# High-order derivatives of the Bessel functions with an application

## TL;DR

This paper analyzes the asymptotic behavior of high-order derivatives of Bessel functions and applies these results to evaluate related Laplace transforms as the order or transform variable becomes large.

## Contribution

It provides the first detailed asymptotic analysis of the derivatives of Bessel functions and their application to incomplete Laplace transforms.

## Key findings

- Asymptotic formulas for the derivatives of $J_
u(a)$ and $K_
u(a)$ as $n o

- Asymptotic evaluation of two incomplete Laplace transforms involving Bessel functions.
- Brief discussion on similar integrals with $Y_
u(t)$ and $I_
u(t)$.

## Abstract

We determine the asymptotic behaviour of the $n$th derivatives of the Bessel functions $J_\nu(a)$ and $K_\nu(a)$, where $a$ is a fixed positive quantity, as $n\to\infty$. These results are applied to the asymptotic evaluation of two incomplete Laplace transforms of these Bessel functions on the interval $[0,a]$ as the transform variable $x\to+\infty$. Similar evaluation of the integrals involving the Bessel functions $Y_\nu(t)$ and $I_\nu(t)$ is briefly mentioned.