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Re: Sqrt of complex number

  • To: mathgroup at smc.vnet.net
  • Subject: [mg126677] Re: Sqrt of complex number
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Wed, 30 May 2012 04:12:28 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <jpspgr$hdj$1@smc.vnet.net> <201205290948.FAA06757@smc.vnet.net>
  • Reply-to: murray at math.umass.edu

"You have been given a bunch of answers, each of which is correct but
unresponsive to your concern.

"Mathematica gives the answer it gives because it was programmed that
way, a way that lacks generality, is incomplete and arguably incorrect."

Nonsense!

There is nothing incomplete or incorrect (arguably or otherwise) about 
Mathematica's behavior with Sqrt. A design decision was made that 
mathematically multi-valued numerical functions should return single 
values; the documentation states that the principal root is returned; 
and that's what happens.

You might _prefer_ that Mathematica return all possible values of a 
multi-valued function, but that would raise a host of difficulties -- 
e.g., what if there are infinitely many values -- that would ripple 
through the system and cause difficulties for most computations by most 
users.

On 5/29/12 5:48 AM, Richard Fateman wrote:
> On 5/27/2012 1:44 AM, Jacare Omoplata wrote:
>> Hi,
>>
>> When I try to find the square root of of a complex number, I get only one answer.
>>
>> In[1]:= Sqrt[3-4 I]
>> Out[1]= 2-I
>>
>> But -2+I is an answer as well.
>>
>> In[2]:= (-2+I)^2
>> Out[2]= 3-4 I
>>
>> Why does Mathematica give the first answer and not the second?
>
> Does it choose the answer with the positive real number?
>
> Is there any way I can get both answers?
>
>    Or do I just have to remember that the negative of the given answer is
> also an answer?
>>
>> Thanks.
>>
>
> ................
>
> You have been given a bunch of answers, each of which is correct but
> unresponsive to your concern.
>
> Mathematica gives the answer it gives because it was programmed that
> way, a way that lacks generality, is incomplete and arguably incorrect.
>    I think it does this probably because it copied earlier systems that
> had similar errors.  If it was addressed specifically in the design,
> then the decision was made to provide a mathematically incomplete
> solution, in the hope that nobody would either not notice or not care.
>
> However, you noticed and appear to care.
>
> What you ask for is an expression that includes all answers: the
> algebraic solution to the equation x^2-(3-4 I)=0.
>
> Notice that you can create the set of the 2 roots of a quadratic, that
> is, both square roots, by typing this.
>
>
> Table[Root[#^2 - (3 - 4 I)&, n], {n, 1, 2}]
>
> but this is a list, not "an algebraic number".
>
> If you want to manipulate "an arbitrary root", that is -2+I OR 2-I
> without specifying which one, it seems that Mathematica could provide
> this facility by allowing you to type, for a symbol n,
>
> y = Root[#^2- (3-4I)&,n]
>
> (actually,  Root[x^2-3+4I,n]  might do just a well and be less obscure).
>
> For example,  we would know the unambiguous single value for y^2, and
> we could perhaps compute  y*Conjugate[y].
>
> Unfortunately, Mathematica's designers/programmers do not allow you to
> write Root[x^2-3+4I,n]  unless n is a specific integer, namely 1 or 2.
>
> Conclusion: Mathematica has a notation for what you want, and it can in
> fact do a few things with Root[], but it is defective in handling
> Root[polynomial,n] for symbolic n.  I would call it a mis-feature. Maybe
> it will be fixed, which would not be easy. (while you are at it,
> consider Root[Exp[I x]-1,n] )
>
> Doing it right would require considerable effort, both to figure out
> what the right features should be, and to implement them.
>
> RJF
>
>
>
>
>
>

-- 
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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