Perturbative calculation of quasi-normal modes of arbitrary spin in Schwarzschild spacetime

We calculate analytically the asymptotic form of quasi-normal modes of perturbations of arbitrary spin of a Schwarzschild black hole including first-order corrections. We use the Teukolsky equation which applies to both bosonic and fermionic modes. Remarkably, we arrive at explicit expressions which coincide with those derived using the Regge-Wheeler equation for integer spin. Our zeroth-order expressions agree with the results of WKB analysis. In the case of Dirac fermions, our results are in good agreement with numerical data.