Riješi k^x=x^x

$$k ^ { x } = x ^ { x }$$

Riješite za k (complex solution)

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

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

Riješite za k

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

Riješite za x

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