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

**References**:**When is x^y = != E^(y*Log[x])***From:*ted.ersek@tqci.net

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