Why are analyticity and convergence related in complex analysis?

[dm4b]

TL;DR Summary

From Proof of Residue Theorem in Mary Boas' Text

Hello,

I am currently reading about the Residue Theorem in complex analysis. As a part of the proof, Mary Boas' text states how the a_n series of the Laurent Series is zero by Cauchy's Theorem, since this part of the Series is analytic. This appears to then be related to convergence of the a_n series.

I suppose convergence and analyticity seem to go together on an intuitive level, but I am having a hard time with the details of why this is so, especially as to how it would relate back to the Cauchy-Riemann conditions for analyticity.

Can anyone offer further insights here for me?

Thanks!
dm4b

 

[r731]

Check out the epsilon-delta definitions of limit and convergence.

 

[Office_Shredder]

I'm not sure what you mean by the a_n series but if you mean the part with positive powers then the proof that the power series is differentiable starts with
- notice that on any closed ball strictly inside the radius of conference, the power series converges uniformly
- if $f_n$ converges to $f$ uniformly in a region, then the derivative passes through the limit, i.e. $f_n'$ converges to $f'$, and in particular $f$ is differentiable.
- the partial sums are all polynomials so are differentiable.