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

**Follow-Ups**:**Re: Re: Re: Reduction of Radicals***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**References**:**Reduction of Radicals***From:*"dimitris" <dimmechan@yahoo.com>

**Re: Reduction of Radicals***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**Re: (revision) NIntegrate that upper limit is infinite**

**Re: (revision) NIntegrate that upper limit is infinite**

**Re: Reduction of Radicals**

**Re: Re: Re: Reduction of Radicals**