I understand that one can in theory analytically continue a function by repeatedly computing new Taylor series. Suppose for example we have an analytic function $f$ defined on some open set $U$ and compute
for some $z_0\in U$ close to the boundary of $U$. If this converges on $V$ where $U\cap V$ is non-empty, we can then compute another Taylor series to extend further:
for some $z_1\in V\setminus U$ etc.
However, it is impossible to compute infinitely many terms and higher derivatives quickly become prone to large amounts of cancellation error.
Furthermore, one must repeated drop the degree of the next series expansion, as demonstrated here, in order for the result to be useful. Otherwise, with the same degree at the new point $z_1$, you will end up recovering the original $T_0$ and fail to approximate $f$ further away.
So how can one actually numerically compute the analytic continuation of a function?
In my specific case, I have a set of data points over a subinterval of $\mathbb R$ and I know some basic behavior about the function $f$'s derivatives (all derivatives are positive over the subinterval and to the right, which is the area I want to continue to) and that it has no singularities to the right of the given subinterval.