# Research in Scientific Computing in Undergraduate Education

### Roots of Taylor polynomials of non-zero functions

The exponential function $e^x$ never takes the value 0, not even for complex values of the variable $x$,  yet its Taylor polynomial of degree $n, p_n(x):=1+x+\frac{1}{2!}x^2+\frac{1}{3!}x^3+\ldots\frac{1}{n!}x^n$, being a polynomial, does have – possibly complex – roots for $n\geq 1$.

As $n$ increases the roots of $p_n(nx)$ -the so-called normalized roots of the Taylor polynomial –  plotted as complex numbers, approach a curve – the Szegö curve:

named  for its discoverer Gabor Szegö who proved this fact in 1924 (G. Szego , Uber eine Eigenschaft der Exponentialreihe, Sitzungsber. Berlin Math. Ges., 23 (1924), pp. 50-64.).

Gabor Szego

The object of this investigation is to extend Szegö’s analysis to other functions that do not take 0 as a value. A prominent example is the function $e^xx^{-x} \textrm{ for } x > 0$:

Because $e^xx^{-x}$ is not defined for $x=0$ it is convenient to expand the Taylor series around the point $x=1$:

$p_n(x):= e-\frac{1}{2!}e(x-1)^2+\frac{1}{3!}e(x-1)^3+\frac{1}{4!}e(x-1)^4+\ldots+\frac{1}{n!}e(x-1)^n$

Note that we can factor the constant term $e$ from $p_n(x)$:

$p_n(x)= e(1-\frac{1}{2!}(x-1)^2+\frac{1}{3!}(x-1)^3+\frac{1}{4!}(x-1)^4+\ldots+\frac{1}{n!}(x-1)^n)$.

A first step is to compute the normalized zeros – the zeros of $p_n(nx)$ – and try to identify a limiting curve. Because $e^xx^{-x}$ approaches 0 as $x\to \infty$, unlike $e^x$ which becomes unbounded, we should also look at the distribution of the zeros of the Taylor polynomials of $e^xx^{-x}$, and not only the normalized zeros.