Solve m^{2}c^2-m^2v^2=m_0^2c^2

$$m ^ { 2 } c ^ { 2 } - m ^ { 2 } v ^ { 2 } = m _ { 0 } ^ { 2 } c ^ { 2 }$$

Solve for c (complex solution)

$$\left\{\begin{matrix}c=-\left(m^{2}-m_{0}^{2}\right)^{-\frac{1}{2}}mv\text{; }c=\left(m^{2}-m_{0}^{2}\right)^{-\frac{1}{2}}mv\text{, }&m\neq -m_{0}\text{ and }m\neq m_{0}\\c\in \mathrm{C}\text{, }&\left(m=0\text{ and }m_{0}=0\right)\text{ or }\left(v=0\text{ and }m=m_{0}\right)\text{ or }\left(v=0\text{ and }m=-m_{0}\right)\end{matrix}\right.$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=-\left(c^{2}-v^{2}\right)^{-\frac{1}{2}}cm_{0}\text{; }m=\left(c^{2}-v^{2}\right)^{-\frac{1}{2}}cm_{0}\text{, }&c\neq -v\text{ and }c\neq v\\m\in \mathrm{C}\text{, }&\left(m_{0}=0\text{ and }c=v\right)\text{ or }\left(m_{0}=0\text{ and }c=-v\right)\text{ or }\left(c=0\text{ and }v=0\right)\end{matrix}\right.$$

Solve for c

$$\left\{\begin{matrix}c=\frac{mv}{\sqrt{m^{2}-m_{0}^{2}}}\text{; }c=-\frac{mv}{\sqrt{m^{2}-m_{0}^{2}}}\text{, }&|m|>|m_{0}|\\c=0\text{, }&\left(m=0\text{ and }m_{0}\neq 0\right)\text{ or }\left(v=0\text{ and }|m|\neq |m_{0}|\right)\\c\in \mathrm{R}\text{, }&\left(m=0\text{ and }m_{0}=0\right)\text{ or }\left(v=0\text{ and }|m|=|m_{0}|\right)\end{matrix}\right.$$

Solve for m

$$\left\{\begin{matrix}m=\frac{cm_{0}}{\sqrt{c^{2}-v^{2}}}\text{; }m=-\frac{cm_{0}}{\sqrt{c^{2}-v^{2}}}\text{, }&|c|>|v|\\m=0\text{, }&\left(m_{0}=0\text{ and }|c|\neq |v|\right)\text{ or }\left(c=0\text{ and }v\neq 0\right)\\m\in \mathrm{R}\text{, }&\left(c=0\text{ and }v=0\right)\text{ or }\left(m_{0}=0\text{ and }|c|=|v|\right)\end{matrix}\right.$$