Solve E=z_{0}sqrt{pq/n}

$$E = z _ { 0 } \sqrt { \frac { p q } { n } }$$

Solve for E (complex solution)

$$E=z_{0}\sqrt{\frac{pq}{n}}$$

$$n\neq 0$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=\left(\frac{z_{0}}{E}\right)^{2}pq\text{, }&z_{0}\neq 0\text{ and }q\neq 0\text{ and }p\neq 0\text{ and }|arg(\sqrt{\frac{E^{2}}{z_{0}^{2}}}z_{0})-arg(E)|<\pi \text{ and }E\neq 0\\n\neq 0\text{, }&\left(z_{0}=0\text{ or }q=0\text{ or }p=0\right)\text{ and }E=0\end{matrix}\right.$$

Solve for E

$$E=z_{0}\sqrt{\frac{pq}{n}}$$

$$\left(q\leq 0\text{ and }p\leq 0\text{ and }n>0\right)\text{ or }\left(q\geq 0\text{ and }p\leq 0\text{ and }n<0\right)\text{ or }\left(p\geq 0\text{ and }q\leq 0\text{ and }n<0\right)\text{ or }\left(p\geq 0\text{ and }q\geq 0\text{ and }n>0\right)$$

Solve for n

$$\left\{\begin{matrix}n=\left(\frac{z_{0}}{E}\right)^{2}pq\text{, }&\left(z_{0}<0\text{ and }E<0\text{ and }q\neq 0\text{ and }p\neq 0\right)\text{ or }\left(z_{0}>0\text{ and }E>0\text{ and }q\neq 0\text{ and }p\neq 0\right)\\n\neq 0\text{, }&\left(q=0\text{ or }p=0\right)\text{ and }\left(z_{0}=0\text{ or }q=0\text{ or }p=0\right)\text{ and }E=0\\n<0\text{, }&\left(q\leq 0\text{ and }p\geq 0\text{ and }E=0\text{ and }z_{0}=0\right)\text{ or }\left(p\leq 0\text{ and }q\geq 0\text{ and }E=0\text{ and }z_{0}=0\right)\\n>0\text{, }&\left(q\geq 0\text{ and }p\geq 0\text{ and }E=0\text{ and }z_{0}=0\right)\text{ or }\left(p\leq 0\text{ and }q\leq 0\text{ and }E=0\text{ and }z_{0}=0\right)\end{matrix}\right.$$