How do you know if a function can be represented as a convergent power series in terms of analyticity and singularities?

Question

I was reading my textbook on Fourier Analysis and it reads,

"Let $F(z)$ be an analytic function of the complex variable $z = x + iy$, without singularities for $|z| \leq 1$. Then $F(z)$ can be expanded in a power series..."

How do you know if a function in the Complex plane can be represented as a convergent power series merely based on analyticity and singularities? What about for a function in the Reals?

Disclaimer: I have looked at similar questions, but they have not answered my particular question.

Answer 1

For complex analyticity search holomorphic functions and complex analysis.

Real analyticity is often proven via estimates for the derivatives.

Have you tried

https://en.wikipedia.org/wiki/Analytic_function#Real_versus_complex_analytic_functions

and there the third reference

http://projecteuclid.org/euclid.pja/1195524081

which deals with real analyticity?

Answer 2

It's a theorem of Complex Analysis that if $F(z)$ is analytic in the disk $|z - a| < r$, then the Taylor series of $F(z)$ at $z=a$ converges in that disk. See e.g. these notes.