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Functional estimates for derivatives of the modified Bessel function and related exponential functions


Falletta, Silvia; Sauter, Stefan A (2014). Functional estimates for derivatives of the modified Bessel function and related exponential functions. Journal of Mathematical Analysis and Applications, 417(2):559-579.

Abstract

Let K0K0 denote the modified Bessel function of second kind and zeroth order. In this paper we will study the function View the MathML sourceω˜n(x):=(−x)nK0(n)(x)n! for positive argument. The function View the MathML sourceω˜n plays an important role for the formulation of the wave equation in two spatial dimensions as a retarded potential integral equation. We will prove that the growth of the derivatives View the MathML sourceω˜n(m) with respect to n can be bounded by O((n+1)m/2)O((n+1)m/2) while for small and large arguments x the growth even becomes independent of n . These estimates are based on an integral representation of K0K0 which involves the function View the MathML sourcegn(t)=tnn!exp(−t) and its derivatives. The estimates then rely on a subtle analysis of gngn and its derivatives which we will also present in this paper.

Abstract

Let K0K0 denote the modified Bessel function of second kind and zeroth order. In this paper we will study the function View the MathML sourceω˜n(x):=(−x)nK0(n)(x)n! for positive argument. The function View the MathML sourceω˜n plays an important role for the formulation of the wave equation in two spatial dimensions as a retarded potential integral equation. We will prove that the growth of the derivatives View the MathML sourceω˜n(m) with respect to n can be bounded by O((n+1)m/2)O((n+1)m/2) while for small and large arguments x the growth even becomes independent of n . These estimates are based on an integral representation of K0K0 which involves the function View the MathML sourcegn(t)=tnn!exp(−t) and its derivatives. The estimates then rely on a subtle analysis of gngn and its derivatives which we will also present in this paper.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2014
Deposited On:21 Jan 2015 16:13
Last Modified:27 Apr 2017 22:40
Publisher:Elsevier
ISSN:0022-247X
Publisher DOI:https://doi.org/10.1016/j.jmaa.2014.03.057

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