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

[John Denker <jsd@av8n.com>]

This issue comes up All The Time in the real world.  Consider
the following DC circuit:

         ________ 
        |        |         I
        |       a|--------->------
        |        |               |
        | device |              load
        |        |               |
        |       b|----------------
        |________|


We are told that the load is passive.  It has an internal
resistance of 1Ω and is dissipating 81 watts of power.  The
question is, what is the current (I)?

This shows the difference between end-of-chapter problems and
the real world.  The proverbial "answer in the back of the
book" is presumably I=9A.  However, the real-world answer
could equally well be I=-9A.  This could have very serious
consequences.  At this point the safe way to proceed is to
try to get additional information, and until then to write
I=±9A or |I|=9A.

Seriously:  In the real world there are lots of ill-posed
problems.  It is not safe to assume that the solution-set
is always a singleton!
  https://www.av8n.com/physics/ill-posed.htm


On 10/28/2016 01:53 PM, Ken Caviness wrote:

> the principal value looks unambiguous, assuming a consistent
> definition of the principal value on the negative real axis in the
> complex plane)

Consistency is a virtue more honored in the breach than in
the observance.

In any other context, if you ask your students to find the cube
root of -1, it seems likely that they will prefer a real-valued
answer (-1) to any complex-valued answer.  Similarly they will
prefer a real-valued answer for the cube root of -3, which is
directly relevant to the original question.  They are taught
this in HS math class.

Mother Nature might (or might not) require a real-valued answer.

Again:  I don't want to take sides in the holy war.  I'm just
pointing out that multiple sides already exist.  For example,
unless I overlooked something, the video did not mention
"principal value".  Instead it talked about "the" value of
the "function".  Some parties to the holy war assume that x^x
implicitly means PV(x^x) while other parties insist that x^x
is not a function unless x is positive, and if you want the
PV you have to write PV(x^x) explicitly.

More to the point, Mother Nature pays no attention to the
mathematicians' failed attempts at consistency.  Suppose we
have a physical situation where we know that
         y^q = b^p             [1]
Then when we solve for y, there is a solution set with q
elements.  It is not the least bit obvious that Mother
Nature will prefer the PV.  The right answer might be
the PV, or it might be any of the other elements of the
solution set.

It is not safe to assume that equation [1] has a unique
solution.  Really not.