If the $G$ transfer function of a die in a reactor can be represented using the plug flow with axial dispersion as:

$G(s)=$exp$\{\frac{Pe}{2}(1-\sqrt[\phantom{2}]{1+\frac{4s}{Pe}})\}$

It is requested to show that the variance is:

${}^2=\frac{2}{Pe}$

Note:

The definition needed for ${}^2$ can be found in the provided curse notes for chapter $1,$ section $1$ in page $20.$