 Alternatively, an analytic function is an infinitely differentiable function such that the Taylor series at any point formula_8 in its domain

converges to formula_6 for formula_7 in a neighborhood of formula_8 pointwise.The set of all real analytic functions on a given set formula_2 is often denoted by formula_15.A function formula_1 defined on some subset of the real line is said to be real analytic at a point formula_7 if there is a neighborhood formula_2 of formula_7 on which formula_1 is real analytic.