Independent proof to the complete asymptotic expansion of the Hankel integral yielding an exact and finite series value for half integer order

Abstract

This paper is motivated by the recent paper on asymptotics where a new representation of the Hankel integral given by F_ν(x) = ∫₀^∞ J_v(xt)/(1+t) dt for half integer ν and x > 0 was found using the distributional approach [Galapon & Martinez, Proc. R. Soc. A 470, 20130529 (2013)]. The paper uses a method that is independent to the distributional approach and exactification of the complete asymptotic expansion of the Hankel integral in question. The derivation uses known properties of other special functions specially their series representations.