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.

Well, in case of power series some criterions do exist. Roughly speaking, one can take the element $$ \sum\limits_{n=1}^{\infty}c_nz^n,\quad z < 1,\label{1}\tag{$\ast$}$$ and consider an analytic function $\phi$, such that $\phi(n)=c_n$ for every $n$. The element \eqref{1} 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.

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.

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.