Ping Li, Jun Cheng, Jiang-he Yang

We show that the photograph of a Schwarzschild black hole can be written in closed form. The image plane is embedded as a plane tangent to a sphere centred on the black hole, and the map between a point source and its image point, which we call the photograph function, is treated as the analytic object. Explicit expressions are obtained for sources on the celestial sphere at infinity and for sources in the equatorial plane, without solving for the intermediate null geodesics. The construction is exact, and it makes the multivalued branch structure, the relation to the photon sphere and to the shadow boundary, and the self-similar higher-order images directly readable.