The x^x Spindle

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It is easy to sketch the graph of for
. For
,
is only defined for rational
with odd denominator, unless complex values are allowed. In this case, set
, where log represents the principal branch of the complex logarithm; other branches differ by an integer multiple of
and these may be specified using the parameter
in the formula. Thus, any
value determines a 3D graph where each
value determines the point in space
. The collection of all of these graphs, or threads, forms the
spindle.
Contributed by: Mark McClure (March 2011)
Open content licensed under CC BY-NC-SA
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Reference: M. D. Meyerson, "The Spindle," Mathematics Magazine, 69(3), 1996 pp. 198–206.
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