  The Regge-Wheeler equation for black-hole gravitational waves is analyzed for
large negative imaginary frequencies, leading to a calculation of the cut
strength for waves outgoing to infinity. In the--limited--region of overlap,
the results agree well with numerical findings [Class. Quantum Grav._20_, L217
(2003)]. Requiring these waves to be outgoing into the horizon as well
subsequently yields an analytic formula for the highly damped Schwarzschild
quasinormal modes,_including_ the leading correction. Just as in the WKB
quantization of, e.g., the harmonic oscillator, solutions in different regions
of space have to be joined through a connection formula, valid near the
boundary between them where WKB breaks down. For the oscillator, this boundary
is given by the classical turning points; fascinatingly, the connection here
involves an expansion around the black-hole singularity r=0.
