# Zeros of incomplete exponential functions

Let $$f_N(z) = \sum_{n=N}^\infty \frac{z^n}{n!},$$ where $N$ is a positive integer.

Where are the (complex) zeros of those functions located?

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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Not mentioning the zero at z=0 and complex conjugates: For N = 3 the smallest solution is 3.838602048 + 8.366815507 i. For N = 4 the smallest solution is 5.439213999 + 9.129463691 i. For N = 5 the smallest solution is 6.952562475 + 9.800729397 i. For N = 6 the smallest solution is 8.407369863 + 10.40707148 i. for N = 7 the smallest solution is 8.28937986 + 8.687044071 i for N = 8 the smallest solution is 9.536388591 + 8.053389728 i but there is a smaller real solution at -9.773234001 – juan Jul 12 '12 at 19:33
Yes, I know, no problem to solve that numerically for fixed $N$. – Daniel Krenn Jul 12 '12 at 19:39
Yes, but I found them while trying to locate the zeros in general. There are an infinite number of zeros that run mainly on a line parallell to the imaginary axis. Of course these zeros are separated one of other approximately in 2 pi i. For example the function for N = 8 has zeros at the points 27.26763038 + 163.7377084 i, 27.52677947 + 170.0540751 i, 27.77668695 + 176.3684616 i, 28.01799005 + 182.6810464 i, 28.25126216 + 188.9919872 i, 28.47702109 + 195.301423 i. – juan Jul 12 '12 at 20:03

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.