On truncated Taylor series and the position of their spurious zeros

Authors:
Søren Christiansen;Per A. Madsen
Affiliations:
Informatics and Mathematical Modelling, The Technical University of Denmark, Building 321, DK-2800 Kgs. Lyngby, Denmark;Department of Mechanical Engineering, The Technical University of Denmark, Building 403, DK-2800 Kgs. Lyngby, Denmark
Venue:
Applied Numerical Mathematics
Year:
2006

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Abstract

A truncated Taylor series, or a Taylor polynomial, which may appear when treating the motion of gravity water waves, is obtained by truncating an infinite Taylor series for a complex, analytical function. For such a polynomial the position of the complex zeros is considered in case the Taylor series has a finite radius of convergence. It is of interest to find whether the moduli of the zeros are close to the radius of convergence. We therefore discuss various upper and lower bounds for the moduli given in the literature and present a new procedure for their estimation. Finally the results obtained are related to an old German paper. It investigates how zeros of partial sums of power series will condensate near the circle of convergence.