Solve varepsilon=m^c^2

$$\varepsilon = m ^ { c ^ { 2 } }$$

Solve for c (complex solution)

$$\left\{\begin{matrix}c=-\ln(m)^{-\frac{1}{2}}\sqrt{\ln(\epsilon )+2\pi n_{1}i}\text{, }n_{1}\in \mathrm{Z}\text{; }c=\ln(m)^{-\frac{1}{2}}\sqrt{\ln(\epsilon )+2\pi n_{1}i}\text{, }n_{1}\in \mathrm{Z}\text{, }&\epsilon \neq 0\text{ and }m\neq 1\text{ and }m\neq 0\\c\in \mathrm{C}\text{, }&\left(m=0\text{ and }\epsilon =0\right)\text{ or }\left(m=1\text{ and }\epsilon =1\right)\end{matrix}\right.$$

Solve for m (complex solution)

$$m=e^{\frac{arg(\epsilon )Im(c^{2})+iarg(\epsilon )Re(c^{2})}{\left(\left(Re(c)\right)^{2}+\left(Im(c)\right)^{2}\right)^{2}}-\frac{2\pi n_{1}iRe(c^{2})}{\left(\left(Re(c)\right)^{2}+\left(Im(c)\right)^{2}\right)^{2}}-\frac{2\pi n_{1}Im(c^{2})}{\left(\left(Re(c)\right)^{2}+\left(Im(c)\right)^{2}\right)^{2}}}\left(|\epsilon |\right)^{\frac{Re(c^{2})-iIm(c^{2})}{\left(\left(Re(c)\right)^{2}+\left(Im(c)\right)^{2}\right)^{2}}}$$

$$n_{1}\in \mathrm{Z}$$

Solve for c

$$\left\{\begin{matrix}c=\sqrt{\log_{m}\left(\epsilon \right)}\text{; }c=-\sqrt{\log_{m}\left(\epsilon \right)}\text{, }&\left(\epsilon \geq 1\text{ and }m>1\right)\text{ or }\left(\epsilon \leq 1\text{ and }\epsilon >0\text{ and }m>0\text{ and }m<1\right)\\c\in \mathrm{R}\text{, }&\left(\epsilon =-1\text{ and }Denominator(c^{2})\text{bmod}2=1\text{ and }Numerator(c^{2})\text{bmod}2=1\text{ and }m=-1\right)\text{ or }\left(\epsilon =1\text{ and }m=1\right)\\c\neq 0\text{, }&m=0\text{ and }\epsilon =0\end{matrix}\right.$$

Solve for m

$$\left\{\begin{matrix}m=\epsilon ^{\frac{1}{c^{2}}}\text{, }&\left(Denominator(\frac{1}{c^{2}})\text{bmod}2=1\text{ and }c\neq 0\text{ and }Numerator(c^{2})\text{bmod}2=1\text{ and }Denominator(c^{2})\text{bmod}2=1\text{ and }\epsilon ^{\frac{1}{c^{2}}}\neq 0\text{ and }\epsilon <0\right)\text{ or }\left(c\neq 0\text{ and }\epsilon \geq 0\right)\\m=-\epsilon ^{\frac{1}{c^{2}}}\text{, }&\left(\epsilon ^{\frac{1}{c^{2}}}<0\text{ and }c\neq 0\text{ and }\epsilon >0\text{ and }Numerator(c^{2})\text{bmod}2=0\right)\text{ or }\left(Numerator(c^{2})\text{bmod}2=0\text{ and }c\neq 0\text{ and }\epsilon =0\right)\text{ or }\left(c\neq 0\text{ and }\epsilon >0\text{ and }Numerator(c^{2})\text{bmod}2=0\text{ and }Denominator(c^{2})\text{bmod}2=1\right)\text{ or }\left(Denominator(\frac{1}{c^{2}})\text{bmod}2=1\text{ and }c\neq 0\text{ and }Numerator(c^{2})\text{bmod}2=0\text{ and }Numerator(c^{2})\text{bmod}2=1\text{ and }Denominator(c^{2})\text{bmod}2=1\text{ and }\epsilon ^{\frac{1}{c^{2}}}\neq 0\text{ and }\epsilon <0\right)\\m\neq 0\text{, }&c=0\text{ and }\epsilon =1\end{matrix}\right.$$