I would like to understand the asymptotic behaviour of the Fourier coefficients of
power type functions 
$f(t) = |t|^{-\alpha} 1\_{[-\pi, \pi]} \qquad 0 < \alpha<1.$
I suppose this is a classic result that I am supposed to know which can be found in many books, but I do not know where to start reading. Can you give me a hint please?



Asymptotics of Fourier coefficients of power-type functions 
$$
\begin{aligned}
&\int\_0^\pi t^\beta e^{iyt}\frac{dt}{t}dt=y^{-\beta}\int\_0^{\pi y} t^{\beta} e^{it}\frac{dt}{t}= 
\cr 
&=y^{-\beta}e^{i\pi\beta/2}\left[\int\_0^{\pi y} t^\beta e^{-t}\frac{dt}{t}+O(t^{\beta-1})\right]=
y^{-\beta}e^{i\pi\beta/2}[\Gamma(\beta)+o(1)].
\end{aligned}
$$


Now just take the real part. If you want more terms, apply Laplace on the arc used for moving the interval of integration to the imaginary axis.


