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The Restricted 3-Body Problem models the motion of a body of negligible massunder the gravitational influence of two massive bodies, called the primaries.If the primaries perform circular motions and the massless body is coplanarwith them, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). Insynodic coordinates, it is a two degrees of freedom Hamiltonian system withfive critical points, L1,..,L5, called the Lagrange points.  The Lagrange point L3 is a saddle-center critical point which is collinearwith the primaries and is located beyond the largest of the two. In this paperand its sequel, we provide an asymptotic formula for the distance between theone dimensional stable and unstable invariant manifolds of L3 when the ratiobetween the masses of the primaries $\mu$ is small. It implies that L3 cannothave one-round homoclinic orbits.  If the mass ratio $\mu$ is small, the hyperbolic eigenvalues are weaker thanthe elliptic ones by factor of order $\sqrt{\mu}$. This implies that thedistance between the invariant manifolds is exponentially small with respect to$\mu$ and, therefore, the classical Poincar\'e--Melnikov method cannot beapplied.  In this first paper, we approximate the RPC3BP by an averaged integrableHamiltonian system which possesses a saddle center with a homoclinic orbit andwe analyze the complex singularities of its time parameterization. We alsoderive and study the inner equation associated to the original perturbedproblem. The difference between certain solutions of the inner equation givesthe leading term of the distance between the stable and unstable manifolds ofL3.  In the sequel we complete the proof of the asymptotic formula for thedistance between the invariant manifolds.

Breakdown of homoclinic orbits to L3 in the RPC3BP (I). Complex singularities and the inner equation