A Liouville function is an analytic function $H: \C \rightarrow \C$ with a Taylor series $\sumn=1\infty xn/an$ such the an’s form a “very fast growing” sequence of integers. In this paper we exhibit the complete first-order theory of the complex field expanded with H.
"The theory of Liouville functions." J. Symbolic Logic 68 (2) 353 - 365, June 2003. https://doi.org/10.2178/jsl/1052669055