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If you map the nth roots of unity $z$ with the function $-W(-z/e)$ you get decent starting points for some root finding algorithm to the roots of the scaled truncated taylor series of $\exp$. Here W is the lambertW function, $e$ is $\exp(1)$ and ‘scaled’ in ‘scaled truncated taylor series of exp’ means the following: say if $$s_5(x) = 1+x+x^2/2+x^3/6+x^4/24+x^5/120$$ is the ‘truncated taylor series of exp’ of degree 5 then we will look at $s_5(5x)$ so we are looking at $s_n(nx)$ in general.

Here is a plot for the case $n=33$ (it only works for uneven $n$).

Using the lambert W function comes from formula (1.1) from paper 221 available from here. This formula is:

$$e^{-nz}s_n(nz)=1-\frac{\sqrt{n}}{\tau_n\sqrt{2\pi}}\int_0^z(\zeta e^{1-\zeta})^n\textrm{d}\zeta,~~z\in \mathbb{C}$$

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$-W(-z/e)$ is the inverse of $ze^{1-z}$.

How to get a better map from the roots of unity to the roots of this polynomial?

Alternatively, is there some infinite sum representation for the roots? There isn’t much difference: “applying” LambertW to some start values is pretty much the same as an infinite series.

Here is an octave script for such a plot as the one above (To use lambertw(), as in the script, install the ‘specfun’ package for octave – or use a more number/function theory centric system than octave).

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You may be interested in the paper *Locating the zeros of partial sums of $\exp(z)$ with Riemann-Hilbert Methods* by T. Kriecherbauer, A.B.J. Kuijlaars, K.D.T-R McLaughlin, and P.D. Miller (arXiv preprint available here). In section 4 they give asymptotic series for the zeros in terms of the images of the roots of unity through the map $z \mapsto -W(-z/e)$.

I’m not familiar with their methods, but I do know of another way to find asymptotic approximations for the zeros of $s_n(nz)$ which stay away from the point $z=1$ (that is, which remain in a compact subset of the punctured plane $\mathbb{C} \setminus \{1\}$ as $n \to \infty$).

The zeros of $s_n(nz)$ satisfy the asymptotic equation

$$

\left(ze^{1-z}\right)^n = \sqrt{2\pi n} \frac{1-z}{z} \Bigl(1+\epsilon_n(z)\Bigr),

\tag{1}

$$

where $\epsilon_n(z) = O(1/n)$ as long as $z$ remains in a compact subset of $\operatorname{Re}(z) < 1$ (at least). By solving this equation for $z$ one may find asymptotic expressions for the individual zeros.

For instance, when $n$ is odd, $s_n(nz)$ has a single real zero $z_n$ which approaches

$$

z=-W(1/e) \approx -0.278465

$$

as $n \to \infty$. For convenience let’s define

$$

w = W(1/e).

$$

According to the paper *On the Zeroes of the Nth Partial Sum of the Exponential Series* by S. Zemyan (JSTOR link), Szegő showed that

$$

z_n = -w – \frac{w}{(1+w)n} \log\left(\sqrt{2\pi n} \frac{1+w}{w}\right) + o\left(\frac{1}{n}\right)

\tag{2}

$$

as $n \to \infty$.

For this result Zemyan cites a book by Pólya and Szegő published in the 60s, though I’m sure Szegő wrote down something like this when he was originally investigating the zeros of these partial sums in the 20s.

In attempting to derive this result myself from equation $(1)$ I found the formula

$$

z_n = -w – \frac{w}{(1+w)n} \log\left(\sqrt{2\pi n} \frac{1+w}{w}\right) – \frac{w}{2(1+w)^3n^2} \left\{\frac{(\log n)^2}{4} + \left[\log\left(\sqrt{2\pi} \frac{1+w}{w}\right)-1\right]\log n\right\} + O\left(\frac{1}{n^2}\right),

\tag{3}

$$

which is a slight improvement on Szegő’s approximation $(2)$. The calculation was tedious, to say the least, but the method can be generalized to find approximations for every such zero of $s_n(nz)$. Begin by writing $z = -W(-\zeta/e) + \delta$, where $\zeta$ is an $n^\text{th}$ root of $-1$, and solve $(1)$ for $\delta$ under the assumption that $\delta$ is small. (Note that in my calculation I chose $\zeta = -1$.)

In a sense this method was used in the paper *Asymptotics for the zeros of the partial sums of $e^z$. I* by A.J. Carpenter, R.S. Varga, and J. Waldvogel (Project Euclid link), though they didn’t carry it through as such. I believe it was actually used prior to that in Carpenter’s doctoral thesis.

Below is a plot of the numerical solutions to $s_{2n+1}((2n+1)z) = 0$ near $z=-W(1/e)$ as black dots, Szegő’s approximation $(2)$ as a blue line, and the approximation in $(3)$ as a red line for $20 \leq n \leq 40$.

The answers to question 4329 over on Math Overflow might be helpful to you.

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