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

The exponential function never takes the value 0, not even for complex values of the variable , yet its Taylor polynomial of degree , being a polynomial, does have – possibly complex – roots for .

As increases the roots of -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.).

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 :

Because is not defined for it is convenient to expand the Taylor series around the point :

Note that we can factor the constant term from :

.

A first step is to compute the normalized zeros – the zeros of – and try to identify a limiting curve. Because approaches 0 as , unlike which becomes unbounded, we should also look at the distribution of the zeros of the Taylor polynomials of , and not only the normalized zeros.

### References & Readings

- Taylor polynomials of the exponential function
- Peter Walker (2003) The Zeros of the Partial Sums of the Exponential Series. The American Mathematical Monthly, Vol. 110, No. 4, (Apr., 2003), pp. 337-339
- Richard S. Varga, Amos J. Carpenter, and Bryan W. Lewis (2008 ) The dynamical motion of the zeros of the partial sums of , and its relationship to discrepancy theory, Electronic Transactions on Numerical Analysis, vol 30 , 28-143. (dynamical_motion_of_the_zeros_of_partial_sums_of_ez)
- I. E. Pritsker and R. S. Varga, Zero distribution, the Szegö curve, and weighted polynomial approximation in the complex plane, Modelling and Computation for Applications in Mathematics, Science and Engineering (J. W. Jerome, ed.), pp. 167-188, Oxford University Press, 1998. (zero_distribution_szego_curve_weighted_polynomial_approximation)
- William M. Y. Goh & Robert Boyer . On the Zero Attractor of the Euler Polynomials (arXiv.org – submitted 4 Sep 2004) (on_the_zero_attractor_of_the_euler_polynomials)
- Robert Boyer and William M. Y. Goh. Asymptotic zero distributions for polynomials from combinatorics. Department of Mathematics, Drexel University, Philadelphia, PA. (asymptotic_zero_distributions_for-polynomials_from_combinatorics)