Large-$x$ asymptotics of the Hankel function $H^{(1)}_\nu$

*The Henkel function of the first kind has the following integral representation: \(H _ { \nu } ^ { ( 1 ) } ( x ) = \frac { 1 } { \pi i } \int _ { 0 + i \epsilon } ^ { - \infty + i \epsilon } \exp \left[ \frac { x } { 2 } \left( z - \frac { 1 } { z } \right) \right] \frac { \mathrm { d } z } { z ^ { \nu + 1 } }\) . The i factor is present since we have a branch cut along the negative real axis. Find the asymptotic behavior as \(x \gg 1\)