Integral representations of high-order derivatives of Bessel functions with respect to the order

We derive integral representations for the $n$-th derivative with respect to the order of Bessel functions of the first, second and third kinds, as well as the modified Bessel function and the Macdonald function. Based on these integral representations, we have developed a highly-efficient algorithm for the numerical evaluation of these derivatives with respect to the order, applicable to any complex argument and any real order.