Επίλυση x^y=y^y

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

Λύση ως προς x (complex solution)

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

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

Λύση ως προς x

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

Λύση ως προς y

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