A natural exponential family $\mathscr{F}$ is characterized by the pair $(V,\Omega)$, called the variance function (VF), where $\Omega$ is the mean domain and $V$ is the variance of $\mathscr{F}$ expressed in terms of the mean. Any VF can be used to construct an exponential dispersion model, thus providing a potential generalized linear model. A problem of increasing interest in the literature is the following: Given an open interval $\Omega$ and a function $V$ defined on $\Omega$, is the pair $(V,\Omega)$ a VF of a natural exponential family? In this paper, we develop a complex analytic approach to this question and focus on VF's having meromorphic mean functions; that is, if $T$ is the Laplace transform of an element of the family, then $T'/T$ is extendable to a meromorphic function on $\mathbb{C}$. We derive properties of such VF's and characterize a class of VF's $(V,\Omega)$, where $V$ admits a unique analytic continuation in $\mathbb{C}$, except for isolated singularities. (Included in this class are VF's having $V$'s that admit meromorphic continuation to $\mathbb{C}$.) We show that this class equals the set of VF's which are at most second degree polynomials. We also investigate the class in which $V$ has the form $P + Q\sqrt{R}$, where $P$ and $Q$ are arbitrary rational functions and $R$ is a polynomial of at most second degree. We characterize all VF's in this class for which the mean function is meromorphic and show that $P = kR$ for some constant $k$ and $Q$ is a polynomial of at most first degree. Throughout the paper, we demonstrate the wide applicability of our results by showing that many classes of simple-form pairs $(V,\Omega)$ can be excluded from being VF's.