I am trying to understand the proof of the Riemann-Roch-Hirzebruch theorem using the index theorem (Heat Kernel and Dirac operators, [BGV]), and at the end they say that since $$TM \otimes \mathbb{C} = T^{1,0}M \oplus T^{0,1}M, $$ with $M$ a Kähler manifold, then
$$ Â(TM) = det \left( \frac{R^+}{e^{\frac{R^+}{2}}-e^{\frac{-R^+}{2}}} \right), $$
with $R^+$ is the curvature of the bundle $T^{1,0}M$ and knowing that the defintion of the Â-genus is
$$ Â(TM) := det^{\frac{1}{2}} \left( \frac{\frac{R}{2}}{sinh \left( \frac{R}{2} \right) } \right), $$
with $R$ the Riemannian curvature of $M$.
But unfortunately, I do not see how that works. Any help would be nice, thank you.