The x^x Spindle

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.