  The Restricted 3-Body Problem models the motion of a body of negligible mass
under the gravitational influence of two massive bodies, called the primaries.
If the primaries perform circular motions and the massless body is coplanar
with them, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In
synodic coordinates, it is a two degrees of freedom Hamiltonian system with
five critical points, L1,..,L5, called the Lagrange points.
  The Lagrange point L3 is a saddle-center critical point which is collinear
with the primaries and is located beyond the largest of the two. In this paper
and its sequel, we provide an asymptotic formula for the distance between the
one dimensional stable and unstable invariant manifolds of L3 when the ratio
between the masses of the primaries $\mu$ is small. It implies that L3 cannot
have one-round homoclinic orbits.
  If the mass ratio $\mu$ is small, the hyperbolic eigenvalues are weaker than
the elliptic ones by factor of order $\sqrt{\mu}$. This implies that the
distance between the invariant manifolds is exponentially small with respect to
$\mu$ and, therefore, the classical Poincar\'e--Melnikov method cannot be
applied.
  In this first paper, we approximate the RPC3BP by an averaged integrable
Hamiltonian system which possesses a saddle center with a homoclinic orbit and
we analyze the complex singularities of its time parameterization. We also
derive and study the inner equation associated to the original perturbed
problem. The difference between certain solutions of the inner equation gives
the leading term of the distance between the stable and unstable manifolds of
L3.
  In the sequel we complete the proof of the asymptotic formula for the
distance between the invariant manifolds.
