# Roots of truncated taylor series of exp and lambertW

If you map the nth roots of unity $z$ with the function $-W(-z/e)$ you get very close approximations 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 picture for the case $n=33$ (it only works for uneven $n$).

I'm not even asking how to make it fit exactly - i think i just have to look more intensively at formula (1.1) from paper 221 available from here - and then i will get it eventually - it probably wont be a simple formula (note that $-W(-z/e)$ is the inverse of $ze^{1-z}$). Yes please show me how it fits exactly.

All i'm asking for is more $\textit useful$ information regarding what this sort of thing ties into - more such connections, applications of this etc. I tried google and i found a lot of stuff that didn't help.

Here is an octave implementation for such a plot as the one above.

You will need a lambertW implementation and i used this one

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## 1 Answer

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

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 yeah thanks - i saw that one as well. It was definitly one of the most helpful links i found (and you also found). They talk about the 'szegö-curve' $|se^{1-s}|=1$ that the roots of the scaled taylor polynomials converge to. But what about the roots for fixed $n$ as opposed to $n\rightarrow$inf?. Why do the roots of unity under $-W(-z/e)$ fit so well for the roots of the taylor polynomials? – Peter Sheldrick Jul 15 '11 at 10:20 maybe i phrased that badly. In the answers there are indeed also helpful ideas about the roots of the taylor polynomials for fixed $n$ - and even what trajectory each roots has as $n$ increases. But how do the roots of unity and lambertW come in the picture? Where can i find more on this? Surely this can't start and end at making observations about taylors theorem applied to $exp$. – Peter Sheldrick Jul 15 '11 at 10:39