# What are the properties of the roots of the incomplete/finite exponential series?

Playing around with the incomplete/finite exponential series

$$f_N(x) := \sum_{k=0}^N \frac{z^k}{k!} \stackrel{N\to\infty}\longrightarrow e^z$$

for some values on alpha (e.g. solve sum_(k=0)^19 z^k/(k!) = 0 for z), I made a few observations:

• The sum of the roots of $f_N$ are $-N$
• The product of the roots of $f_N$ are $(-1)^N\cdot N!$
• Their imaginary part seems to lie between $\pm10$
• The zeros seem to form an interesting shape:

Patterns for $N=17, 18, 19$

Now the sum and product part are clear, since

\begin{align} f_N(x) &= \frac1{N!}\left(z^N + N z^{N-1} + N(N-1)z^{N-2} + ... + N!\right) \\ &= \frac1{N!}(z-z_{N0})(z-z_{N1})\cdots(z-z_{NN}) \\ &= \frac1{N!}\left(z^N - \left(\sum_{k=0}^Nz_{Nk}\right) z^{N-1} + ... + (-1)^N\prod_{k=0}^N z_{NK}\right) \end{align}

and since $e^z=0 \Leftarrow \Re z\to-\infty$ it is clear that the roots tend towards real parts with negative infinity, but I'm still intrigued by the questions

what ($N$-dependent) curve do the zeros of $f_N(z)$ lie on, does that curve maintain its shape for varying $N$ and merely translate or also deform; and what other properties of the zeros (e.g. absolute value) can be derived?

• @martini Thanks, how did you fix thi-- I see, you escaped most characters, did you use a tool for that or do it manually? Oct 22, 2013 at 15:19
• I used the "add a link" feature of the edit/answer window here on SE. Just click on the "chain" and paste the link there ... Oct 22, 2013 at 15:20
• @martini m-/ I never used that feature until now, always manually linked via []()... Thanks! Oct 22, 2013 at 15:21
• You may also be interested in these three questions: one, two, three. Nov 5, 2013 at 21:59
• @AntonioVargas Thanks, interesting indeed! Nov 6, 2013 at 7:32

The zeros of the scaled functions $f_N(Nz)$ do converge to an airfoil-like curve. See an animation here.

• Perhaps an interesting relation: there is a proposition of a function as 2-fold iteration $f_c(z)=c^z \text{ with } c,z \in \mathbb C$ and a method to find periodic-points $f^{o2}_c(z)=z$ with a function called "HyperW()" or $HW()$ in an article as a generalization of the Lambert-W. The method to actually approximate periodic points means to solve the truncated taylor-series of $f^{o2}_c(z)$ . This is a strong relation to the subject here. I've done some investigation. See question math.stackexchange.com/q/3706909 and one answer math.stackexchange.com/a/3713978 . Jun 11, 2020 at 10:03
One more picture; here I rescale the radial distances from the origin to their logarithm; the roots of the polynomials $f_{16},f_{32},f_{64},f_{128}$ are shown, the magenta line is that for $f_{16}$. I find it interesting, that the radial positions fit nicely together, see the straight lines from the origin (the roots do not exactly match with the lines but remarkably good)
• So in other words, you plot on a log-log scale, right? Wow, that's an interesting observation! Also interesting: The amount of roots between two such lines appears to be of the form $2^n-1$ Oct 30, 2013 at 8:49
• Well, that's not quite the correct transformation. In fact, if I do separately rescale the x-axis and the y-axis, then the curves approximate squares (you might try this using an asinh-scaling instead, which becomes quite similar to a log-log-scaling for values >1). No, here I compute the polar coordinates of the roots, rescale the distance to the origin $\sqrt{\Re(\rho)^2+\Im(\rho)^2}= |\rho|$ by logarithmizing and recalc the rectangular coordinates from that. Oct 30, 2013 at 9:10