[Sorry for the long quote, but context is important] Dan Piponi wrote:
It's fairly standard practice, when documenting functions of a complex variable, to specify precisely which 'branch cuts' are being used. Here's a quote from the Mathematica documentation describing their Log function: "Log[z] has a branch cut discontinuity in the complex z plane running from -infinity to 0".
With this interpretation, (-1)^(1/3) = 0.5 + sqrt(3)/2 * i. If you go with the real solution (-1) you might need to do so carefully in order to preserve other useful properties of ^, like continuity.
You can guarantee this by making sure you make the right 'cuts'.
If only it were that simple. Up until rather recently, it was not even known if there was a set of "right cuts" for ln and all the arc-trig functions that would work "together" properly. See `"According to Abramowitz and Stegun" or arccoth needn't be uncouth', by Corless, Jeffrey, Watt and Davenport, available from http://citeseer.ist.psu.edu/corless00according.html or http://www.apmaths.uwo.ca/~djeffrey/Offprints/couth.pdf This is by no means a solved problem. For example, there is still no consensus as to where the branch cuts for the various versions of the LegendreQ function should go. Jacques http://www.apmaths.uwo.ca/%7Edjeffrey/Offprints/couth.pdf