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Re: Re: Re: Reduction of Radicals
No disagreement!
Andrzej Kozlowski wrote:
> *This message was transferred with a trial version of CommuniGate(tm) Pro*
> Of course this is completely true. But perhaps it has somewhat obscured
> the point I was trying to make, which was meant to answer Dimitris's
> question: "why is Mathematica's convention about principal parts in
> variance with those of classical algebraist's?" It seems to me that the
> answer is simply that they (the classical algebraists and perhaps some
> of their modern successors) did not care about the relation with
> "principal values" of multivalued functions in analysis and did not even
> know about them (Vieta, for one, live a whole century before Euler).
> Even in certain modern texts on algebra the relationship between
> radicals and logarithms is not relevant, so it is not surprising that
> some authors even today may find it convenient even today to use
> different "principal values' when working in a purely algebraic setting.
> The point I was making about computer algebra programs like Mathematica
> is that with their universal scope why have to be consistent across both
> algebra and analysis in a way that a book on, say, Galois theory, need
> not be.
>
> Andrzej Kozlowski
>
> On 5 Dec 2006, at 20:04, Murray Eisenberg wrote:
>
>> That Mathematica gives value I Pi for Log[-1] is consistent with the
>> most common convention is that the principal argument, Arg, of a nonzero
>> complex number z satisfies -Pi < Arg[z] <= Pi.
>>
>> Then the usual definition of the principal logarithm, Log, is
>>
>> Log[z] = Log[Abs[z]] + I Arg[z],
>>
>> and the multi-valued argument function, arg, would be given as:
>>
>> arg[z] = set of all Arg[z] + n 2 Pi I (n an integer)
>>
>> The multi-valued logarithm, log, would be given as
>>
>> log[z] = Log[Abs[z]] + I arg[z].
>>
>> In this case one can define
>>
>> z^w = Exp[w log[z]]
>>
>> and the principal value of this as Exp[w Log[z]].
>>
>> Andrzej Kozlowski wrote:
>>> ...
>>> The issue of which branch of a multivalued function should be chosen
>>> as the so called "principal branch" is, of course, a matter of
>>> convention. Since Mathematica defines Power[x,y] as Exp[y Log[x]],
>>> the issue of what is (-1)^(1/3) is equivalent to choosing the value
>>> of Log[-1]. Mathematica chooses the value
>>>
>>> Log[-1]
>>>
>>> I*Pi
>>>
>>> Of course once that is decided, everything else follows:
>>>
>>> FullSimplify[(-1)^(1/3) - Exp[(1/3)*I*Pi]]
>>> 0
>>>
>>>
>>> It seems to me (though it is not something that lies within the scope
>>> of my "professional" interest), that before the advent of computer
>>> algebra there no need was felt for a uniform way of choosing
>>> principal values for various multivalued functions that occur in
>>> algebra and analysis. In other words, the relation
>>>
>>> x^y = Exp[y,Log[x]]
>>>
>>> was not treated as the definition of x^y, but as a relation that held
>>> only up to the choice of branches of the multivalued functions
>>> involved. So it seems to me that it was always thought that the
>>> natural choice for Log[-1] is I Pi, but before computer algebra
>>> systems appeared it was not necessarily felt that the "principal
>>> value" of x^(1/3) is the one that makes x^y = Exp[y,Log[x]] hold.
>>> Note that to keep this relation true and to have the principal value
>>> of the cube root of -1 equal to -1, one would have do choose 3 I Pi
>>> as the principal value of Log[-1], which does not seem very natural.
>>
>> --Murray Eisenberg murray at math.umass.edu
>> Mathematics & Statistics Dept.
>> Lederle Graduate Research Tower phone 413 549-1020 (H)
>> University of Massachusetts 413 545-2859 (W)
>> 710 North Pleasant Street fax 413 545-1801
>> Amherst, MA 01003-9305
>>
>
>
--
Murray Eisenberg murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower phone 413 549-1020 (H)
University of Massachusetts 413 545-2859 (W)
710 North Pleasant Street fax 413 545-1801
Amherst, MA 01003-9305
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