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RE: [Axiom-developer] Complex exponentiation and 0
From: |
Martin Rubey |
Subject: |
RE: [Axiom-developer] Complex exponentiation and 0 |
Date: |
Mon, 21 Jun 2004 12:19:45 +0000 |
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