Behavior of the Fourier coefficients at $\infty$ for a particular function $f$

Question

• I know that Fourier coefficients $c_{k}$ tend to zero for $k \to \infty$, given some necessary conditions on the function.
• I am trying to prove that given $\mathrm{f}(x) \equiv \left\vert 1 - \mathrm{e}^{-\mathrm{i}x}\right\vert^{-2d} \,\mathrm{h}(x)$, with $x \in [-\pi, \pi]$, $d \in (-1/2,1/2) \setminus \{0\}$ and $\mathrm{h}$ even, positive and continuous on $[-\pi,\pi]$, and differentiable on $[-\pi,\pi] \setminus \{0\}$.
• There are other hypothesis but I don't think they are necessary for this ( I mean, there is other stuff going on later which requires further hypothesis, but I am not interested in that ).
• I am trying to prove that, for $k \to \infty$, $c_k \approx k^{2d - 1}$. I tried integration by parts, but it didn't seem to work.