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

TL;DR

This paper derives closed-form expressions for derivatives of Bessel and modified Bessel functions with respect to their order, using integral representations and special functions, also calculating two new integrals.

Contribution

It provides new closed-form formulas for derivatives of Bessel functions with respect to order, expanding the mathematical tools available for these special functions.

Findings

Closed-form expressions in terms of hypergeometric and Meijer-G functions.

Integral representations for derivatives of Bessel functions.

Calculation of two non-tabulated integrals.

Abstract

We have used recent integral representations of the derivatives of the Bessel functions with respect to the order to obtain closed-form expressions in terms of generalized hypergeometric functions and Meijer-$G$ functions. Also, we have carried out similar calculations for the derivatives of the modified Bessel functions with respect to the order, obtaining closed-form expressions as well. For this purpose, we have obtained integral representations of the derivatives of the modified Bessel functions with respect to the order. As by-products, we have calculated two non-tabulated integrals.