Well, in case of power series some criterions do exist. Roughly speaking, one can take the element
$$\sum\limits_{n=1}^{\infty}c_nz^n,\quad z < 1,\qquad\qquad (*)$$
and consider an analytic function $\phi$, such that $\phi(n)=c_n$ for every $n$. $(*)$ can be analytically extended onto some angular domain iff $\phi(z)$ has finite exponential growth.
Let $E\subset \mathbb C$ be a closed unbounded domain and let $H(E)$ denote the set of functions such that each of them is analytic in a neighborhood of $E$.
For a function $\phi\in H(E)$, the exponential type of $\phi$ on $E$ is defined as
$$\sigma_\phi(E)=\limsup\limits_{z\to\infty,\\ z\in E}\frac{\log^+|\phi(z)|}{|z|}.$$
The following result is due to LeRoy and Lindelöf.
Theorem 1. Let $\Pi=\{z\in\mathbb C|\ \Re z \geq 0\}$. Assume that $\phi\in H(\Pi)$ is of finite exponential type $\sigma<\pi$. Then the series
$$f(z)=\sum\limits_{n=1}^{\infty}\phi(n)z^n$$
can be analytically extended onto the angular domain $\{z\in\mathbb C|
\ |\arg z|>\sigma\}$.
The LeRoy-Lindelöf theorem gives only a sufficient condition. A criterion can be obtained if we relax a bit the condition that $\phi$ is of finite exponential type.
Let $\Omega=\{z\in\mathbb C|\ \Re z > 0\}$ be the interior of $\Pi$. An analytic function $\phi\in H(\Omega)$
is said to be of (finite) interior exponential type iff
$$\sigma_\phi^\Omega=\sup\limits_{\{\Delta\}}\sigma_\phi(\Delta)< \infty,$$
where $\{\Delta\}$ is the set of all closed angular domains such that $\Delta\subset \Omega\cup \{0\}.$
Theorem 2. The element
$$\sum\limits_{n=1}^{\infty}c_nz^n,\quad z < 1,$$
can be analytically extended onto the angular domain ${{{{}}{}}}\{z\in\mathbb C|
\ |\arg z|>\sigma\}$ for some $\sigma\in[0,\pi)$ iff there is a function $\phi\in H(\Pi)$ of interior exponential type less or equal to $\sigma$ such that $$c_n=\phi(n),\quad n=0,1,2,\dots.$$
You might be interested in this article.
As I said in the remark there is no book comparable to the Russian edition of
Hurwitz-Courant
(Evgrafov was the editor of the Russian translation who improved the original very much).
There are 2 comprehensive Russian books covering much of geometric theory; both exist in
English translation: Markushevich and Goluzin.
Another book which covers a lot of geometric theory is Caratheodory (2 vols).
None of these has the theory of compact Riemann surfaces,
but Shabat (which you like)
also does not have it. I would say that Markushevich is a good replacement of Shabat.
Goluzin can serve as a source of graduate courses.
Exposition of compact Riemann surfaces in Courant is unique, on my opinion.
Best Answer
English books are Hardy, Divergent series, and P. Dienes, Taylor series: an introduction to the theory of functions of a complex variable. Dover Publications, Inc., New York, 1957. (The title is somewhat misleading. This is a large book that indeed contains an "introduction to complex variables" but it is also the most comprehensive book in English on analytic continuation). But the best book on my opinion is L. Bieberbach, Analytische Fortsetzung, Springer Berlin, 1955, which is available in German and Russian only. (Russian translation by Evgrafov is better because many mistakes are corrected, and some proofs simplified).