Inverses of Complex Power Functions
Just as in the case of functions of a real variable ![[LINK]](../../../../buttons/link.gif)
, we can rotate the graph of a complex function to exhibit
the graph of its inverse relation. Since the inverse relation also lies in
four-space, we will use projections to view it.
Since z = x + yi and w =
u + vi, the graph in four-space consists of the points
(x,y,u,v). There are two natural projections
to use, one that uses the points (x,u,v) by dropping the
y coordinate, and the other using (y,u,v) by
dropping the x coordinate. The first of these yields the real part
of the inverse relationship, and the second yields its imaginary part.
Other projections of interest combine the two. There is a series of
projections of the form (cos(a) x + sin(a)
y, u, v) as a varies from 0 to
pi/2. These correspond to looking at the
inverse relationship from different viewpoint in four-space, or
alternatively, to looking at the inverse relationship from a constant
viewpoint while rotating it in four-space. In this case, the rotation
occurs in the xy-plane in four-space through an angle of 90 degrees.
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The complex square root function
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The complex cube root function
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