解決W=a^t |Microsoft數學求解器

$$W = a ^ { t }$$

解 a (復數求解)

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

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

解 W

$$W=a^{t}$$

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

解 a

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