A function f(x) is described as analytic around a point a, when f(x)  can be expressed as a power series ∑Cn (x-a)^n around a with a positive radius of convergence R.

This means that at any point c if |c-a|≤R, then the series  ∑Cn (c-a)^n converges.

Ex. f(x)=e^x  can be expressed as ∑((x-0)^n )/n! center of convergence 0.

By using ratio test on ∑(x^n)/n!, the radius of convergence is found to be R=∞

Then f(x)=e^x is an analytic function around 0.

What are the implications/uses of this function property?

If f(x) can be expressed as a power series around a point, then arithmetic operations can be made term by term in the series, as well as differentiation and integration. One use of  these operations  can be found when the power series of a given function cannot be described, but  the power series of the derivative or integral exists.

Ex. Evaluating ∫e^x² dx can be “easily” done by knowing

e^x=∑ (x^n) /n! which is analytic around 0 (from the previous example). Because it is analytical we can operate on it.

So: e^x²=∑ (x^2n) /n! =1+x²+(x^4)/2!+(x^6)/3!+…+x^(2n)/n!

And we can take the integral of every term in 1+x²+(x^4)/2!+(x^6)/3!+…+x^(2n)/n! ->  ∫e^x² dx