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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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