Gravitational lensing from compact bodies: Analytical results for strong and weak deflection limits

We develop a nonperturbative method that yields analytical expressions for the deflection angle of light in a general static and spherically symmetric metric. The method works by introducing into the problem an artificial parameter, called δ, and by performing an expansion in this parameter to a given order. The results obtained are analytical and nonperturbative because they do not correspond to a polynomial expression in the physical parameters. Already to first order in δ the analytical formulas obtained using our method provide at the same time accurate approximations both at large distances (weak deflection limit) and at distances close to the photon sphere (strong deflection limit). We have applied our technique to different metrics and verified that the error is at most 0.5% for all regimes. We have also proposed an alternative approach which provides simpler formulas, although with larger errors.