A note on the order derivatives of Kelvin functions
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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.
Keywords
Kelvin functions bessel functions generalized hypergeometric function meijer-G functionMathematics Subject Classification
33C10 33C20 33E20Notes
Acknowledgements
It is a pleasure to thank Prof. A. Apelblat for the literature and wise comments given to the author.
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