why integral causes hankel function

[yefj]

Hello,i have the following current density.
i am integrating it as shown bellow to get vector potential.
Why does th result is HANKEL function of the second order?
could you please shown me the steps which gices us this hankel function result?

Thanks.

 

[MuroSamuro]

First, note that J is a vector which raises the question, what direction is n? (J_n).
Let us assume a Z directed line source located at the origin. The Z directed magnetic vector potential is given on your second formula (you should have a single integral from -inf to +inf). You then change variables into cylindrical coordinates and use hankel function identities to obtain the desired result. As a last step you take care of the shift (r_n in your equation).