[Haskell-cafe] How odd...

[Jacques Carette]

[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>