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Re: When is x^y = != E^(y*Log[x])
*To*: mathgroup at smc.vnet.net
*Subject*: [mg66270] Re: [mg66236] When is x^y = != E^(y*Log[x])
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Sat, 6 May 2006 01:54:57 -0400 (EDT)
*References*: <200605050902.FAA28575@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
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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