A note on the order derivatives of Kelvin functions

TL;DR

This paper derives simplified formulas for the derivatives of Kelvin functions with respect to order, introduces novel expressions for certain derivatives, and computes new related integrals in closed-form.

Contribution

It provides more straightforward expressions for order derivatives of Kelvin functions, including novel formulas for the derivatives of $ ext{ker}_ u$ and $ ext{kei}_ u$, and calculates new integrals involving these functions.

Findings

Simplified formulas for derivatives of $ ext{ber}_ u$ and $ ext{bei}_ u$ functions.

Novel expressions for derivatives of $ ext{ker}_ u$ and $ ext{kei}_ u$ functions.

Closed-form expressions for new integrals involving Kelvin functions.

Abstract

We calculate the derivative of the $\mathrm{ber}_{\nu }$, $\,\mathrm{bei}_{\nu }$, $\mathrm{ker}_{\nu }$, and $\,\mathrm{kei}_{\nu }$ functions with respect to the order $\nu $ in closed-form for $\nu \in \mathbb{R}$. Unlike the expressions found in the literature for order derivatives of the $\mathrm{ber}_{\nu }$ and $\,\mathrm{bei}_{\nu }$ functions, we provide much more simple expressions that are also applicable for negative integral order. The expressions for the order derivatives of the $\mathrm{ker}_{\nu }$ and $\,\mathrm{kei}_{\nu }$ functions seem to be novel. Also, as a by-product, we calculate some new integrals involving the $\mathrm{ber}_{\nu }$ and $\,\mathrm{bei}_{\nu }$ functions in closed-form. Finally, we include a simple derivation of some integral representations of the $\mathrm{ber}$ and $\,\mathrm{bei}$ functions.