Functional Estimates for Derivatives of the Modified Bessel Function $K_0$ and related Exponential Functions

Let $K_0$ denote the modified Bessel function of second kind and zeroth order. In this paper we will studying the function $\tilde{\omega}{n}\left( x\right) :=\frac{\left( -x\right) ^{n}K{0}^{\left( n\right) }\left( x\right) }{n!}$forpositiveargument\mathrm{.}Thefunction$\tilde{\omega}{n}$ 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 ${\tilde{\omega }}_n^{\left(m\right)}$ with respect to $n$ can be bounded by $O\left({(n+1)}^{m\mathrm{/}2}\right)$ while for small and large arguments $x$ the growth even becomes independent of $n$. These estimates are based on an integral representation of $K_0$ which involves the function $g{n}\left( t\right) =\frac{t^{n}}{n!}\exp\left( -t\right) $ and their derivatives. The estimates then rely on a subtle analysis of $g_n$ and its derivatives which we will also present in this paper.