Bulk-to-Boundary propagator How can I show that the bulk-to-boundary propagator
$$
K(z,x;x')~=~\frac{z^{\Delta}}{[z^2+(x-x')^2]^{\Delta}} \tag{1}
$$
goes as a delta function near the boundary
$$
K(z,x;x')~\sim ~z^{d-\Delta}\delta(x-x') ~~?\tag{2}
$$
 A: In order to show that some function is the Delta distribution, you have to show that


*

*It is zero except for where the argument of the function vanishes.

*It integrates to 1 when integrated over the full coordinate range.


We can see these two properties explicitly.


*

*If $x-x'\not=0$, then 
$$
\lim_{z\to0}K(z,x,x')=\lim_{z\to0}\frac{z^\Delta}{(x-x')^{2\Delta}}=0.
$$
So near the boundary, $K(z,x,x')=0$ if $x-x'\not=0$.

*We can do the integral
$$
\begin{align}
\int d^dx\, K(z,x,x')&=\int d^dx\, K(z,x,0)\\
  &= \int d^dx\,\frac{z^\Delta}{(z^2+x^{2})^\Delta}\\
  &= z^\Delta*S^{d-1}\int_0^\infty dr\,\frac{r^{d-1}}{(z^2+r^{2})^\Delta}\\
  &= z^\Delta*\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d}{2}\right)}*z^{d-2\Delta}\frac{\Gamma\left(\frac{d}{2}\right)\Gamma\left(\Delta-\frac{d}{2}\right)}{2\Gamma(\Delta)}\\
  &=\frac{\pi^{\frac{d}{2}}\Gamma\left(\Delta-\frac{d}{2}\right)}{\Gamma(\Delta)}z^{d-\Delta}
\end{align}
$$
In the first step we have shifted $x$. In the third step, we switched to spherical coordinates. As the integrad only depends on the radius, we simply get a factor of the surface of the unit sphere. The resulting integral can be explicitly done.


The result is
$$
\lim_{z\to0}z^{\Delta-d}K(z,x,x')=\frac{\pi^{\frac{d}{2}}\Gamma\left(\Delta-\frac{d}{2}\right)}{\Gamma(\Delta)}\delta^d(x-x')
$$
which is what is meant by the statement
$$
K(z,x;x')~\sim ~z^{d-\Delta}\delta(x-x').
$$
A: *

*Within distribution theory, a mathematically rigorous formulation of OP's eq. (2) is
$$ \lim_{z\to 0^+} z^{\Delta-d}K_{\Delta}(z,x)~=~\delta^d(x),  \tag{A}$$
where
$$ K_{\Delta}(z,x)~:=~ \frac{\Gamma(\Delta)}{\pi^{\frac{d}{2}} \Gamma(\Delta\!-\!\frac{d}{2})} \left(  \frac{z}{z^2+x^2}\right)^{\Delta},\qquad x~\in~\mathbb{R}^d, \qquad  \Delta >\frac{d}{2},\tag{B} $$
is the normalized propagator, see e.g. Ref. 1. The proof of formula (A) is a straightforward application of e.g. Lebesgue's dominated convergence theorem after introducing test functions.

*For comparison, note that
$$ K_{\Delta}(z,x)~\longrightarrow~\left\{\begin{array}{ccl} 
0\text{ almost everywhere} & \text{if} & \Delta~<~d  \cr 
 \delta^d(x) &\text{if} & \Delta~=~d\cr
\text{too singular}& \text{if} & \Delta~>~d\end{array} \right\} \text{  for  } z~\to~ 0^+. \tag{C}$$
The last case $\Delta>d$ is too singular to make sense as a distribution.
References:


*

*D.Z. Freedman, S.D. Mathur, A. Matusis & L. Rastelli, Correlation functions in the $CFT_d/AdS_{d+1}$ correpondence, Nucl. Phys. B546 (1999) 96, arXiv:hep-th/9804058; eqs. (11)-(12).

