Suppose we want to analytically extend a function $g(z)$ that is defined as an analytic function everywhere in the complex $z$ plane except the negative $z$ axis (for example, via an integral). Would its analytic extension always have some kind of poles or branch points on the negative $z$ axis? Is it possible that it continues to an analytic function? (Is there a way to argue for this using Liouville's theorem?)
As pointed out in a comment below there are some trivial examples that extend to analytic functions. The example I am interested in is the function $g(z)$ defined via the integral $$g(z)=\int_0^{\infty}ds\ e^{-zs}f(s),$$
with $f(s)$ having poles or branch points on the imaginary $s$ axis (along with other properties required for the convergence of this integral). A theorem for Laplace transforms says this is holomorphic for $\mathrm{Re}(z)>0$. We can then vary the contour of integration between $\mathrm{arg}(s)\in(-\pi/2,\pi/2)$ to define a holomorphic function in the complex $z$ plane everywhere except the negative $z$ axis. Would the continuation of $g(z)$ to the entire complex $z$ plane then have poles or branch points on the negative $z$ axis?