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.
|Titolo:||Functional estimates for derivatives of the modified Bessel function K0 and related exponential functions|
|Data di pubblicazione:||2014|
|Digital Object Identifier (DOI):||10.1016/j.jmaa.2014.03.057|
|Appare nelle tipologie:||1.1 Articolo in rivista|