  We compute the semiclassical decay rate for Kerr black hole by deriving a
one-way connection formula, relating the near horizon solution to the outgoing
solution at infinity. In particular, we discuss the relevance of the Stokes
phenomenon and show how it leads to a Boltzmann-like thermal weight factor by
making use of the Stokes diagrams. We also give the exact result for the
semiclassical greybody factor $e^{-2S}, $ where $S$ is the leading order WKB
action. We contrast our results with Maldacena and Strominger's work [Phys.
Rev. D 56, 4975 (1997)], where the emission spectrum for a rotating black hole
was computed locally via asymptotic matching. We find that the relative error
of semiclassical decay rate with respect to asymptotic matching formula
diminishes in the limit of large angular momentum, $l$, as expected. In this
limit, the action assumes a compact form: $2S \sim (2 l+1) (\text{Log} \left(
\frac{16 }{z_0}\right)-1)$, where $z_0$ is the cross ratio formed by the
critical points (zeroes) of the scattering potential.
