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Let $$ f_N(z) = \sum_{n=N}^\infty \frac{z^n}{n!},$$ where $N$ is a positive integer.
It would be sufficient for me to know what the smallest (in absolute value) non-zero zeros are for a general $N$. Maybe an approximation of those zeros would also help. Up to now I just found something about the zeros of the polynomial $e(z) - f_N(z)$ (i.e. the truncated taylor expansion of $e(z)$). The keyword here is "Szegö curve". |
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This is studied in the Appendix of the paper, Hassen and Nguyen, Hypergeometric zeta functions, available here. Let $$T_N(z)=\sum_{k=0}^N{z^k\over k!}$$ and let $$z=x+iy=re^{i\theta}$$ be a root of $$e^z-T_N(z)=0$$ with $y\gt0$. They give an asymptotic approximation of the roots: $$x\sim N\log\bigl(2q\pi+(\pi/2)N-\log(N!)\bigr),\qquad y\sim(N!)^{1/N}e^{x/N}$$ I think "asymptotic" here means asymptotic in $r$, so this gives the big zeros, not the small ones that were requested. However, there is rather more in this appendix, and you may find something of use in it. |
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See two surveys of I. V. Ostrovskii, on zeros of "tails" of power series, MR1890545, MR1771769. |
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