To compute the residue of a pole of very high order $M$ at $z=0$, one needs to compute
$\frac{d^M}{dz^M} g(z)$
Suppose that $g(z)$ is a reasonable but not trivial function, that itself may depend on $M$. At the moment I'm interested in
$g(z) = \left(\frac{f(z)}{h(z)}\right)^{\gamma-k M}$
where $f(z)$, $h(z)$ are polynomials, and $\gamma$ and $k$ are rational but not integers, but I would like to generalize later.
Are there any tricks to get the asymptotic behavior for this quantity?
I know the following:
(1) I can Laurent expand everything in sight, collect sums and pick out the $M^{th}$ term. This works but then appears to require a lot of Stirling approximations to see the dominant terms.
(2) In principle, I could find the Fourier transform of $g$ and look at its large $q$ behavior. This looks intractable, but would at least allow saddle-point methods to be applied.
A general method is to use Cauchy's integral formula
$\frac{d^M}{dz^M}g(z) = \frac{M!}{2\pi i}\oint_\gamma \frac{g(w)}{(w-z)^{1+M}}$,
where $\gamma$ is a curve encircling $z$ in the anticlockwise direction. For large $M$ the saddle-point method effectively extracts the leading behaviour.
For example, if $g(z)=F(z)^{kM}$, then the saddle-point equation is $0 = \frac{d}{dw} [k\log F(w)-\log(w-z)] = k \frac{F'(w)}{F(w)} - \frac{1}{w-z}$
and to leading nontrivial exponential order (in M) we have
\begin{align}\log\frac{d^M}{dz^M}g(z) & \sim \log\left[ \frac{M!}{2\pi i} g(w_*)(kF'(w_*)/F(w_*)^M \right] + O(1) \\ & \sim M \log (M/e) - M \log k + M(k-1) \log F(w_*) + M \log F'(w_*) + O(1) \end{align} where $w_*(z)$ is the saddle point. If there are multiple saddle points, we should sum over all of them, in principle also checking that the integration path $\gamma$ can be continuously deformed to go through them in the correct orientation.