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:

Szego 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

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.

References & Readings

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