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 + j y , x , y ∈ R , j 2 = + 1 , u ( z ) , v ( z ) ∈ R . {\displaystyle f(z)=u(z)+j\ v(z),\ z=x+jy,\ x,y\in R,\quad j^{2}=+1,\quad u(z),v(z)\in R.}
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, and more generally in hypercomplex analysis.
Elementary functions Let D = { z = x + j y : x , y ∈ R } {\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 u = exp ( a j ) = cosh a + j sinh a {\displaystyle u=\exp(aj)=\cosh a+j\sinh a} is combined with translation to produce the affine transformation
f ( z ) = u z + c {\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
f ( z ) = z 2 {\displaystyle f(z)=z^{2}\ } and note that f ( − 1 ) = f ( j ) = f ( − j ) = 1. {\displaystyle f(-1)=f(j)=f(-j)=1.\ }
The result is that the four quadrants are mapped into one, the identity component:
U 1 = { z ∈ D :∣ y ∣< x } {\displaystyle U_{1}=\{z\in D:\mid y\mid <x\}} , and there are four square roots for elements of this component but no square roots for elements of the other three components. Note that z z ∗ = 1 {\displaystyle zz^{*}=1\ } forms the unit hyperbola x 2 − y 2 = 1 {\displaystyle x^{2}-y^{2}=1} . Thus, the reciprocation
f ( z ) = 1 / z = z ∗ / ∣ z ∣ 2 where ∣ z ∣ 2 = z z ∗ {\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.
Linear fractional transformations 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 ] ( a c b d ) = [ a z + b : c z + d ] , {\displaystyle [z:1]{\begin{pmatrix}a&c\\b&d\end{pmatrix}}=[az+b:cz+d],} sometimes written
f ( z ) = a z + b c z + d , {\displaystyle f(z)={\frac {az+b}{cz+d}},} provided cz + d is a unit in D. Elementary linear fractional transformations include
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