On Oct 22, 2009, at 3:03 PM, Francis Clarke wrote:
The following article has interesting remarks on this question,
particularly pages 407--408:
\bib{MR1163629}{article}{
author={Knuth, Donald E.},
title={Two notes on notation},
journal={Amer. Math. Monthly},
volume={99},
date={1992},
number={5},
pages={403--422},
}
Among the arguments given in favour of 0^0 = 1 are (1) that we might
like the binomial expansions to hold in general; (2) that there is
precisely one function from the empty set to itself.
I know of at least one case where 0^0 needs to be undefined in order to
get the right answer to a problem. In Mechanics of Solids, there are
singularity functions where,
<x-a>^n = 0 , x < a
= (x-a)^n , x >= a
if n > 0. n = 0 gives the Heaviside function, n = -1 is the Dirac delta
and n = -2 is the unit doublet which is the derivative of the delta
function.
There's also integration rules, but they're unimportant for this
discussion.
So, 0^0 amounts to defining that the Heaviside function = 1 at x = a. I prefer to think of it as undefined and define it strictly in terms of left and right
limits. Cheers, Tim. --- Tim Lahey PhD Candidate, Systems Design Engineering University of Waterloo http://www.linkedin.com/in/timlahey
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