Solve x^x=y^z

$${ x }^{ x } = { y }^{ z }$$

Solve for y (complex solution)

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

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

Solve for y

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