[Math] Analytic continuation of holomorphic functions

Question

Analytic/meromorphic continuation is a difficult problem in general. For "motivic L-functions", the idea of proving their analytic continuations by first proving their modularity goes back, I guess, to Riemann.

Here I just want to ask a purely complex-analytic question. Let's restrict ourselves to the case of one variable functions. Let $U$ be a region in the complex plane, and let $f$ be a holomorphic function on $U.$ Is there any criterion for $f$ to have analytic continuation to a larger region? And what is this "maximal domain of regularity"?

Feel free to assume $U$ and $f$ to have the shape you like, e.g. a power series on an open disk or a Dirichlet series on some half plane. I guess even if $U$ is an annulus or a punctured disk, where one can compute (theoretically or numerically) the value of the extended function (if exists) at the points inside the inner loop by Cauchy's formula, it is still difficult to decide if this extension is continuous or analytic.

Answer 1

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 $$c_n=\phi(n),\quad n=0,1,2,\dots.$$

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