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I was recently introduced to analytic combinatorics, and found the method of removing poles astonishing. More precisely, I was reading the last chapter of the popular "generatingfunctionology", in which the author treated the basic examples of asymptotic analysis of generating functions.

There, the author only treated for those that have poles and algebraic singularities, missing out the essential singularities. I wonder if it was just too hard to do it or there are some deeper reason. I imagine if there's a way to characterize essential singularities, then we can also remove them and still get a good approximation of the growth of sequence of interest.

My questions are

  1. Is my imagination correct?

  2. Are there some examples?

  3. If characterizing essential singularities is really hard, what else can we do when we do encounter one?

  4. If $g(z)=\Sigma a_n (z-1)^n$ is analytic about $1$ and the nearest singularity is at $0$, is there a way to tell what kind of singularity it is based on the coefficients $a_n$?

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  • $\begingroup$ The typo has been changed from $\Sigma a_n z^n$ to $\Sigma a_n (z-1)^n$. $\endgroup$
    – Student
    Commented Apr 28, 2019 at 14:54
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    $\begingroup$ See math.stackexchange.com/questions/390875/…. I think there is no hope of classifying essential singularities. $\endgroup$
    – Sam Hopkins
    Commented Apr 28, 2019 at 15:01
  • $\begingroup$ Thank you for pointing that out. Then a softer hope might be: given that the nearest singularity is essential, is there a way to cancel it so that we can still get asymptotic behavior of the coefficients? $\endgroup$
    – Student
    Commented Apr 28, 2019 at 21:26

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