Solve r^mu+r^prime=V

$$r ^ { \mu } + r ^ { \prime } = V$$

Solve for V (complex solution)

$$V=r^{\mu }$$

Solve for r (complex solution)

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

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

Solve for V

$$V=r^{\mu }$$

$$\left(r<0\text{ and }Denominator(\mu )\text{bmod}2=1\right)\text{ or }\left(r=0\text{ and }\mu >0\right)\text{ or }r>0$$

Solve for r

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