Novel exact and asymptotic series with error functions, for a function involved in diffraction theory: the incomplete Bessel function

The incomplete Bessel function, closely related to incomplete Lipschitz-Hankel integrals, is a well known known special function commonly encountered in many problems of physics, in particular in wave propagation and diffraction [1]-[5]. We present here novel exact and asymptotic series with error functions, for arbitrary complex arguments and integer order, derived from our recent publication [5].  

[1] Shimoda M, Iwaki R, Miyoshi M, Tretyakov OA, 'Wiener-hopf analysis of transient phenomenon caused by time-varying resistive screen in waveguide', IEICE transactions on electronics, vol. E85C, 10, pp.1800-1807, 2002
[2] DS Jones, 'Incomplete Bessel functions. I', proceedings of the Edinburgh Mathematical Society, 50, pp 173-183, 2007
[3] MM Agrest, MM Rikenglaz, 'Incomplete Lipshitz-Hankel integrals', USSR Comp. Math. and Math Phys., vol 7, 6, pp.206-211, 1967
[4] MM Agrest amd MS Maksimov, 'Theory of incomplete cylinder functions and their applications', Springer, 1971.
[5] JML Bernard, 'Propagation over a constant impedance plane: arbitrary primary sources and impedance, analysis of cut in active case, exact series, and complete asymptotics', IEEE TAP, vol. 66, 12, 2018