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

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.