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Re: When is x^y = != E^(y*Log[x])

  • To: mathgroup at smc.vnet.net
  • Subject: [mg66271] Re: [mg66236] When is x^y = != E^(y*Log[x])
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sat, 6 May 2006 01:55:02 -0400 (EDT)
  • References: <200605050902.FAA28575@smc.vnet.net> <79EE0E94-5B77-4F09-81B8-E37030934CE4@mimuw.edu.pl>
  • Sender: owner-wri-mathgroup at wolfram.com

I neeed to make one correction. I wrote:


> So to return to your question: the sufficient condition is that x  
> and y and both sides of your "identity" be complex numbers.

What I should have said is (in a fuller form):

x^y = the principal value of  ( E^(y*Log[x] )

if x and y are both complex numbers and the right hand side is also a  
complex number (not any kind of infinity or Infeterminate).

This can be taken as a definition or if you prefer as a sufficient  
condition.

Andrzej

On 6 May 2006, at 11:23, Andrzej Kozlowski wrote:

>
> On 5 May 2006, at 18:02, ted.ersek at tqci.net wrote:
>
>> The Mathematica documentation for Power says:
>> For complex numbers (x^y) gives the principal value of  ( E^(y*Log 
>> [x] ).
>> This is consistent with reference books.
>>
>> I wanted to see where in the extended complex plane this identity  
>> applies.
>> Also when it doesn't apply how do we determine (x^y).
>> So consider the following:
>>
>>
>> In[1]:=
>>   x=0; y=-3;
>>   {x^y, E^(y*Log[x])}
>>
>> Out[3]=
>>   {ComplexInfinity, Infinity}
>>
>>
>> The documentation wasn't wrong in the above example because 0, -3  
>> are not
>> complex numbers.
>> However, I have seen books that imply the identity above works for  
>> any (x,y).
>> Well I can see the above identity doesn't apply when x=0 and y is  
>> negative.
>> The above identity doesn't apply in the following cases either.
>>
>> In[4]:=
>>   x=-2;  y=(2+I)*Infinity;
>>   {x^y, E^(y*Log[x])}
>>
>> Out[6]=
>>   {Indeterminate, 0}
>>
>>
>>
>> In[7]:=
>>   x=5-6*I;  y=-Infinity*I;
>>   {x^y, E^(y*Log[x])}
>>
>> Out[9]=
>>   {Indeterminate, 0}
>>
>>
>>
>> In[10]:=
>>   x=y=Infinity;
>>   {x^y, E^(y*Log[x])}
>>
>> Out[11]=
>>   {ComplexInfinity, Infinity}
>>
>>
>> Then you might say the above identity doesn't apply when Abs[y] 
>> ==Infinity,
>> but the above identity does apply in the next example.
>>
>> In[12]:=
>>   x=(1+2*I)*Infinity;  y=(-1+2*I)*Infinity;
>>   {x^y, E^(y*Log[x])}
>>
>> Out[14]=
>>   {0, 0}
>>
>>
>> Could somebody provide conditions on (x,y) that are necessary and
>> sufficient for
>> E^(y*Log[x])  to return the same thing as (x^y).
>>
>> -------------
>>   Ted Ersek
>>
>>
>
>
> This issue has been discussed here in the past (see the recent  
> reply by David Cantrell to an earlier post of yours) and I do not  
> want to repeat it all here again, but so will try to confine myself  
> to what I think are the main points.
>
> You can only speak of an "identity" in mathematics if you have at  
> least two well defined objects in the same "mathematical context".  
> Only then you can speak of them as being identically equal. Objects  
> belonging to different contexts can't be compared.
>
> To say that x^y is defined to be the principal value of  ( E^(y*Log 
> [x] ) says exactly that: the left hand side is defined to be the  
> principal value of the right hand side whenever x and y are complex  
> numbers and when both sides are well defined complex numbers. That  
> means in particular that no 'Infinities" are included in this  
> definition.
>  In fact,  0 and -3 are complex numbers, but of Infinity or  
> ComplexInfinty are not. The issue of what should be returned when  
> something in the above expression is "infinite" is a separate one  
> from the above definition. It is partly a question of what model of  
> "extended complex plane" should be used. This is the main source of  
> confusion (but not the only one).
>
> Some of the inconsistencies that appear in your examples are all  
> due to the fact that Mathematica uses simultaneously two different  
> "compactifications" of the complex plane: the one point  
> compactification (where there is only one infinity, denoted by  
> ComplexInfinity and the compactified complex plane is topologically  
> just the 2-sphere) and "infinitely many point compactification",  
> where you have lots of infinities of the form DirectedIfinity[z],  
> where z is any complex number. The complex plane with this  
> compactification is topologically the closed 2-dimensional disk.
> These two models represent two different mathematical contexts, in  
> other words they do not fit together into any consistent  
> mathematical object. Or to put in yet another way, Infinity and  
> ComplexInfinity do not "live in the same world": one lives on the  
> disc the other on the sphere.  Moreover, the "disk model" itself  
> has serious problems, since analytic functions cannot be extended  
> continuously to the disk so inevitably one will sometimes obtain  
> contradictory results if one assumes that they do.
>
> If you only allow one  Infinity: ComplexInfinity - then all the  
> above computations will return the same result, although in most  
> cases it will be the rather useless Indeterminate. This approach,  
> although mathematically the most satisfactory and the one that is  
> almost always used in texts on complex analysis, is not fully  
> satisfactory when you consider Mathematica as a computational tool  
> rather than as a logical system. It can be argued that having  
> ComplexInfinity as the only kind of Infinity would involve  
> "loosing" useful information in certain situations, and as a result  
> Mathematica would not be able to solve some problem that it can do  
> now. (On the other hand it would not get into the kind of  
> contradictions that it does get into now). Also, the "many  
> infinities" disc model allows one to "embed" the standard model of  
> the extended real line (the two point compactification = closed  
> interval) in the extended complex plane, by thinking of Infinity as  
> DirectedInfinity[1], and -Infinity as DirectedInfinity[-1], which  
> is attractive in a program like Mathematica where you do not  
> declare in advance the mathematical context you are working in.
>
> To summarise: x^y === principal part of E^(y*Log[x]) for complex  
> numbers is not an identity: it is a definition. What values either  
> side should take when something becomes "infinite" is a matter of  
> the model of the complex plane that you use and of how you decide  
> to extend the definition. There is no unique or fully satisfactory  
> way of doing this. Having chosen a model, one tries to find the  
> most useful definition, which means that the function will be "as  
> continuous as possible". Note however that even for genuine complex  
> numbers the function x^y will never be continuous everywhere since  
> even the function x^(1/2)  can't be defined continuously on the  
> standard complex plane.
>
> As I already mentioned above: the standard model of the extended  
> complex plane called the Riemann sphere, has only one infinity:  
> ComplexInfinity. We know that on the Riemann sphere the functions z- 
> >z^a can be defined continuously for all real a>0, so we have  
> ComplexInfinity^a = ComplexInfinity for a>0 and ComplexInfinity^a=0  
> for a<0. ComplexInfinity^0 would be best defined as 1, but as that  
> has a discontinuity with respect to the exponent Mathematica  
> chooses to define it as Indeterminate. On the other hand  
> a^ComplexInfinity does not have any definition with good properties  
> so Mathematica leaves it undefined.
>
> When you use the other model of the extended complex plane you into  
> a confusing situation, which is due to the fact that there are more  
> "useful" extensions for various kinds of infinite expressions, but  
> they often tend to be inconsistent with one another. In fact, the  
> answers that you get often depend on such things as the order in  
> which you perform arithmetical operations (in other words the usual  
> laws of arithmetic do not hold). Some of them are hard to explain:  
> I can't see any good reason at all why Infinity^Infinity is  
> ComplexInfinity, and it seems to contradict the the most basic rule  
> that x^y is always real when x and y are positive reals. Besides,  
> as I mentioned earlier, Infinity and ComplexInfinity do not belong  
> together in any topological model known to me (you need a  
> "topological model" to be able to consider the issue of continuity)  
> and should never appear in the same formula. I can only consider  
> this as a bug, and a rather silly one.
>
> But the fact is that there is no accepted standard way in  
> mathematics of dealing with such matters and I do not know of any  
> way that would be satisfactory. In Mathematica in such cases  
> pragmatic computational considerations tend to decide the issue.
>
> So to return to your question: the sufficient condition is that x  
> and y and both sides of your "identity" be complex numbers. As for  
> the necessary condition the question is not really well defined if  
> you allow infinite quantities, since there is neither a single  
> accepted model of the extended complex plane nor, for any given  
> model, a standard accepted way of extending x^y or E^(y*Log[x]) to  
> take account of various kinds of infinities.
>
> Andrzej Kozlowski


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