\[\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}J_{\nu}\left((z^{2}-\zeta%
^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{\zeta\cos\theta}\cos%
\left(z\sin\theta-\nu\theta\right)\,\mathrm{d}\theta-\frac{\sin\left(\nu\pi%
\right)}{\pi}\int_{0}^{\infty}e^{-\zeta\cosh t-z\sinh t-\nu t}\,\mathrm{d}t,\]