During a talk I was at today, the speaker mentioned that if you truncate the Taylor series for e^x - 1, you'll get lots of roots with nonzero real part, even though the full Taylor series only has pure imaginary roots.
If you plot the roots of truncations of e^x - 1 (or check out the ready-made plots in this Mathematica notebook) you can see lots of cool features. I'd like to know where they come from! I know there's a vast literature on polynomials, but I'm a total beginner, and I don't know where to start.
Here are a few specific questions:
The roots of a high-degree truncation seem to fall into two categories: roots that lie very close to the imaginary axis, and roots that lie on a C-shaped curve. (Another interpretation is that all of the roots lie on a curve, which has a very sharp kink near the imaginary axis.) Can you write down an equation for the curve?
If you put the roots of a lot of consecutive truncations together on the same plot, you'll see definite "stripes" to the right of the imaginary axis. Once a stripe appears, each higher-degree truncation sticks another root onto the end, making the stripe grow outward. Can you write down equations for the stripes?
If k is odd, the truncation of degree k has no nonzero real roots. If k is even, the truncation of degree k has one nonzero real root. The location of this root depends almost linearly on k. Why is the dependence so close to linear? Does it get more linear as k increases, or less?
Can roots be given identities that persist across time? That is, as k increases, can you point to a sequence of roots and say, "those are all the same individual, which was born at k = so-forth, is following such-and-such trajectory, and will grow up to become the root (2 pi i whatever) of e^x - 1"?