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arXiv · 1806.09164

A note on the order derivatives of Kelvin functions

J. L. González-Santander
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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.

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J. L. González-Santander. 2018-06-24. A note on the order derivatives of Kelvin functions. https://doi.org/10.1007/s00025-019-0957-3

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