[seqfan] Re: Without any discussion

[David Wilson]

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.