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?

Being analytic, and especially being complex-analytic, is a really useful property to have, because


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*It's very restrictive. Complex-analytic functions integrate to zero around closed contours, are constant if bounded and analytic throughout $\mathbb{C}$ (or if their absolute value has a local maximum inside a domain), preserve angles locally (they are conformal), and have isolated zeros. Analyticity is also preserved in uniform limits.

*Most of the functions we obtain from basic algebraic operations, as well as the elementary transcendental functions, (and, indeed, solutions to linear differential equations), are analytic at almost every point of their domain, so the surprising restrictiveness of being an analytic function does not stop the class of functions that are analytic from containing many interesting and useful examples. Proving something about analytic functions tells you a something about all of these functions.


Being real-analytic is rather less exciting (in particular, there is no notion of conformality and its related phenomena). Most properties of real-analytic functions can be deduced from restricting local properties of complex-analytic ones anyway, due to this characterisation. So we still have isolation of zeros, and various other properties, but nowhere near as much (and uniform limits are no longer analytic).
