  In conventional scattering theory, to obtain an explicit result, one imposes
a precondition that the distance between target and observer is infinite. With
the help of this precondition, one can asymptotically replace the Hankel
function and the Bessel function with the sine functions so that one can
achieve an explicit result. Nevertheless, after such a treatment, the
information of the distance between target and observer is inevitably lost. In
this paper, we show that such a precondition is not necessary: without losing
any information of distance, one can still obtain an explicit result of a
scattering rigorously. In other words, we give an rigorous explicit scattering
result which contains the information of distance between target and observer.
We show that at a finite distance, a modification factor --- the Bessel
polynomial --- appears in the scattering amplitude, and, consequently, the
cross section depends on the distance, the outgoing wave-front surface is no
longer a sphere, and, besides the phase shift, there is an additional phase
(the argument of the Bessel polynomial) appears in the scattering wave
function.
