The consensus is to use the definition 0 0 = 1, although there are textbooks that refrain from defining 0 0. A (real) polynomial is an expression of the form a 0 x 0 + ⋅⋅⋅ + a n x n, where x is an indeterminate, and the coefficients a i are real numbers. The combinatorial interpretation of b 0 is the number of 0-tuples of elements from a b-element set; there is exactly one 0-tuple. Muller, Jean-Michel; Brisebarre, Nicolas; de Dinechin, Florent; Jeannerod, Claude-Pierre; Lefèvre, Vincent; Melquiond, Guillaume; Revol, Nathalie; Stehlé, Damien; Torres, Serge (2010).

On the other hand, if f and g are analytic functions on an open neighborhood of a number c, then f( t) g( t) → 1 as t approaches c from any side on which f is positive. This does not define 0 w since there is no branch of log z defined at z = 0, let alone in a neighborhood of 0. Thus, the two-variable function x y, though continuous on the set {( x, y): x> 0}, cannot be extended to a continuous function on {( x, y): x> 0} ∪ {(0, 0)}, no matter how one chooses to define 0 0. Euler, when setting 0 0 = 1, mentioned that consequently the values of the function 0 x take a "huge jump", from ∞ for x< 0, to 1 at x = 0, to 0 for x> 0.Other authors leave 0 0 undefined because 0 0 is an indeterminate form: f( t), g( t) → 0 does not imply f( t) g( t) → 1. Cauchy, Augustin-Louis (1821), Cours d'Analyse de l'École Royale Polytechnique, Oeuvres Complètes: 2 (in French), vol. This and more general results can be obtained by studying the limiting behavior of the function ln( f( t) g( t)) = g( t) ln f( t).

An anonymous commentator pointed out the unjustified step; [21] then another commentator who signed his name simply as "S" provided the explicit counterexamples ( e −1/ x) x → e −1 and ( e −1/ x) 2 x → e −2 as x → 0 + and expressed the situation by writing that " 0 0 can have many different values". Some languages document that their exponentiation operation corresponds to the pow function from the C mathematical library; this is the case with Lua [33] and Perl's ** operator [34] (where it is explicitly mentioned that the result of 0**0 is platform-dependent). The multiplicative identity of R[ x] is the polynomial x 0; that is, x 0 times any polynomial p( x) is just p( x).In the complex domain, the function z w may be defined for nonzero z by choosing a branch of log z and defining z w as e w log z.