  The Riesz $z$-energy of a manifold $X$ is the integration of the distance
between two points to the power $z$ over the product space $X\times X$.
Considered as a function of a complex variable $z$, it can be generalized to a
meromorphic function by analytic continuation, which we will call the
meromorphic energy function of $X$. It has only simple poles at some negative
integers. The residues of a manifold $X$ are the residues of the meromorphic
energy function. For example, the volume and the Willmore energy for surfaces
in $\mathbb{R}^3$ can be obtained as residues.
  In this paper we first show the M\"obius invariance of the residue at
$z=-2\dim X$ of a closed submanifold or a compact body in a Euclidean space. We
introduce the relative residues for compact bodies and weighted residues, and
show that the scalar curvature and the mean curvature as well as the Euler
characteristic of compact bodies of dimension less than $4$ can be expressed in
terms of residues and local residues. We study the order of differentiaion of a
local defining function of $X$ that is necessary to obtain the (global)
residues when $X$ is a closed submanifold of a Euclidean space. We also show
the inclusion-exclusion principle.
  Residues appear to be similar to quantities obtained by asymptotic expansion
such as intrinsic volumes (Lipschitz-Killing curvatures), spectra of Laplacian,
and the Graham-Witten energy. We show that residues are independent from them.
Finally we introduce a \M invariant principal curvature energy for
$4$-dimensional hypersurfaces in $\mathbb{R}^5$, and express the Graham-Witten
energy in terms of the residues, Weyl tensor, and this M\"obius invariant
principal curvature energy.
