\[K_{\nu}\left(x\right)=\sec\left(\tfrac{1}{2}\nu\pi\right)\int_{0}^{\infty}\cos%
\left(x\sinh t\right)\cosh\left(\nu t\right)\,\mathrm{d}t=\csc\left(\tfrac{1}{%
2}\nu\pi\right)\int_{0}^{\infty}\sin\left(x\sinh t\right)\sinh\left(\nu t%
\right)\,\mathrm{d}t,\]