Solve N=N_0^t

$$N = N _ { 0 } ^ { t }$$

Solve for N_0 (complex solution)

$$N_{0}=e^{\frac{Im(t)arg(N)+iRe(t)arg(N)}{\left(Re(t)\right)^{2}+\left(Im(t)\right)^{2}}-\frac{2\pi n_{1}iRe(t)}{\left(Re(t)\right)^{2}+\left(Im(t)\right)^{2}}-\frac{2\pi n_{1}Im(t)}{\left(Re(t)\right)^{2}+\left(Im(t)\right)^{2}}}\left(|N|\right)^{\frac{Re(t)-iIm(t)}{\left(Re(t)\right)^{2}+\left(Im(t)\right)^{2}}}$$

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

Solve for N

$$N=N_{0}^{t}$$

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

Solve for N_0

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