
#1
Aug2411, 12:13 AM

P: 135

From my lecture notes I was given, the definiton 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 xxo which converges absolutely for [xxo]<R What I undestand is that for all x values, xxo 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 ? 



#2
Aug2411, 02:31 AM

Emeritus
Sci Advisor
PF Gold
P: 9,010

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 [itex]\mathbb C[/itex], a series is convergent if and only if it's absolutely convergent. So if you're talking about functions from [itex]\mathbb C[/itex] into [itex]\mathbb C[/itex], 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 xx_{0}<R, there's a series [tex]\sum_{n=0}^\infty a_n \left( xx_0 \right)^n[/tex] that's convergent and =f(x). I like Wikipedia's definitions by the way. Link. 


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