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Say I have a power series expansion of a meromorphic function $f(z)$ which converges on a set $\mathcal S$:

$$f(z) = \sum_{k=0}^\infty c_kz^k (\text{ on } \mathcal S \text{ } )$$ and a truncated version:

$$f_e(z) = \sum_{k=0}^e c_kz^k = c_0 + c_1z +\cdots+ c_ez^e$$

Is there some way to estimate the effect of truncating a power series expansion on $\mathcal S$?

Can we make sure to avoid falling into traps of fake zeros which will invariably be produced by the fundamental theorems of algebra?

Is there maybe some result showing that those zeros must exist outside of the convergence radius or something?

Nosrati
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mathreadler
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    See https://math.stackexchange.com/questions/535720/what-are-the-properties-of-the-roots-of-the-incomplete-finite-exponential-series for a striking example. – lhf May 06 '17 at 16:06
  • @lhf: Nice pictures. I realize now maybe I should instead have asked about some way to calculate zero-free zones (as function of the order). But it feels rude to change the question now that it is already answered. – mathreadler May 06 '17 at 16:25
  • See also the Jentzsch-Szego theorem. – Antonio Vargas Aug 19 '17 at 11:27

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Suppose you have convergence on a disk $D(z_0,R)$. Then for any $r<R$, if $e$ is large enough (depending on $r$), there will be no "fake zeroes" in $D(z_0,r)$.

This follows directly from the uniform convergence on compact subsets of $\mathcal{S}$ of the truncated sums.

In general, you can't reallly get anything more than that.

In particular, if you take $f$ to be the exponential function, you can see that you will automatically get "fake zeroes" by truncating (by the fundamental theorem of algebra and the fact that exp has no zeroes). However, as $e$ tends to infinity, those zeroes will also go to infinity since here the domain of (uniform on all compacts) convergence is the whole plane.

Albert
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  • Yes, you are right in the case of the exponentials of course there will always be false roots but they will likely be pushed further away by each new additive action by the series. Maybe I should have asked for some way to calculate "zero free zone" or something like that. Well +1 anyway. – mathreadler May 06 '17 at 16:23
  • @mathreadler, you may be interested in this answer. – Antonio Vargas May 06 '17 at 21:48
  • @AntonioVargas: Yes! Something like that but for general series expansions would interest me greatly. +1 – mathreadler May 07 '17 at 01:38
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    @mathreadler, as far as I know the only general theory in that vein is due to Rosenbloom and his 1944 thesis On sequences of polynomials, especially sections of power series. I work on the related topic of what happens on the boundary of the zero free zone. – Antonio Vargas May 07 '17 at 12:41
  • @AntonioVargas Ok thanks, I will check out it. I guess I should increase the bounty and add to an edit if it is so difficult. – mathreadler May 10 '17 at 16:13
  • Would increase the bounty and extend if possible, but seems not possible so you may end up getting it automatically. – mathreadler May 13 '17 at 21:04