[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>
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