Location of zeros of truncated taylor series of exp(x)

[woelen]

Hello,

 

Suppose we have a truncated taylor series for exp(x), truncated at the

term, just beyond the N-th power. Let's call these truncated series

trunc(N, x). Some examples:

 

trunc(0, x) = 1

trunc(1, x) = 1 + x

trunc(2, x) = 1 + x + x^2/2

...

trunc(N, x) = 1 + x + x^2/2! + .... x^N/N!

 

Here N! means the factorial of the integer N.

Here, trunc(N, x) is an N-th degree polynomial over the complex numbers

with variable x.

 

 

The zeros of trunc(N, x) are on a very regularly shaped curve. The

shape of the curve hardly depends on N.

 

As an example I'll show the zeros for trunc(400, x) as a gif image,

computed by means of my polynomial solving routine (which uses adaptive

multiple precision arithmetic).

 

http://www.woelen.nl/zeros-trunc400-exp.gif

 

The image shows the distribution of the zeros of trunc(400, x) in the

complex plane. Each red dot represents a single zero. As the image shows, the zeros are nicely distributed along a curve.

 

My question is, is there a closed analytic expression for the curve, as

function of N, or is there a closed analytic expression for the

limiting curve, for N going towards infinity?

 

When a plot is made for other values of N, then the curve looks very

similar. For increasing N, the curve tends to blow up, but its basic

shape hardly changes.

 

 

If some of you has any idea about an expression for the curve, then I

would be pleased to read about that.

 

Thanks,

 

Wilco

 

 

PS: Source code of the polynomial equation solver, together with some test programs is available at http://woelen.scheikunde.net/science/software/mpsolve.html