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MathGroup Archive 2006

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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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