Let K0 denote the modified Bessel function of second kind and zeroth order. In this paper we will study the function ω~n(x):=(-x)nK0(n)(x)n! for positive argument. The function ω~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 ω~n(m) with respect to n can be bounded by 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 K0 which involves the function gn(t)=tnn!exp(-t) and its derivatives. The estimates then rely on a subtle analysis of gn and its derivatives which we will also present in this paper.
Functional estimates for derivatives of the modified Bessel function K0 and related exponential functions / Falletta, Silvia; Sauter, S. A.. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - 417:(2014), pp. 559-579. [10.1016/j.jmaa.2014.03.057]
Functional estimates for derivatives of the modified Bessel function K0 and related exponential functions
FALLETTA, SILVIA;
2014
Abstract
Let K0 denote the modified Bessel function of second kind and zeroth order. In this paper we will study the function ω~n(x):=(-x)nK0(n)(x)n! for positive argument. The function ω~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 ω~n(m) with respect to n can be bounded by 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 K0 which involves the function gn(t)=tnn!exp(-t) and its derivatives. The estimates then rely on a subtle analysis of gn and its derivatives which we will also present in this paper.File | Dimensione | Formato | |
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https://hdl.handle.net/11583/2584483
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