Solve u=-z/sqrt{x^2+y^2}

$$u = - \frac { z } { \sqrt { x ^ { 2 } + y ^ { 2 } } }$$

Solve for u (complex solution)

$$u=-\left(x^{2}+y^{2}\right)^{-\frac{1}{2}}z$$

$$x\neq iy\text{ and }x\neq -iy$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\sqrt{-y^{2}+\left(\frac{z}{u}\right)^{2}}\text{; }x=\sqrt{-y^{2}+\left(\frac{z}{u}\right)^{2}}\text{, }&|arg(\sqrt{\left(\frac{z}{u}\right)^{2}}u)-arg(-z)|<\pi \text{ and }z\neq 0\text{ and }u\neq 0\\x\in \mathrm{C}\setminus -iy,iy\text{, }&u=0\text{ and }z=0\end{matrix}\right.$$

Solve for u

$$u=-\frac{z}{\sqrt{x^{2}+y^{2}}}$$

$$y\neq 0\text{ or }x\neq 0$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{z^{2}-\left(uy\right)^{2}}}{|u|}\text{; }x=-\frac{\sqrt{z^{2}-\left(uy\right)^{2}}}{|u|}\text{, }&\left(y\neq 0\text{ and }z<0\text{ and }u>0\text{ and }u\leq -\frac{z}{|y|}\right)\text{ or }\left(y\neq 0\text{ and }u<0\text{ and }z>0\text{ and }u\geq -\frac{z}{|y|}\right)\text{ or }\left(u<-\frac{z}{|y|}\text{ and }y\neq 0\text{ and }z<-u|y|\text{ and }z<0\text{ and }u>0\right)\text{ or }\left(u>-\frac{z}{|y|}\text{ and }y\neq 0\text{ and }z>-u|y|\text{ and }u<0\text{ and }z>0\right)\text{ or }\left(y=0\text{ and }u>0\text{ and }z<0\right)\text{ or }\left(y=0\text{ and }u<0\text{ and }z>0\right)\\x\in \mathrm{R}\text{, }&y\neq 0\text{ and }u=0\text{ and }z=0\\x\neq 0\text{, }&u=0\text{ and }z=0\end{matrix}\right.$$