Exp, Log, and Square Root
The exponential function carries the whole plane D into U1:
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Thus when x = bj, then ex is a hyperbolic versor. For the general motor variable z = a + bj, one has
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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 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.
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