\log r/r_+ +\cdots\;.
\end{split}$$ Note that there are logarithmic terms and that the coefficient $a_{T(2)}$ or $a_{L(2)}$ can be shifted by $a_{T(\log)}$ and $a_{L(\log)}$ if we change the scale of the radial coordinate $r$. This corresponds to the choice of a renormalization scale, and should not affect physical quantities. The behavior of the equations near the $AdS_5$ boundary $r=\infty$ fixes $a_{T(\log)}$ and $a_{L(\log)}$ to be $$\label{E:logTerms}
a_{T(\log)}=\frac 1 2 (\omega^2 - k^2) a_{T(0)}\;,\qquad a_{L(\log)} = \frac{\omega^2}{2} a_{L(0)}\;.$$ Since we are solving a set of linear differential equations, the coefficients $a_{T(2)}$ and $a_{L(2)}$ are determined linearly from $a_{T(0)}$ and $a_{L(0)}$. That is, $$\label{E:omega}
\begin{pmatrix} a_{T(2)} \\ a_{L(2)} \end{pmatrix} = \Omega  \begin{pmatrix} a_{T(0)} \\ a_{L(0)} \end{pmatrix}\;,$$ for some $2\times 2$ complex matrix $\Omega$. From the prescription in [@Son:2002sd], and analogously to [@Horowitz:2008bn], the retarded Green’s functions for fields $a_T$ and $a_{L}$ are given by $$G^R = 2 \Omega + \begin{pmatrix} \left(\omega^2-k^2\right)\left( \log \frac{r}{r_+}- \frac 1 2 \right) & 0 \\ 0 & \omega^2\left(\log \frac{r}{r_+}- \frac 1 2\right)\end{pmatrix}\;,$$ where the last term is due to the logarithmically divergent terms in whose coefficients are given in . These divergent parts can be removed by adding a suitable boundary counterterm to the gravity action. After 