The analytic conformal compactification of the Schwarzschild spacetime

Among the coordinates used to construct a conformal compactification of the Schwarzschild spacetime, none of them simultaneously extend smoothly both through an event horizon and beyond null infinity. To construct such coordinates, instead of starting with the Kruskal-Szekeres coordinates, we assume direct analytic transformation between Schwarzschild and compactified coordinates and determine their behavior on the event horizon and at null infinity. We then propose an example of such coordinates and illustrate the way they cover the conformally extended Schwarzschild spacetime as well as their suitability for numerical applications.