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On truncated Taylor series and the position of their spurious zeros. (English) Zbl 1084.76062

Summary: 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 the case when 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 [R. Jentzsch, Acta Math. 41, 219–251 (1917; JFM 46.0516.03)]. It investigates how zeros of partial sums of power series will condensate near the circle of convergence.

MSC:

76M40 Complex variables methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
30B10 Power series (including lacunary series) in one complex variable

Citations:

JFM 46.0516.03

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References:

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