Real-analytic function

A function on an open subset of $\mathbb R^n$ is real analytic if near every point $a$ it is represented by a convergent power series

f(x)=\sum_{\alpha\in\mathbb N^n}c_\alpha(x-a)^\alpha.

The series converges absolutely in a neighborhood of $a$ ; its coefficients are $c_\alpha=\partial^\alpha f(a)/\alpha!$ . For a vector-valued map the definition applies to each component.

Smoothness and complex extension

Real analyticity implies smoothness. Locally the same convergent series gives a holomorphic function of complex variables. A smooth function need not be analytic: the function $e^{-1/x^2}$ for $x\ne0$ , extended by zero at zero, has every derivative zero there but is positive nearby. Therefore specifying all Taylor coefficients need not determine a smooth function.