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Re: [Phys-L] y= x^x



Executive summary: Just because some number crawled out
of a software library doesn't mean it's the whole story.

On 10/26/2016 07:36 AM, Bob Sciamanda wrote:

A curious presentation of y = x^x :

http://screencast.com/t/m4fmGwmkrT9

Curious indeed.
Points awarded for data visualization.
Points deducted for disastrous mathematical unsophistication.

Holy wars are fought over the meaning of b^c when the base (b)
is negative and the exponent (c) is not an integer.

As always, I do not wish to take sides in holy wars, but we
can perhaps take note of some of the possibilities. As a
/subset/ of the problem, consider the case where the exponent
is a rational number. We write c = p/q in lowest terms, so
that p and q are relatively prime integers.

So the first question is, what do we mean by y = b^(p/q) ???
It could mean:
-- Find all y such that y^q = b^p.
There will be q different elements in the solution-set.
Multiply any element by a qth root of unity to find another.
-- Find some y such that y^q = b^p.
That is, choose any element from the aforementioned solution-set.
-- Find the principal y such that y^q = b^p
based on some specified set of preferences.

Again, I don't want to take sides in holy wars, but here is one
possible way of choosing the preferred "principal" value (PV).
Note that b is still negative.

*) If p and q are both odd, then b^p is negative and we can
choose PV(y) to be a negative real number.

*) If p is even and q is odd, then b^p is positive and we can
choose PV(y) to be a positive real number.

*) If p is odd and q is even, then b^p is negative and every
y is a nontrivial complex number.

*) We can't have both p and q even.

This is just one possibility. For an arbitrary piece of software
operating on an arbitrary floating-point number, it is not the
least bit obvious what it will choose as "the" principal value.

If we generalize to the case where the exponent (c) is irrational,
then I have no idea how to make sense of b^c when b is negative.

=================

Now fix p=1000 and consider the sequence q=1001, 1003, 1005,
et cetera. Let's set x=-p/q and take a look at

y = (x)^(x)
= (-p/q)^(-p/q)
= (-q/p)^(p/q)

Since p is even, the aforementioned PV(y) is a positive real
number in all these cases. That means PV(y) is a positive
real number at infinitely many places on the x-interval
from -1 to 0. It's a positive real number at 500 different
places just between -1 and -0.5.

This is not what is shown in the video. Not even close.

If the presenter had chosen slightly different x-values,
and/or chosen a different software package, the results
would have been wildly different.

If the presenter had considered all elements of the solution-
set (not just some ill-defined principal value) then the
results would have been a spectacular mess.

Bottom line: Just because some number crawled out of a
software library doesn't mean it's the whole story.