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?

**Answer**

Analytic functions have several nice properties, including but not limited to:

- They are C^\infty functions.
- If, near x_0, we havef(x)=a_0+a_1(x-x_0)+a_2(x-x_0)^2+a_3(x-x_0)^3+\cdots,thenf'(x)=a_1+2a_2(x-x_0)+3a_3(x-x_0)^2+4a_4(x-x_0)^3+\cdotsand you can start all over again. That is, you can differentiate them as if they were polynomials.
- The fact that you can express them locally as sums of power series allows you to compute fast approximate values of the function.
- 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).

**Attribution***Source : Link , Question Author : Code Complete , Answer Author : T_M*