Solve 0=z^x-(z/z-varepsilon)

$$0 = z ^ { x } - ( z / z - \varepsilon )$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{2i\pi n_{1}}{\ln(z)}+\log_{z}\left(1-\epsilon \right)\text{, }n_{1}\in \mathrm{Z}\text{, }&\epsilon \neq 1\text{ and }z\neq 1\text{ and }z\neq 0\\x\in \mathrm{C}\text{, }&\epsilon =0\text{ and }z=1\end{matrix}\right.$$

Solve for z (complex solution)

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

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

Solve for x

$$\left\{\begin{matrix}x=\log_{z}\left(1-\epsilon \right)\text{, }&\epsilon <1\text{ and }z\neq 1\text{ and }z>0\\x\in \mathrm{R}\text{, }&\left(\epsilon =2\text{ and }Denominator(x)\text{bmod}2=1\text{ and }Numerator(x)\text{bmod}2=1\text{ and }z=-1\right)\text{ or }\left(\epsilon =0\text{ and }z=1\right)\end{matrix}\right.$$

Solve for z

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