Motor variable

Let D = ${\displaystyle \{z=x+jy:x,y\in R\}}$, the split-complex plane. The following exemplar functions f have domain and range in D:

The action of a hyperbolic versor ${\displaystyle u=\exp(aj)=\cosh a+j\sinh a}$ is combined with translation to produce the affine transformation

${\displaystyle f(z)=uz+c\ }$. When c = 0, the function is equivalent to a squeeze mapping.

The squaring function has no analogy in ordinary complex arithmetic. Let

${\displaystyle f(z)=z^{2}\ }$ and note that ${\displaystyle f(-1)=f(j)=f(-j)=1.\ }$

The result is that the four quadrants are mapped into one, the identity component:

${\displaystyle U_{1}=\{z\in D:\mid y\mid <x\}}$.

Note that ${\displaystyle zz^{*}=1\ }$ forms the unit hyperbola ${\displaystyle x^{2}-y^{2}=1}$. Thus the reciprocation

${\displaystyle f(z)=1/z=z^{*}/\mid z\mid ^{2}{\text{where}}\mid z\mid ^{2}=zz^{*}}$

involves the hyperbola as curve of reference as opposed to the circle in C.

On the extended complex plane one has the class of functions called Möbius transformations:

${\displaystyle f(z)={\frac {az+b}{cz+d}}.}$

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.

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 into a rectangle provides a comparable bounding action:

${\displaystyle f(z)={\frac {1}{z+1/2}},\quad f:U_{1}\to T}$

where T = {z = x + jy : |y| < x < 1 or |y| < 2 – x when 1 ≤ x <2}.

The exponential function carries the whole plane D into U₁:

${\displaystyle e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}=\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}+\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}=\cosh x+\sinh x}$.

Thus when x = bj, then e^x is a hyperbolic versor. For the general motor variable z = a + bj, one has

${\displaystyle e^{z}=e^{a}(\cosh b+j\ \sinh b)\ }$.

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 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 of the squaring function and the exponential. Thus it is the 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 inverses of the squaring function and the exponential function.

Graphic description of the logarithm of D is given by Motter & Rosa in their article "Hyperbolic Calculus" (1998).[1]

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

${\displaystyle 0\ =\ ({\partial \over \partial x}-j{\partial \over \partial y})(u+jv)}$

${\displaystyle =\ u_{x}-j^{2}v_{y}+j(v_{x}-u_{y}).}$

By considering real and imaginary components, a D-holomorphic function satisfies

${\displaystyle u_{x}=v_{y},\quad v_{x}=u_{y}.}$

These equations were published[2] in 1893 by Georg Scheffers, so they have been called "Scheffers' conditions".[3]

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.[6] Naturally the subalgebra of real tessarines arose and came to be called the bireal numbers.

In 1946 U. Bencivenga published an essay[7] 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

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.

Fox focuses on "bilinear transformations" ${\displaystyle w={\frac {\alpha z+\beta }{\gamma z+\delta }}}$, where ${\displaystyle \alpha ,\beta ,\gamma ,\delta }$ are bireal constants. To cope with singularity he augments the plane with a single point at infinity (page 73).

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 < |k| < (a + b)²

the hyperbolas

|z| = a² and |z − k| = b²

do not intersect (form an interlocked system). He then shows that this property is preserved by bilinear transformations of a bireal variable.

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.[8] 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.[9] They prove three versions of the fundamental theorem of algebra where a polynomial of degree n has n² roots counting 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

${\displaystyle (z-1)(z+1)=z^{2}-1=(z-j)(z+j)}$

exhibiting the set {1, −1, j, −j } of four roots to the second degree polynomial. Another example is

${\displaystyle (z+2)(z+3)=z^{2}+5z+6=(z+{\frac {5}{2}}-{\frac {1}{2}}j)(z+{\frac {5}{2}}+{\frac {1}{2}}j)}$

In general, a quadratic polynomial with two real roots can be factored in two ways as follows:

${\displaystyle a(z+{\frac {b-{\sqrt {b^{2}-4ac}}}{2a}})(z+{\frac {b+{\sqrt {b^{2}-4ac}}}{2a}})=az^{2}+bz+c=a(z+{\frac {b}{2a}}-{\frac {\sqrt {b^{2}-4ac}}{2a}}j)(z+{\frac {b}{2a}}+{\frac {\sqrt {b^{2}-4ac}}{2a}}j)}$

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: ${\displaystyle H=\{(x,y,z):z^{2}+x^{2}-y^{2}=1\}.}$ As with the Riemann sphere, the method is stereographic projection from P = (0, 0, 1) through t = (x, y, 0) to the hyperboloid. The line L = Pt is parametrized by s in ${\displaystyle L=\{(sx,sy,1-s):s\in R\}}$ so that it passes P when s is zero and t when s is one.

From H ∩ L it follows that

${\displaystyle (1-s)^{2}+(sx)^{2}-(sy)^{2}=1,{\text{ so that}}\quad s={\frac {2}{1+x^{2}-y^{2}}}.}$

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 ${\displaystyle y^{2}>1+x^{2},}$ s is negative. The implication is that the back-ray through P to t provides the point on H. These points t are above and below the hyperbola conjugate to the unit hyperbola.

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 ${\displaystyle \{(w,x,y,z)\in P^{3}R:z^{2}+x^{2}=y^{2}+w^{2}\},}$ which is a compact space.

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.[10] 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.[11]

• Francesco Catoni, Dino Boccaletti, & Roberto Cannata (2008) Mathematics of Minkowski Space-Time, Birkhäuser Verlag, Basel. Chapter 7: Functions of a hyperbolic variable.