Asymptotics of Fourier coefficients of power-type functions  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?
 A: $$
\begin{aligned}
&\int_0^\pi t^\beta e^{iyt}\frac{dt}{t}dt=y^{-\beta}\int_0^{\pi y} t^{\beta} e^{it}\frac{dt}{t}= 
\cr 
&=y^{-\beta}e^{i\pi\beta/2}\left[\int_0^{\pi y} t^\beta e^{-t}\frac{dt}{t}+O(t^{\beta-1})\right]=
y^{-\beta}e^{i\pi\beta/2}[\Gamma(\beta)+o(1)].
\end{aligned}
$$
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.
A: This is a cosine transform of $t^{-a} \mathbb{1}[0, 1],$ which can be evaluated explicitly using the trusty mathematica, which gives:
$
\frac{t^{a+1} \,
   _1F_2\left(\frac{a}{2}+\frac{1}{2};\frac{1}{2},\frac{a}{2}+\frac{3}{2};-\frac{1}{4}
   k^2 t^2\right)}{a+1}
$
If you want the asymptotic of the above expression in $k$ (the transform variable), you can use the mathematica command Series[your_favorite_expression, {k, Infinity, 10}] (10 gives you the first ten terms in the power series, feel free to use your favorite integer). If you use 1 instead of 10 (for ease of typesetting), you get this.
(sorry, easier to use mathurl than do line breaks by hand).
