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MathGroup Archive 2006

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Re: Re: Re: Reduction of Radicals


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
>


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