Laplace coefficients are Fourier coefficients used in Celestial mechanics calculations
$ b^n_s (\alpha) \equiv {1 \over \pi} \int_0^{2\pi} {\cos n \phi \over (1 - 2 \alpha \cos \phi + \alpha^2)^s} d \phi $
with $s = i + 1/2$ (a half integer) and $0<\alpha <1$.
The function $(1 - 2\alpha \cos \phi + \alpha^2)^{-s}$ is analytic (locally) so the Fourier coefficients decay rapidly. For large $n$ and $\alpha$ not small, I think that $b^n_{1/2} \sim c_1 e^{-c_2(1-\alpha)n}$ (with constants $c_1,c_2$). How do I show that this is true and how do I derive constants $c_1,c_2$? If this was a bad guess is there an exponential function that does approximate the coefficients at large $n$?