Solve z=y^x^2

$$z = y ^ { x ^ { 2 } }$$

Solve for x (complex solution)

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

Solve for y (complex solution)

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

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

Solve for x

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

Solve for y

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