For questions about analytic functions, which are real or complex functions locally given by a convergent power series.
An analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions. Besides, not all infinitely differentiable real function are analytic; for instance the fonction $f\colon\mathbb{R}\longrightarrow\mathbb{R}$ defined by $f(x)=\exp\left(-\frac1{x^2}\right)$ if $x\neq0$ and such that $f(0)=0$ is infinitely differentiable, but not analytic. On the other hand, every differentiable function from an open non-empty subset of $\mathbb C$ into $\mathbb C$ is analytic.
A function is analytic if and only if its Taylor series about $x_0$ converges to the function in some neighborhood for every $x_0$ in its domain.
