A note on the order derivatives of 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.