Partial Laplace Transform Inversion

Question

I am trying to invert the function $$ F(s) = \frac{s^{\beta/2-1}}{ 2 \pi } K_0 ( y s^{\beta/2} ) \quad 0<\beta \leq 1 $$ I am using the Basset representation of the Bessel K function as $$ \frac{1}{2}\int_{-\infty }^{\infty }{\left. \frac{{{ e}^{ i {{s}^{\frac{\beta}{2}}} u z}}}{\sqrt{{{u}^{2}}+1}}du\right.} = \frac{1}{2} \int_{-\infty }^{\infty }{\left. {{ e}^{ i {{s}^{\frac{\beta}{2}}} z\, \sinh(y)}}d y\right.} $$ Is it possible to switch places of the integrals in the double integral representation: $$ \frac{1}{2} \oint_{Ha} ds {{s}^{\beta-1}}\, \int_{-\infty }^{\infty }{\left. {{ e}^{ i {{s}^{\frac{\beta}{2}}}\, \operatorname{sinh}(y) z+s t}}dy\right.} $$