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| In [[mathematics]], a '''function of a motor variable''' is a [[function (mathematics)|function]] with arguments and values in the [[split-complex number]] plane, much as functions of a [[complex variable]] involve ordinary [[complex number]]s. [[William Kingdon Clifford]] coined the term '''motor''' for a kinematic operator in his "Preliminary Sketch of Biquaternions" (1873). He used split-complex numbers for scalars in his [[split-biquaternion]]s. ''Motor variable'' is used here in place of ''split-complex variable'' for euphony and tradition.
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| For example,
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| :<math>f(z) = u(z)+ \jmath \ v(z) ,\ z = x +\jmath y ,\ x,y \in R ,\quad j^2 = +1,\quad u(z),v(z) \in R.</math>
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| Functions of a motor variable provide a context to extend [[real analysis]] and provide compact representation of mappings of the plane. However, the theory falls well short of function theory on the ordinary complex plane. Nevertheless, some of the aspects of conventional complex analysis have an interpretation given with motor variables.
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| ==Elementary functions of a motor variable==
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| Let '''D''' = <math>\{ z = x + y \jmath : x,y \in R \}</math>, the split-complex plane. The following exemplar functions ''f'' have domain and range in '''D''':
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| The action of a [[versor#Hyperbolic versor|hyperbolic versor]] <math>u = \exp(aj) = \cosh a + \jmath \sinh a</math> is combined with [[translation (geometry)|translation]] to produce the [[affine transformation]]
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| :<math>f(z) = uz + c \ </math> .When ''c'' = 0, the function is equivalent to a [[squeeze mapping]].
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| The squaring function has no analogy in ordinary complex arithmetic. Let
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| :<math> f(z) = z^2 \ </math> and note that <math>f(-1)=f(j)= f(-j) = 1. \ </math>
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| The result is that the four quadrants are mapped into one, the [[identity component]]:
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| :<math>U_1 = \{z \in D : \mid y \mid < x \}</math>.
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| Note that <math>z z^* = 1 \ </math> forms the [[unit hyperbola]] <math>x^2 - y^2 = 1 </math>. Thus the
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| [[multiplicative inverse|reciprocation]]
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| :<math>f(z) = 1/z = z^*/\mid z \mid^2 \text{where} \mid z \mid^2 = z z^* </math>
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| involves the hyperbola as curve of reference as opposed to the circle in C.
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| On the extended complex plane one has the class of functions called [[Möbius transformation]]s:
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| :<math>f(z) = \frac {az + b} {cz + d}.</math>
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| Using the concept of a [[projective line over a ring]], the projective line P('''D''') is formed and acted on by the group of homographies GL(2,'''D'''). The construction uses [[homogeneous coordinates]] with split-complex number components.
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| On the ordinary complex plane, the [[Cayley transform#Conformal map|Cayley transform]] carries the upper half-plane to the [[unit disk]], thus bounding it. A mapping of the identity component U<sub>1</sub> into a rectangle provides a comparable bounding action:
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| :<math>f(z) = \frac {1}{z + 1/2}, \quad f:U_1 \to T </math>
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| where T = {z = x + j y : |y| < x < 1 or |y| < x − 1 when 1<x<2}.
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| ==Exp, log, and square root==
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| The [[exponential function]] carries the whole plane ''D'' into ''U''<sub>1</sub>:
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| :<math>e^x = \sum_{n=0}^\infty {x^n \over n!} = \sum_{n=0}^\infty \frac {x^{2n}} {(2n)!} + \sum_{n=0}^\infty \frac {x^{2n+1}} {(2n+1)!} = \cosh x + \sinh x </math>.
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| Thus when ''x'' = ''b''j, then e<sup>x</sup> is a hyperbolic versor. For the general motor variable ''z'' = ''a'' + ''b''j, one has
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| :<math>e^z = e^a (\cosh b + j \ \sinh b) \ </math>.
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| In the theory of functions of a motor variable special attention should be called to the square root and logarithm functions. In particular, the plane of split-complex numbers consists of four [[Connected component (topology)|connected components]] and the set of singular points that have no inverse: the diagonals ''z'' = ''x'' ± ''x'' j, ''x'' ∈ '''R'''. The [[identity component]], namely {''z'' : ''x'' > |''y''| }, is the [[range (mathematics)|range]] of the squaring function and the exponential. Thus it is the [[domain (function)|domain]] of the square root and logarithm functions. The other three quadrants do not belong in the domain because square root and logarithm are defined as [[one to one correspondence|one-to-one]] inverses of the squaring function and the exponential function.
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| Graphic description of the logarithm of '''D''' is given by Motter & Rosa in their article "Hyperbolic Calculus" (1998).
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| ==D-holomorphic functions==
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| The [[Cauchy-Riemann equations]] that characterize [[holomorphic function]]s on a [[domain (mathematical analysis)|domain]] in the [[complex plane]] have an analogue for functions of a motor variable. An approach to D-holomorphic functions using a [[Wirtinger derivative]] was given by Motter & Rossa:<ref>A.E. Motter & M.A.F. Rosa (1998) "Hyperbolic Calculus", [[Advances in Applied Clifford Algebras]] 8(1):109–28</ref>
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| The function ''f'' = ''u'' + j ''v'' is called '''D-holomorphic''' when
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| :<math>0 \ = \ ({\partial \over \partial x} - j {\partial \over \partial y}) (u + j v)</math>
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| :<math>= \ u_x - j^2 v_y + j (v_x - u_y).</math>
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| By considering real and imaginary components, a D-holomorphic function satisfies
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| :<math>u_x = v_y, \quad v_x = u_y.</math>
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| These equations were published<ref>[[Georg Scheffers]] (1893) "Verallgemeinerung der Grundlagen der gewohnlichen komplexen Funktionen", ''Sitzungsberichte Sachs. Ges. Wiss, Math-phys Klasse'' Bd 45 S. 828-42</ref> in 1893 by [[Georg Scheffers]], so they have been called "Scheffers' conditions"<ref>[[Isaak Yaglom]] (1988) ''Felix Klein & Sophus Lie, The Evolution of the Idea of Symmetry in the Nineteenth Century'', [[Birkhäuser Verlag]], p. 203</ref>
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| The comparable approach in [[harmonic function]] theory can be viewed in a text by Peter Duren<ref>Peter Duren (2004) ''Harmonic Mappings in the Plane'', pp. 3,4, [[Cambridge University Press]]</ref>
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| It is apparent that the components ''u''
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| and ''v'' of a D-holomorphic function ''f'' satisfy the [[wave equation]], associated with
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| [[D'Alembert]], whereas components of C-holomorphic functions satisfy [[Laplace's equation]].
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| ==La Plata lessons==
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| 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.<ref>Vignaux, J.C. & A. Durañona y Vedia (1935) "Sobre la teoría de las funciones de una variable compleja hiperbólica", ''Contribución al Estudio de las Ciencias Físicas y Matemáticas'', pp. 139–184, [[Universidad Nacional de La Plata]], República Argentina</ref> 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):
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| :This system of hyperbolic complex numbers [motor variables] is the [[direct sum of modules#Direct sum of algebras|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.
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| He then proceeds, for example, to generalize theorems due to Cauchy, Abel, Mertens, and Hardy to the domain of the motor variable.
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| 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:
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| :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.<p>
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| 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.
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| ==Bireal variable==
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| In 1892 [[Corrado Segre]] recalled the [[tessarine]] algebra as ''bicomplex numbers''.<ref>G. Baley Price (1991) ''An introduction to multicomplex spaces and functions'', [[Marcel Dekker]] ISBN 0-8247-8345-X</ref> Naturally the subalgebra of real tessarines arose and came to be called the '''bireal numbers'''
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| In 1946 U. Bencivenga published an essay<ref>Bencivenga, U. (1946) "Sulla Rappresentazione Geometrica Della Algebre Doppie Dotate Di Modulo", ''Atti. Accad. Sci. Napoli'' Ser(3) v.2 No 7</ref> on the [[dual number]]s and the split-complex numbers where he used the term bireal number. He also described some of the function theory of the bireal variable. The essay was studied at [[University of British Columbia]] in 1949 when Geoffry Fox wrote his master’s thesis "Elementary function theory of a hypercomplex variable and the theory of conformal mapping in the hyperbolic plane". On page 46 Fox reports "Bencivenga has shown that a function of a bireal variable maps the hyperbolic plane into itself in such a manner that, at those points for which the derivative of a function exists and does not vanish, [[hyperbolic angle]]s are preserved in the mapping".
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| G. Fox proceeds to provide the [[polar decomposition#Alternative planar decompositions|polar decomposition]] of a bireal variable and discusses [[hyperbolic orthogonality]]. Starting from a different definition he proves on page 57
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| :Theorem 3.42 : Two vectors are mutually orthogonal if and only if their unit vectors are mutually reflections of one another in one or another of the diagonal lines through 0.
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| Fox focuses on "bilinear transformations" <math> w = \frac {\alpha z + \beta} {\gamma z + \delta} \quad \text{where} \quad \alpha, \beta, \gamma, \delta </math>
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| are bireal constants. To cope with singularity he augments the plane with a single point at infinity (page 73).
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| Among his novel contributions to function theory is the concept of an ''interlocked system''. Fox shows that for a bireal ''k'' satisfying
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| : (a - b)<sup>2</sup> < |k| < (a + b)<sup>2</sup>
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| the hyperbolas
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| :| z | = a<sup>2</sup> and | z – k | = b<sup>2</sup>
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| do not intersect (form an interlocked system). He then shows that this property is preserved by bilinear transformations of a bireal variable.
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| ==Polynomial factorization==
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| Two staples of introductory algebra include [[factorization of polynomials]] and the [[fundamental theorem of algebra]]. With the adoption of motor variables the traditional expectations are countered.<ref>Note that similar adjustments to traditional expectations are necessary for the idea of the [[square root of a matrix]].</ref> The reason is that ('''D''', +, × ) does ''not'' form a [[unique factorization domain]]. Substitute structures for the motor plane were provided by Poodiack and LeClair in 2009.<ref>Robert D. Poodiack & Kevin J. LeClair (2009) "Fundamental theorems of algebra for the perplexes", [[The College Mathematics Journal]] 40(5):322–35</ref> They prove three versions of the fundamental theorem of algebra where a polynomial of degree ''n'' has ''n''<sup>2</sup> [[root of a function|roots]] counting [[multiplicity (mathematics)#Multiplicity of a root of a polynomial|multiplicity]]. To provide an appropriate concept for multiplicity, they construct a matrix which contains all the roots of a polynomial. Furthermore, their method allows derivation of a similar theorem for polynomials with [[tessarine]] coefficients. The article in [[The College Mathematics Journal]] uses the term "perplex number" for a motor variable, and the term "hyperbolic number" for a tessarine. A basic example of the non-unique factorization is
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| :<math>(z - 1)(z + 1) = z^2 - 1 = (z - j)(z + j)</math>
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| exhibiting the set {1, −1, j, −j } of four roots to the second degree polynomial.
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| ==References==
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| {{Reflist}}
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| * Francesco Catoni, Dino Boccaletti, & Roberto Cannata (2008) ''Mathematics of Minkowski Space-Time'', [[Birkhäuser Verlag]], Basel. Chapter 7: Functions of a hyperbolic variable.
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| [[Category:Complex analysis]]
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| [[Category:Functions and mappings]]
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