My supervisor briefly showed me a statement of something she called "Darboux's theorem," but I am having trouble finding more information about it on the internet. Here is what I have written down (some details may be incorrect):
Let function $R$ be analytic in a neighborhood about $z=0$ of radius $\rho$. Suppose it has exactly one singularity at distance $\rho$ from $z=0$ (not sure about this sentence). If $R$ can be written in the form $$R(z) = \left(1-\frac zp\right)^{-s} G(z) + H(z)$$ where $s \notin \{0, -1, -2,\ldots\}$, where $G$ and $H$ are analytic in neighborhoods about $z=0$ of radius larger than $\rho$, and $G(\rho)\ne 0$, then the coefficient of $z^n$ in the power expansion of $R(z)$ follows this asymptotic description $$[z^n] R(z) = \rho^{-n} n^{s-1} \frac{G(s)}{\Gamma(s)} \left(1 + O\left(\frac 1n\right)\right)$$
My guess is that it's called "Darboux's method" in English, but still I only have these two links (here and here), that don't have a statement that is exactly the above, although they seem close. Can anyone comment or help me find what I'm looking for? Also any corrections to what I've written above would be appreciated.
You should take a look at the book Analytic Combinatorics, the section about analytic method contains an extensive discussion of Darboux's and other related methods.
The book is available here http://algo.inria.fr/flajolet/Publications/books.html
In general these types of results are known as "Tauberian theorems".