The interesting part is where pi/2 < cos(x) < 3pi/2 because here the values of cos(x) are negative (between -1 and 0), causing negative numbers to be raised to negative number.

This can cause complex or imaginary results: such as (-.5)^(-.5) = -1.414i, which can not be graphed on the real coordinates system (don't just graph the real part of the solution)

....or negative results such as (-1/3)^(-1/3) = -1.4422

....or positive results such as (-2/3)^(-2/3) = 1.3104

It all depends on the numerator and denominator being odd or even, and the irrationality of the numerator and denominator.

So, depending on the value of cos(x) in this region the graph should bounce from positive and negatives, but not be connected anywhere due to the complex results.

The trick was to get a graph to show this behavior. Some calculators will only show the complex values (or roots), and others give the real valued roots.

For instance, the value of (-2/3)^(-2/3) will either be real or complex depending on the calculator or math software: Hp's and TI's give the real root, while Mathematica and the Casio Fx9860g will only give the complex root.

The trick was to find the values of x that produces real roots (both positive and negative). After some trial and error, this is what I came up with:

The middle portion is the non continuous positive-negative-complex behavior. It should be filled-in completely, but I only used fractions as large as -38/39 (i.e. (-38/39)^(-38/39) which gives a positive value) and this will occur for radians of about 2.91 and 3.368, hence the large gap. Had I used values closer to 1, i.e., 257/259 the gap would have been smaller.

Anyway, fun problem, but I think I spent way to much time kicking it around. :)

Have great weekend.

CHUCK

*Edited: 4 Dec 2009, 7:52 p.m. *