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Here is a nice derivation for Green's function of a Laplacian in cylindrical coordinates.

For the $r$ coordinate, the equation looks like this:

$$\frac{1}{r}\,\frac{d}{dr}\!\left(r\,\frac{dg_m}{dr}\right)-\left(k^{\,2} + \frac{m^{\,2}}{r^{\,2}}\right) g_m= \frac{1}{r}\,\delta(r-r')$$

And the solution like this:

$$g_m(r,r') = -\,I_m(k\,r_<)\,K_m(k\,r_>)$$

Where $r_< = \min(r,r'), \quad r_> = \max(r,r')$.

It's all perfectly clear to me, however I think that because the equation is even in $k$, the solution should be:

$$g_m(r,r') = -\,I_m(|k| \,r_<)\,K_m(|k| \,r_>)$$

Am I correct?

It's not an idle question since later we have to integrate from $k=-\infty$ to $+\infty$.

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