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Re: Re: Reduction of Radicals
*To*: mathgroup at smc.vnet.net
*Subject*: [mg71948] Re: [mg71932] Re: [mg71902] Reduction of Radicals
*From*: Murray Eisenberg <murray at math.umass.edu>
*Date*: Tue, 5 Dec 2006 06:04:50 -0500 (EST)
*Organization*: Mathematics & Statistics, Univ. of Mass./Amherst
*References*: <200612031126.GAA08075@smc.vnet.net> <200612041139.GAA02996@smc.vnet.net>
*Reply-to*: murray at math.umass.edu
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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