Fourier cosine transform and K-Bessel function I am looking for a reference on the link between the asymptotic expansion of a functions and its Fourier cosine transform asymptotics.
More precisely I would like to find a reference to have the expansion in zero and infinity of (for $Re(a)<\frac{1}{2}$):
$$F(y)= \int_0^{\infty} \frac{1}{\sqrt{x}} K_a(\frac{1}{x}) \cos(xy) dx$$
By integration by parts it is easy to show that for $n$ positive interger $F(y)= o(\frac{1}{y^n})$ for $y \to \infty$, but any reference to use the fact that $\frac{1}{\sqrt{x}} K_a(\frac{1}{x}) \sim_0 \sqrt{\frac{\pi}{2}}e^{-\frac{1}{x}}$ for $x \to 0$ and show that $F(y)$ decreases like $e^{-c \sqrt{y}}$ for $y\to0$?
And to show that $F(y) \sim k_1 y^{-\frac{1}{2}+a} + k_2 y^{-\frac{1}{2}-a}$ for $y \to 0$ using that $\frac{1}{\sqrt{x}} K_a(\frac{1}{x}) \sim c_1 x^{-\frac{1}{2}+a} + c_2 x^{-\frac{1}{2}-a}$ for $x \to \infty$.
I found only a reference to deduce the asymptotic expansion at infinity of a cosine transform depending of serie expansion in zero of initial function.(Here there is no serie expansion near zero as $\frac{1}{\sqrt{x}} K_a(\frac{1}{x}) \sim_0 \sqrt{\frac{\pi}{2}}e^{-\frac{1}{x}}$ as no serie expansion in zero...)

To obtain the asymptotic behaviour of this integral near $y=0$, the Mellin transform method can be used. One has
\begin{align}
\mathcal{M} \left[\cos \left(z\right),s\right]&=\Gamma (s)\cos\left( \frac{\pi}{2}s \right)\\
\mathcal{M} \left[x^{-1/2}K_a\left( x^{-1} \right),1-s\right]&=2^{-5/2+s}\Gamma (\frac{s}{2}-\frac{1}{4}+\frac{a}{2})\Gamma (\frac{s}{2}-\frac{1}{4}-\frac{a}{2})
\end{align}
The second expression is obtained by transformation of the well known Mellin transform of the Bessel function (multiplication by $x^{-1/2}$ and change $x\to x^{-1}$). The first identity is valid for $0<s<1$ and the second for $s>a+\tfrac{1}{2}$. It comes
\begin{equation}
F(y)=\frac{1}{2i\pi}\int_{c-i\infty}^{c+i\infty}x^{-s}\Gamma (s)\cos\left( \frac{\pi}{2}s \right)2^{-5/2+s}\Gamma (\frac{s}{2}-\frac{1}{4}+\frac{a}{2})\Gamma (\frac{s}{2}-\frac{1}{4}-\frac{a}{2})\,ds
\end{equation} 
where $s+\frac{1}{2}<s<1$.
To evaluate this integral, one may close the contour by a large semi-circle on the left side of the vertical line $s=c$. It gives the asymptotic expansion of the integral, as the poles are situated at $s=\frac{1}{2}+a-2n, \frac{1}{2}-a-2n,-2n$, with $n=0,1,2...$ Taking into account the first three contributions ($n=0$), one obtains
\begin{equation}
F(y)\sim K_1x^{-\tfrac{1}{2}-a}+K_1x^{-\tfrac{1}{2}+a}+K_3
\end{equation} 
Expressions for the constants are obtain from the calculus of the residues
\begin{align}
K_1&=2^{-a-\frac{1}{2}}\sqrt{\pi}\Gamma(2a)\left( \cos(\frac{\pi}{2}a)-\sin(\frac{\pi}{2}a)\right)\\
K_2&=2^{a-\frac{1}{2}}\sqrt{\pi}\Gamma(-2a)\left( \cos(\frac{\pi}{2}a)+\sin(\frac{\pi}{2}a)\right)\\
K_3&=2^{3/2}\frac{2\Gamma(\frac{3+2a}{4})\Gamma(\frac{3-2a}{4})}
{1-4a^2}
\end{align}
