Let $K_{ 0}$ denote the modified Bessel function of second kind and zeroth order. In this paper we will studying the function $\tilde{ \omega}_{ n}\left( x\right) :=\frac{ \left( -x\right) ^{ n}K_{ 0}^{ \left( n\right) }\left( x\right) }{ n!}$ for positive argument. The function $\tilde{ \omega}_{ n}$ 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( \left( n+1\right) ^{ m/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.
Categories: math.CA