# Why is the notion of analytic function so important?

I think I have some understanding of what an analytic function is — it is a function that can be approximated by a Taylor power series. But why is the notion of “analytic function” so important?

I guess being analytic entails some more interesting knowledge rather than just that it can be approximated by Taylor power series, right?

Or, maybe I don’t understand (underestimate) how a Taylor power series is important? Is it more than just a means of approximation?

1. They are $C^\infty$ functions.
2. If, near $x_0$, we havethenand you can start all over again. That is, you can differentiate them as if they were polynomials.
4. When the domain is connected, the whole function $f$ becomes determined by its behaviour in a very small region. For instance, if $f\colon\mathbb{R}\longrightarrow\mathbb R$ is analytic and you know the sequence $\left(f\left(\frac1n\right)\right)_{n\in\mathbb N}$, then this knowledge completely determines the whole function (the identity theorem).