130 Chapter 3
- M. Murrill [12] has introduced a so-called complex hyperanalytic
geometry of four dimensions with a frame of reference consisting of four co-
ordinate axes x, y, u, v, which are mutually perpendicular, by definition, at
a common origin 0. Equal scales are used on all four axes, and as a matter
of practical representation in two dimensions, the particular asymmetrical
arrangement of axes shown in Fig. 3.3 is adopted. In this arrangement the
viewpoint is such that the projections of the positive x-and u-axes are 150°
degrees apart. Now, for a given function w = f(z ), consider in the xy-plane
the z-vector representing a certain value of z in the domain D of f, as well
as the corresponding w-vector in the uv-plane. The vector sum t = z +w,
which has two degrees of freedom [in view of u = u(x, y), v = v(x, y)J is
called the transformation vector. As z describes D, the tip T of that vec-
tor traces in the four-dimensional space a two-dimensional surface (called
the mirror surface or the t-surface ), which is taken as the graph of the
complex function w = f(z) (Fig. 3.3).
3.4 Functions Associated with a Given Function
By composition of a given complex function with certain simple functions,
in particular, with J(z) = z and K(z) = -z, associated functions arise
that are of interest in subsequent developments. Thus, using composition
with J in different ways (and with K), we obtain:
- J = J o f (the conjugate function)
2. f* = f o J (the function of the conjugate)
- r = J 0 f 0 J (the conjugate of the function of the conjugate)
- r = J 0 f 0 J{ 0 J (the paraconjugate of the function)
The functions J and J are defined only if the domain D off is symmet-
ric with respect to the x-axis (or, on a subset of D that is symmetric with
respect to the x-axis). The function f1' is defined only if D is symmetric
with respect to the y-axis.
The mappings defined by these functions are illustrated geometrically
in Fig. 3.4. Clearly,
J*(z) = (f o J)(z) = f(z)
J*(z) =(Jo f o J)(z) = J(z)
J"(z) =(Jo f o I< o J)(z) = ](-z)