V^x=x^y lösen

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

Nach V auflösen (komplexe Lösung)

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

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

Nach V auflösen

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