Saddle-point asymptotics of $\int_{-\infty}^\infty \frac{e^{-t(z^2-1/4)}\cos tz}{1+z^2}\,dz$

Use the saddle point method to approximate \(f ( t ) = \int _ { - \infty } ^ { \infty } \frac { e ^ { - t ( z ^ { 2 } - 1 / 4 ) } \cos t z } { 1 + z ^ { 2 } } \mathrm { d } z\) for \(t \gg 1\)