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

Falletta, Silvia; Sauter, Stefan A (2014). Functional estimates for derivatives of the modified Bessel function and related exponential functions. Journal of Mathematical Analysis and Applications, 417(2):559-579.

## 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.

## 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.

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