[Haskell-cafe] How odd...
Jacques Carette
carette at mcmaster.ca
Sat Aug 11 10:21:15 EDT 2007
[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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