Solve F_{R}=sqrt{Fx^2+Fy^2}

$$F _ { R } = \sqrt { F x ^ { 2 } + F y ^ { 2 } }$$

Solve for F (complex solution)

$$\left\{\begin{matrix}F=\frac{F_{R}^{2}}{x^{2}+y^{2}}\text{, }&x\neq iy\text{ and }x\neq -iy\text{ and }\left(F_{R}=0\text{ or }arg(F_{R})<\pi \right)\\F\in \mathrm{C}\text{, }&\left(x=-iy\text{ or }x=iy\right)\text{ and }F_{R}=0\end{matrix}\right.$$

Solve for F_R (complex solution)

$$F_{R}=\sqrt{F\left(x^{2}+y^{2}\right)}$$

Solve for F

$$\left\{\begin{matrix}F=\frac{F_{R}^{2}}{x^{2}+y^{2}}\text{, }&\left(y\neq 0\text{ or }x\neq 0\right)\text{ and }F_{R}\geq 0\\F\in \mathrm{R}\text{, }&F_{R}=0\text{ and }x=0\text{ and }y=0\end{matrix}\right.$$

Solve for F_R

$$F_{R}=\sqrt{F\left(x^{2}+y^{2}\right)}$$

$$\left(y=0\text{ and }x=0\right)\text{ or }F\geq 0$$