In Impulse Invariance Transformation, the mapping of points from the s-plane to the z-plane is given by the relation $z=e^{p_{i} T}=e^{s T},$ where $p_{i}$ are poles of analog filter and T is sampling frequency.

If $s=\sigma+j \Omega$ then $z=e^{(\sigma+j \Omega) T}$

Let $s=\sigma+j\left(\Omega+\frac{2 \pi k}{T}\right)$

$\therefore z=e^{\left[\sigma+j\left(\Omega+\frac{2 \pi k}{T}\right)\right] T}$ …