The Brylinski beta function of a double layer

An analogue of Brylinski’s knot beta function is defined for a compactly supported (Schwartz) distribution $T$ on $d$-dimensional Euclidean space. This is a holomorphic function on a right half-plane. If $T$ is a (uniform) double-layer on a compact smooth hypersurface, then the beta function has an analytic continuation to the complex plane as a meromorphic function, and the residues are integrals of invariants of the second fundamental form. The first few residues are computed when $d=2$ and $d=3$.