[seqfan] Re: Without any discussion
David Wilson
davidwwilson at comcast.net
Tue Sep 1 07:04:22 CEST 2009
The reason I asked this question is because NJAS took issue with a b-file I
submitted for A061862 which treated 0^0 as 1, e.g, I included elements such
as
308 = 3^5 + 0^0 + 8^2
evaluating 0^0 = 1. NJAS requested that I recompute the b-file, disallowing
0^0 = 1.
I guess I do not agree. Generally, I agree with Wikipedia article
"Exponentiation".
Basically, the article suggests that, for discreet exponents, 0^0 = 1 is the
natural choice. The empty product, combinatorial, and set theoretic
interpretations mentioned in that article all dictate 0^0 = 1, and that
choice uniquely simplifies many expressions involving discreet exponents. I
would also add that the recursive definition of integer exponents:
x^0 = 1; x^(n+1) = x * x^n
which is clearly valid x != 0, generalizes to x = 0, where it implies 0^0 =
1 if n = 0; 0 if n > 0; undefined if n < 0.
This definition is a special case of integer exponentiation over ring(?) R:
x^0 = multiplcative identity of R; x^(n+1) = x * x^n.
which suggests that "0^0 = multiplicative identity of base type" is the
proper generalization.
The difficulties with defining 0^0 almost exclusively arise with continuous
exponents. Specifically, the definition
x^y = exp(y log x)
fails to define 0^0, and approaching (x, y) = (0, 0) along various curves
leads to different values of 0^0. For that reason, 0^0 is indeterminate when
the exponents are continuous. However, I might add that when (x, y)
approaches (0, 0) along an analytic curve, x^y approaches 1 (I think), so
even here 0^0 = 1 is an appropriate choice for a wide array of problems.
And yes, the domain does make a difference, in this case at least. 0^0
should be interpreted differently as an integer expression (where it should
be 1) and a real expression (where it should be left indeterminate). Note
that indeterminate and undefined are not the same thing. Undefined means
that no appropriate value can be assigned, as with 1/0. Indeterminate means
that many appropriate values can be defined, according to application.
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