  The tunneling method for stationary black holes in the Hamilton-Jacobi
variant is reconsidered in the light of various critiques that have been moved
against. It is shown that once the tunneling trajectories have been correctly
identified the method isfree from internal inconsistencies, it is manifestly
covariant, it allows for the extension to spinning particles and it can even be
used without solving the Hamilton-Jacobi equation. These conclusions borrow
support on a simple analytic continuation of the classical action of a
pointlike particle, made possible by the unique assumption that it should be
analytic in complexified Schwarzschild or Kerr-Newman spacetimes. A more
general version of the Parikh-Wilczek method will also be proposed along these
lines.
