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.


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