Solve u=[varphi(x)]^varphi(y)

$$u = [ \varphi ( x ) ] ^ { \varphi ( y ) }$$

Solve for u (complex solution)

$$u=\left(x\phi \right)^{y\phi }$$

Solve for x (complex solution)

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

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

Solve for u

$$u=\left(x\phi \right)^{y\phi }$$

$$\left(\phi <0\text{ and }x<0\right)\text{ or }\left(\phi >0\text{ and }x>0\right)\text{ or }\left(x=0\text{ and }\phi >0\text{ and }y>0\right)\text{ or }\left(x=0\text{ and }\phi <0\text{ and }y<0\right)\text{ or }\left(\phi <0\text{ and }x>0\text{ and }Denominator(y\phi )\text{bmod}2=1\right)\text{ or }\left(\phi >0\text{ and }x<0\text{ and }Denominator(y\phi )\text{bmod}2=1\right)$$