Whakaoti i te n=frac{(Z_{alpha}+Z_{beta})^2+3}{15ln[(1+r)/(1-r)]}

$$n = \frac { ( Z _ { \alpha } + Z _ { \beta } ) ^ { 2 } + 3 } { 15 \ln [ ( 1 + r ) / ( 1 - r ) ] }$$

Whakaoti mō Z_α (complex solution)

$$Z_{α}=-\left(Z_{β}+\sqrt{3\left(5\ln(\frac{r+1}{1-r})n-1\right)}\right)$$

$$Z_{α}=-\left(Z_{β}-\sqrt{3\left(5\ln(\frac{r+1}{1-r})n-1\right)}\right)\text{, }r\neq -1\text{ and }r\neq 1\text{ and }r\neq 0$$

Whakaoti mō Z_β (complex solution)

$$Z_{β}=-\left(Z_{α}+\sqrt{3\left(5\ln(\frac{r+1}{1-r})n-1\right)}\right)$$

$$Z_{β}=-\left(Z_{α}-\sqrt{3\left(5\ln(\frac{r+1}{1-r})n-1\right)}\right)\text{, }r\neq -1\text{ and }r\neq 1\text{ and }r\neq 0$$

Whakaoti mō Z_α

$$\left\{\begin{matrix}Z_{α}=-\left(Z_{β}+\sqrt{3\left(5\ln(\frac{r+1}{1-r})n-1\right)}\right)\text{; }Z_{α}=-\left(Z_{β}-\sqrt{3\left(5\ln(\frac{r+1}{1-r})n-1\right)}\right)\text{, }&\left(r>-1\text{ and }r<0\text{ and }n\leq \frac{1}{5\ln(\frac{r+1}{1-r})}\right)\text{ or }\left(r>0\text{ and }r<1\text{ and }n\geq \frac{1}{5\ln(\frac{r+1}{1-r})}\right)\\Z_{α}=-Z_{β}\text{, }&r\neq 0\text{ and }n=\frac{1}{5\ln(\frac{r+1}{1-r})}\text{ and }r>-1\text{ and }|r|<1\end{matrix}\right.$$

Whakaoti mō Z_β

$$\left\{\begin{matrix}Z_{β}=-\left(Z_{α}+\sqrt{3\left(5\ln(\frac{r+1}{1-r})n-1\right)}\right)\text{; }Z_{β}=-\left(Z_{α}-\sqrt{3\left(5\ln(\frac{r+1}{1-r})n-1\right)}\right)\text{, }&\left(r>-1\text{ and }r<0\text{ and }n\leq \frac{1}{5\ln(\frac{r+1}{1-r})}\right)\text{ or }\left(r>0\text{ and }r<1\text{ and }n\geq \frac{1}{5\ln(\frac{r+1}{1-r})}\right)\\Z_{β}=-Z_{α}\text{, }&r\neq 0\text{ and }n=\frac{1}{5\ln(\frac{r+1}{1-r})}\text{ and }r>-1\text{ and }|r|<1\end{matrix}\right.$$