KK-Bessel functions (awaiting review)

For $x\in\mathbb{R}_{>0}$ and $\nu\in\mathbb{C}$ we define the $K$-Bessel functions $K_\nu(x)=\frac{1}{2}\int_0^\infty t^{\nu-1}e^{-(x/2)(t+1/t)}dt.$