Revisiting the orthogonality of Bessel functions of the first kind on an infinite interval

The rigorous proof of the orthogonality integral integral(infinity)(0) rho J(v) (k rho)J(v)(k'rho) d rho = delta(k-k'/k), for v >= -1, is laborious and requires the use of mathematical techniques that, probably, are unfamiliar to most physics students, even at the graduate level. In physics, we are used to the argument that it may be proved by the use of Hankel transforms. However, the logic of the matter is the opposite, i.e., the existence of the inverse Hankel transform is a consequence of the orthogonality integral. The goal of this work is to prove this integral without circular reasoning. In this paper, using elementary properties of Bessel functions, we give a simple analytical derivation of this integral for the case where v is an integer, zero, or half-integer not less than -1/2. Then, using the asymptotic behaviour of J(v) (x), we extend the result to any v >= -1. This work is of a pedagogical nature. Therefore, to add educational value to the discussion, we do not skip the details of the calculations.