Understanding Analytic Functions and Convergence

[JamesGoh]

From my lecture notes I was given, the definition of an analytic function was as follows:

A function f is analytic at xo if there exists a radius of convergence bigger than 0 such that f has a power series representation in x-xo which converges absolutely for [x-xo]<R

What I undestand is that for all x values, |x-xo| must be less than R (radius of convergence) in order for f to be analytic at xo.

Convergence in a general sense is when the sequence of partial sums in a series approaches a limit

Is my understanding of convergence and analytic functions correct ?

 

[Fredrik]

What I undestand is that for all x values, |x-xo| must be less than R (radius of convergence) in order for f to be analytic at xo.

What you're saying here would imply that the truth value ("true" or "false") of the statement "f is analytic at x₀" depends on the value of some variable x. It certainly doesn't. It depends only on f and x₀. (What you said is actually that if |x-x₀|≥R, then f is not analytic at x₀).

I'm a bit surprised that your definition says "converges absolutely". I don't think the word "absolutely" is supposed to be there. But then, in $\mathbb C$, a series is convergent if and only if it's absolutely convergent. So if you're talking about functions from $\mathbb C$ into $\mathbb C$, then it makes no difference if the word "absolutely" is included or not.

What the definition is saying is that there needs to exist a real number R>0 such that for all x with |x-x₀|<R, there's a series $$\sum_{n=0}^\infty a_n \left( x-x_0 \right)^n$$ that's convergent and =f(x).

I like Wikipedia's definitions by the way. Link.