  Let $\Gamma\subset\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the
first kind acting by fractional linear transformations on the upper half-plane
$\mathbb{H}$, and let $M=\Gamma\backslash\mathbb{H}$ be the associated finite
volume hyperbolic Riemann surface. Associated to any cusp of $M$, there is the
classically studied non-holomorphic (parabolic) Eisenstein series. In 1979,
Kudla and Millson studied non-holomorphic (hyperbolic) Eisenstein series
associated to any closed geodesic on $\Gamma\backslash\mathbb{H}$. In 2004,
Jorgenson and Kramer introduced so-called elliptic Eisenstein series associated
to any elliptic fixed point of $M$. In the present article, we prove the
meromorphic continuation of the elliptic Eisenstein series and we explicitly
compute its poles and residues. Further, we derive a Kronecker limit type
formula for elliptic Eisenstein series for general $\Gamma$. Finally, for the
full modular group $\mathrm{PSL}_{2}(\mathbb{Z})$, we give an explicit formula
for the Kronecker's limit functions in terms of holomorphic modular forms.
