In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. 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-biquaternions. Motor variable is used here in place of split-complex variable for euphony and tradition.
For example,
f(z)=u(z)+j v(z), z=x+jy, x,y\inR, j2=+1, u(z),v(z)\inR.
Let D =
\{z=x+jy:x,y\inR\}
u=\exp(aj)=\cosha+j\sinha
f(z)=uz+c
The squaring function has no analogy in ordinary complex arithmetic. Let
f(z)=z2
f(-1)=f(j)=f(-j)=1.
U1=\{z\inD:\midy\mid<x\}
Note that
zz*=1
x2-y2=1
f(z)=1/z=z*/\midz\mid2where\midz\mid2=zz*
Using the concept of a projective line over a ring, the projective line P(D) is formed. The construction uses homogeneous coordinates with split-complex number components. The projective line P(D) is transformed by linear fractional transformations:
[z:1]\begin{pmatrix}a&c\ b&d\end{pmatrix}=[az+b:cz+d],
f(z)=
| az+b | |
| cz+d |
,
Elementary linear fractional transformations include
\begin{pmatrix}u&0\ 0&1\end{pmatrix},
\begin{pmatrix}1&0\ t&1\end{pmatrix},
\begin{pmatrix}0&1\ 1&0\end{pmatrix}.
Transformations bounding regions can be compared: For example, on the ordinary complex plane, the Cayley transform carries the upper half-plane to the unit disk, thus bounding it. A mapping of the identity component U1 of D into a rectangle provides a comparable bounding action:
f(z)=
| 1 | |
| z+1/2 |
, f:U1\toT
To realize the linear fractional transformations as bijections on the projective line a compactification of D is used. See the section given below.
The exponential function carries the whole plane D into U1:
ex=
| infty | |
| \sum | |
| n=0 |
{xn\overn!}=
| infty | |
| \sum | |
| n=0 |
| x2n | |
| (2n)! |
+
| infty | |
| \sum | |
| n=0 |
| x2n+1 | |
| (2n+1)! |
=\coshx+\sinhx
ez=ea(\coshb+j \sinhb)
\{U1,-U1,jU1,-jU1\},
Graphic description of the logarithm of D is given by Motter & Rosa in their article "Hyperbolic Calculus" (1998).
The Cauchy–Riemann equations that characterize holomorphic functions on a 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:[1]
The function f = u + j v is called D-holomorphic when
0 = \left({\partial\over\partialx}-j{\partial\over\partialy}\right)(u+jv)= ux-j2vy+j(vx-uy).
ux=vy, vx=uy.
The comparable approach in harmonic function theory can be viewed in a text by Peter Duren.[4] 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.
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.[5] 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 since its sides are on isotropic lines.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.
In 1892 Corrado Segre recalled the tessarine algebra as bicomplex numbers. Naturally the subalgebra of real tessarines arose and came to be called the bireal numbers.
In 1946 U. Bencivenga published an essay[6] on the dual numbers 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 Geoffrey 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 angles are preserved in the mapping".
G. Fox proceeds to provide the polar decomposition of a bireal variable and discusses hyperbolic orthogonality. Starting from a different definition he proves on page 57
w=
| \alphaz+\beta | |
| \gammaz+\delta |
\alpha,\beta,\gamma,\delta
Among his novel contributions to function theory is the concept of an interlocked system. Fox shows that for a bireal k satisfying
(a − b)2 < < (a + b)2the hyperbolas
= a2 and = b2do not intersect (form an interlocked system). He then shows that this property is preserved by bilinear transformations of a bireal variable.
The multiplicative inverse function is so important that extreme measures are taken to include it in the mappings of differential geometry. For instance, the complex plane is rolled up to the Riemann sphere for ordinary complex arithmetic. For split-complex arithmetic a hyperboloid is used instead of a sphere:
H=\{(x,y,z):z2+x2-y2=1\}.
L=\{(sx,sy,1-s):s\inR\}
From H ∩ L it follows that
(1-s)2+(sx)2-(sy)2=1,sothat s=
| 2 | |
| 1+x2-y2 |
.
If t is on the null cone, then s = 2 and (2x, ±2x, – 1) is on H, the opposite points (2x, ±2x, 1) make up the light cone at infinity that is the image of the null cone under inversion.
Note that for t with
y2>1+x2,
The compactification must be completed in P3R with homogeneous coordinates (w, x, y, z) where w = 1 specifies the affine space (x, y, z) used so far. Hyperboloid H is absorbed into the projective conic
\{(w,x,y,z)\inP3R:z2+x2=y2+w2\},
Walter Benz performed the compactification by using a mapping due to Hans Beck. Isaak Yaglom illustrated a two-step compactification as above, but with the split-complex plane tangent to the hyperboloid.[7] In 2015 Emanuello & Nolder performed the compactification by first embedding the motor plane into a torus, and then making it projective by identifying antipodal points.[8]
. Isaak Yaglom . Abe Shenitzer (translator) . A simple non-Euclidean geometry and its physical basis : an elementary account of Galilean geometry and the Galilean principle of relativity . 1979 . Springer-Verlag . New York . 0-387-90332-1 . registration .