  A fundamental relation in celestial mechanics is Kepler's equation, linking
an orbit's mean anomaly to its eccentric anomaly and eccentricity. Being
transcendental, the equation cannot be directly solved for eccentric anomaly by
conventional treatments; much work has been devoted to approximate methods.
Here, we give an explicit integral solution, utilizing methods recently applied
to the 'geometric goat problem' and to the dynamics of spherical collapse. The
solution is given as a ratio of contour integrals; these can be efficiently
computed via numerical integration for arbitrary eccentricities. The method is
found to be highly accurate in practice, with our C++ implementation
outperforming conventional root-finding and series approaches by a factor
greater than two.
