On the order derivatives of Bessel functions

TL;DR

This paper investigates the derivatives of Bessel functions with respect to their order, deriving integral and series representations, and developing asymptotic approximations involving Airy functions for large order.

Contribution

It introduces new integral representations and asymptotic approximations for the order derivatives of Bessel functions, enhancing understanding of their behavior for large orders.

Findings

Derived integral representations involving products of Bessel functions

Obtained series expansions for the integrals

Constructed asymptotic approximations with Airy functions for large order

Abstract

The derivatives with respect to order {\nu} for the Bessel functions of argument x (real or complex) are studied. Representations are derived in terms of integrals that involve the products pairs of Bessel functions, and in turn series expansions are obtained for these integrals. From the new integral representations, asymptotic approximations involving Airy functions are constructed for the order derivatives, for {\nu} large and uniformly valid for unbounded positive real x.