A note on the order derivatives of Kelvin functions

Abstract

We calculate the derivative of the ${\mathrm{ber}}_{\nu }$, ${\mathrm{bei}}_{\nu }$, ${\ker }_{\nu }$, 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 ${\ker }_{\nu }$ 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.

References in corpus (1)

• Closed form expressions for derivatives of Bessel functions with respect to the order