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. In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real William Kingdon Clifford FRS ( May 4, 1845 &ndash March 3, 1879) was an English Mathematician and Let
- f(z) = u(z) + j v(z) , z = x + j y , x, y ∈ R , j2 = +1 , u(z), v(z) ∈ R.
Analogous to planar mapping with ordinary complex number variables, functions depending on a motor variable provide a tool for mathematical models. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Note The term model has a different meaning in Model theory, a branch of Mathematical logic.
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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. In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector Translation is the interpreting of the meaning of a text and the subsequent production of an equivalent text likewise called a " translation
- 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. Möbius transformations should not be confused with the Möbius transform or the Möbius function. In Mathematics, inversive ring geometry is the extension to the context of Associative rings of the concepts of Projective line, Homogeneous
- 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. In Mathematics, the Cayley transform, named after Arthur Cayley, has a cluster of related meanings )
D-holomorphic functions
The function f is called D-holomorphic when
- 0 = (∂/∂x − j ∂/∂y) ( u + j v)
- = ux − j2 vy + j (vx − uy ).
The consequent partial differential equations are called "Scheffer's conditions" by Isaak Yaglom who credits George Scheffer's work of 1893. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i Isaak Moiseevich Yaglom (Иссак Моисеевич Яглом ( 6 March 1921, Kharkov — 17 April 1988, Moscow) was 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. The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties
La Plata lessons
At the National University of La Plata in 1935, J. The National University of La Plata ( Spanish: Universidad Nacional de La Plata, UNLP is an Argentine National university and the most important C. Vignaux, an expert in convergence of infinite series, contributed four articles on the motor variable to the university’s annual periodical. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with 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. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction
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. In the mathematical field of Numerical analysis, a Bernstein polynomial, named after Sergei Natanovich Bernstein, is a Polynomial in the 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. Félix Édouard Justin Émile Borel ( January 7, 1871 in Saint-Affrique, France &ndash February 3, 1956 in Paris
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.
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