  We present an analytic calculation of the layer (parallel) susceptibility at
the extraordinary transition in a semi-infinite system with a flat boundary.
Using the method of integral transforms put forward by McAvity and Osborn
[Nucl. Phys. B 455 (1995) 522] in the boundary CFT we derive the
coordinate-space representation of the mean-field propagator at the transition
point. The simple algebraic structure of this function provides a practical
possibility of higher-order calculations. Thus we calculate the explicit
expression for the layer susceptibility at the extraordinary transition in the
one-loop approximation. Our result is correct up to order $O(\varepsilon)$ of
the $\varepsilon=4-d$ expansion and holds for arbitrary width of the layer and
its position in the half-space. We discuss the general structure of our result
and consider the limiting cases related to the boundary operator expansion and
(bulk) operator product expansion. We compare our findings with previously
known results and less complicated formulas in the case of the ordinary
transition. We believe that analytic results for layer susceptibilities could
be a good starting point for efficient calculations of two-point correlation
functions. This possibility would be of great importance given the recent
breakthrough in bulk and boundary conformal field theories in general
dimensions.
