Realanalytic

Realanalytic, or real-analytic, describes a class of functions and maps in real analysis. A real-valued function f defined on an open subset of real space is real-analytic if, around every point in its domain, it equals a convergent power series in the variables (a multivariable Taylor series). In one variable, this means that around each point there is a neighborhood where f has a Taylor series with positive radius of convergence that sums to f.

Consequently, real-analytic functions are infinitely differentiable, and their values are completely determined by their Taylor series in a neighborhood of each point. However, not every smooth (C∞) function is real-analytic; a classic counterexample is e^{-1/x^2} for x ≠ 0, with f(0) = 0, which is smooth but not analytic at 0. Real-analytic functions include polynomials, the exponential function, sine and cosine, and many rational and elementary functions on domains where they are defined.

Analyticity is a local property: a real-analytic function is locally given by convergent power series around each point, and its local behavior is governed by these series. Real-analytic maps also arise between real-analytic manifolds, where coordinate changes and maps are given by real-analytic functions. The class is closed under addition, multiplication, differentiation, and composition (where defined). Real-analytic functions can often be extended to holomorphic (complex-analytic) functions in a neighborhood of their domain in complex space, though global extensions are not guaranteed.