A real analytic function can always be continued holomorphically to some domain. However, the holomorphic continuations of definable functions in an o-minimal structure may not be definable. I will present joint work with P. Speissegger in which we study holomorphic continuations of functions definable in two o-minimal expansions of the real field. I will also discuss how to apply these results to the complex Gamma function and Riemann zeta function.[-]

The existential closedness problem for a function $f$ is to show that a system of complex polynomials in $2 n$ variables always has solutions in the graph of $f$, except when there is some geometric obstruction. Special cases have be proven for exp, Weierstrass $\wp$ functions, the Klein $j$ function, and other important functions in arithmetic geometry using a variety of techniques. Recently, some special cases have also been studied for well-known solutions of difference equations using different methods. There is potential to expand on these results by adapting the strategies used to prove existential closedness results for functions in arithmetic geometry to work for analytic solutions of difference equations.[-]