.

[edit] References

• Duren, Peter (2004) Harmonic Mappings in the Plane, pp. 3,4, Cambridge U. Press.
• Scheffers, George (1893) "Verallgemeinerung der Grundlagen der gewohnlichen komplexen Funktionen", Sitzungsberichte Sachs. Ges. Wiss, Math-phys Klasse Bd 45 S. 828-42.
• Vignaux, J.C. & A. Durañona y Vedia (1935) "Sobre la teoria de las functiones de una variable compleja hiperbolica", Contribucion al Estudio de las Ciencias Fisicas y Matematicas, pp. 139-184, Universidad Nacional de La Plata, Republica Argentina.
• Yaglom, I.M. (1988) Felix Klein & Sophus Lie, The Evolution of the Idea of Symmetry in the Nineteenth Century, Birkhauser, p. 203.