If $$z = \frac{\pi }{4}{\left( {1 + i} \right)^4}\left( {\frac{{1 - \sqrt \pi i}}{{\sqrt \pi + i}} + \frac{{\sqrt \pi - i}}{{1 + \sqrt \pi i}}} \right),$$        then $$\left( {\frac{{\left| z \right|}}{{amp\left( z \right)}}} \right)$$  equals

Solution :
$$\eqalign{ & z = \frac{\pi }{4}{\left( {1 + i} \right)^4}\left( {\frac{{1 - \sqrt \pi i}}{{\sqrt \pi + i}} + \frac{{\sqrt \pi - i}}{{1 + \sqrt \pi i}}} \right) \cr & = \frac{\pi }{4}{\left( {1 + i} \right)^4}\left[ {\frac{{1 + \pi + \pi + 1}}{{\left( {\sqrt \pi + i} \right)\left( {1 + \sqrt \pi i} \right)}}} \right] = \frac{\pi }{4}{\left( {1 + i} \right)^4}\frac{2}{i} \cr & = \frac{\pi }{4}{\left( {2i} \right)^2}\frac{2}{i} = 2\pi i \cr & \therefore \left( {\frac{{\left| z \right|}}{{amp\left( z \right)}}} \right) = \frac{{2\pi }}{{\frac{\pi }{2}}} = 4 \cr} $$