$U (-p^ 2/ 4k^ 2 , 0, k^2 z^ 2) $. The boundary condition can be trivially satisfied following ref. [@Afonin:2011ff; @Afonin:2012xq] and writing: $$v (p,z ) \, = \, \frac{ U (-p^ 2/ 4k^ 2 , 0, k^2 z^ 2 ) }{U (-p^ 2/ 4k^ 2 , 0, k^2 z_0^ 2 )}\,.
 \label{bulktoboundary2}$$ The decay constants appear in the two point function: $$\Pi (p^2)  = \sum_{n=1}^\infty \, \frac{f_n^ 2}{(- p^ 2) - m_n^ 2 + i \epsilon} \,.
\label{2point}$$ On the other hand, the two point function is related to the current current correlator $$\left( p^2 \eta_{\mu\nu} -  p_\mu p_\nu   \right) \, \Pi ( p^2 ) 
\, =\, \int d^4x \,\, e^{-ip\cdot x} \langle 0 \vert \, J_\mu (x) J_\nu (0) \,  \vert 0 \rangle  \, , 
\label{correlatorand2pointfunction}$$ that can be obtained holographically by differentiating the on shell action by the boundary values of the fields, with the result: $$\Pi ( p^2 )  \, =\,   \frac{1}{{\tilde g}_5^ 2 \, (-p^ 2) } \left[  \frac{ e^{ -k^2 z^2 }  \,  v (p,z) \partial_z v (p,z)  }{  \, z  } 
  \right]_{_{ \! z \to z_0 }} \, .  
 \label{hol2point}$$

The expression (\[hol2point\]) has simple poles, although it does not have the 