User:FactSpewer/Draft of Exponentiation

Zero to the zero power
Most authors agree with the statements related to 00 in the two lists below, but come to differing conclusions when it comes to defining 00 or not: see the next subsection.

In most settings not involving continuity, interpreting 00 as 1 simplifies formulas and eliminates the need for special cases in theorems (see the next paragraph for some settings that do involve continuity). For example:
 * Regarding 00 as an empty product of zeros suggests a value of 1.
 * The combinatorial interpretation of 00 is the number of empty tuples of elements from the empty set. There is exactly one empty tuple.
 * Equivalently, the set-theoretic interpretation of 00 is the number of functions from the empty set to the empty set. There is exactly one such function, the empty function.
 * It greatly simplifies the theory of polynomials and power series that a constant term can be written ax0 for an arbitrary x. For example:
 * The formula for the coefficients of a product of polynomials would lose much of its simplicity if constant terms had to be treated specially.
 * Identities like $$\textstyle \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$$ and $$\textstyle e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$ are not valid for x = 0 unless 00 = 1.
 * The binomial theorem $$\textstyle(1+x)^n = \sum_{k = 0}^n \binom{n}{k} x^k$$ is not valid for x = 0 unless 00 = 1.
 * In differential calculus, the power rule $$\textstyle\frac{d}{dx} x^n = nx^{n-1}$$ is not valid for n = 1 at x = 0 unless 00 = 1.

On the other hand, 00 must be handled as an indeterminate form in settings where the exponent varies:
 * When f(x) and g(x) are positive-valued functions approaching 0 (as x approaches a real number or ∞), the function f(x)g(x) need not approach 1. In fact, depending on f and g, the limit of f(x)g(x) can be any real number between 0 and 1, or it can be undefined.  One expresses this ambiguity by saying that 00 is an indeterminate form.  The source of this ambiguity is the fact that the two-variable function xy, continuous on the region defined by x ≥ 0, y ≥ 0, (x,y) ≠ (0,0), cannot be extended to a continuous function on a region including (0,0), no matter how 00 is defined.   The rule in calculus that limx →a f(x)g(x) = (limx→a f(x))limx→a g(x) whenever both sides of the equation are defined would fail if 00 were defined.
 * The function zz is defined for nonzero complex numbers z by choosing a branch of log z and setting zz := ez log z, but there is no branch of log z defined at z = 0, let alone in a neighborhood of 0. There is no holomorphic function defined in a neighborhood of 0 that agrees with zz for all positive real numbers z.

History of differing points of view
Different authors interpret the situation above in different ways:
 * Some argue that the best value for 00 depends on context, and hence that defining it once and for all is problematic. According to Benson (1999), "The choice whether to define 00 is based on convenience, not on correctness."
 * Others argue that 00 is 1. According to p. 408 of Knuth (1992), it "has to be 1".

The debate has been going on at least since the early 1800s. At that time, most mathematicians agreed that 00 = 1, but in 1821 Cauchy listed 00 along with expressions like 0/0 in a table of undefined forms. In the 1830s Libri published an unconvincing argument for 00 = 1, and Möbius sided with him, erroneously claiming that $$\lim_{x \to 0^+} f(x)^{g(x)} = 1$$ whenever $$\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} g(x) = 0.$$ A commentator who signed his name simply as "S" provided a counterexample, and this quieted the debate for some time, with the apparent conclusion of this episode being that 00 should be undefined. More details can be found in Knuth (1992).

Treatment in programming languages and calculators
Computer programming languages that evaluate 00 to 1 include bc, Haskell, J, Java, MATLAB, ML, Perl, Python, R, Ruby, Scheme, and SQL. In the .NET Framework, the method  treats 00 to be 1.

Microsoft Excel issues an error when it evaluates 00.

Microsoft Windows' Calculator and the calculator in Google search evaluate 00 to 1.

Maple simplifies a0 to 1 and 0a to 0, even if no constraints are placed on a, and evaluates 00 to 1.

Mathematica simplifies a0 to 1, even if no constraints are placed on a. It does not simplify 0a, and it takes 00 to be an indeterminate form.