MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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".

share|cite|improve this question
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.

share|cite|improve this answer

See two surveys of I. V. Ostrovskii, on zeros of "tails" of power series, MR1890545, MR1771769.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.