the function $\psi$ has the asymptotic near horizon solutions $\psi_{in/out} = e^{ \mp i \omega  r_\ast} $ representing incoming and outgoing waves respectively.

Expanding the incoming wave solution near the horizon as $$\psi_{in} =  e^{ - i \omega  r_\ast} \left[ 1 + a_1 (z - z_h) + a_2 (z - z_h )^2 + ... \right] \,,$$ and inserting in the equation of motion one finds the relevant coefficient: $$a_1 =   \frac{ 1 + 2k^2 z_h^2 }{ z_h (  i \omega z_h - 2 ) } \,.
 \label{a1}$$ In order to implement the incoming wave condition we write the bulk to boundary propagator as $$v (\omega, z) \,=\,e^{ - i \omega  r_\ast} \, F(\omega,z) \,,
\label{vdecomp}$$ so that the function $F$ takes the form: $$F(\omega,z) = \sqrt{z} e^{\frac{k^2z^2}{2}}  \left[ 1 + a_1 (z - z_h) + a_2 (z - z_h )^2 + ... \right]
\label{BCF}$$ and the derivative of $F $ at the horizon is obtained from this expansion and the expression for $a_1$ in eq. (\[a1\]).

Finally, the spectral function for spatial components $\rho^{ii} $ with the choice of momentum $q^\mu = (\omega , \vec 0 ) $ and written in terms of $F $ takes the form (omitting the 