$p=-\sqrt{\frac{1-\left(qz\right)^{2}}{z^{4}}}$
$p=\sqrt{\frac{1-\left(qz\right)^{2}}{z^{4}}}\text{, }z\neq 0$

$q=-\sqrt{-\left(pz\right)^{2}+\frac{1}{z^{2}}}$
$q=\sqrt{-\left(pz\right)^{2}+\frac{1}{z^{2}}}\text{, }z\neq 0$

$p=\frac{|z|\sqrt{1-\left(qz\right)^{2}}}{z^{3}}$
$p=-\frac{|z|\sqrt{1-\left(qz\right)^{2}}}{z^{3}}\text{, }z\neq 0\text{ and }\left(|z|\leq \frac{1}{|q|}\text{ or }q=0\right)$

$q=\frac{\sqrt{1-p^{2}z^{4}}}{|z|}$
$q=-\frac{\sqrt{1-p^{2}z^{4}}}{|z|}\text{, }\left(p=0\text{ or }|z|\leq \frac{1}{\sqrt{|p|}}\right)\text{ and }z\neq 0$