RE: [Axiom-developer] Complex exponentiation and 0

[Martin Rubey]

This should have gone to axiom-devel also...

Martin Rubey writes:
 > Bill Page writes:
 >  > On Sunday, June 20, 2004 7:20 AM David MENTRE wrote:
 >  > > ... 
 >  > > Is your bug report related to:
 >  > >   [bugs #9313] 0^0 handled inconsistently
 >  > >   http://savannah.nongnu.org/bugs/?func=detailitem&item_id=9313
 >  > > 
 >  > > According to Martin comment, 0^0 is not mathematically defined.
 >  > > 
 >  >
 > 
 > The problem is that the function f(x,y) = x^y is not continuous at x=y=0: 
 > 
 > (1) -> limit(x^y,y=0)
 > 
 >    (1)  1
 >                         Type: Union(OrderedCompletion Expression Integer,...)
 > (2) -> limit(x^y,x=0)
 > 
 >    (2)  "failed"
 > 
 > (which is correct, note that y might be less than (or equal to)
 > zero... Unfortunately, I see no way to tell axiom to assume y > 0 here. Such 
 > a
 > facility would be very nice.)
 > 
 > (3) -> limit(x^1,x=0)
 > 
 >    (3)  0
 > 
 > (very nice:)
 > 
 > (4) -> limit(x^(-1),x=0)
 > 
 >    (4)  [leftHandLimit= - infinity,rightHandLimit= + infinity]
 > 
 > Type: Union(Record(leftHandLimit: Union(OrderedCompletion Fraction Polynomial
 > Integer,"failed"),rightHandLimit: Union(OrderedCompletion Fraction Polynomial
 > Integer,"failed")),...)
 > 
 > ------------------------------------------------------------------------------
 > 
 > I think it's dangerous to say that 0^0=1, although it's natural in many 
 > cases:
 > 
 > http://db.uwaterloo.ca/~alopez-o/math-faq/mathtext/node14.html
 > 
 > In fact, I'm not sure what we would gain if axiom assumes 0^0=1 throughout. I
 > think, before we decide to adopt this strategy, we should have examples which
 > were otherwise cumbersome to deal with.
 > 
 > Maybe as a guide:
 > 
 > Mathematica 5.0 for Linux
 > Copyright 1988-2003 Wolfram Research, Inc.
 >  -- Motif graphics initialized -- 
 > 
 > In[1]:= 0^0
 > 
 >                                         0
 > Power::indet: Indeterminate expression 0  encountered.
 > 
 > Out[1]= Indeterminate
 > 
 > In[2]:= Limit[x^y,y->0]
 > 
 > Out[2]= 1
 > 
 > It seems that MMA can't do the following either:
 > 
 > In[3]:= FullSimplify[Limit[x^y,x->0],y>0]
 > 
 >                y
 > Out[3]= Limit[x , x -> 0]
 > 
 > 
 > 
 >     |\^/|     Maple 8 (IBM INTEL LINUX)
 > ._|\|   |/|_. Copyright (c) 2002 by Waterloo Maple Inc.
 >  \  MAPLE  /  All rights reserved. Maple is a registered trademark of
 >  <____ ____>  Waterloo Maple Inc.
 >       |       Type ? for help.
 > 0^0;
 > > 0^0;
 >                                        1
 > 
 > limit(x^y,y=0);
 > > limit(x^y,y=0);
 >                                        1
 > 
 > limit(x^y,x=0);
 > > limit(x^y,x=0);
 >                                             y
 >                                     lim    x
 >                                    x -> 0
 > 
 > assume(y>0):limit(x^y,x=0);
 > > assume(y>0):limit(x^y,x=0);
 >                                        0
 > 
 > MuPad also says 0^0=1