Analytical formulas for gravitational lensing: Higher order calculation

We extend to higher order a recently published method for calculating the deflection angle of light in a general static and spherically symmetric metric. We have tested our method on the metric of Schwarzschild and Reissner-Nordstr\"om black holes, on the metric of a charged black hole coupled to Born-Infeld electrodynamics and on the metric of Weyl gravity. Since our method is geometrically convergent, as proved in our previous work, our analytical formulas obtained working to fourth order are sufficient to reach errors of few percents even in proximity of the photon sphere.

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