Solve pi=pi-beta/omega_{nsqrt{1-varepsilon^2}}

$$\pi = \frac { \pi - \beta } { \omega _ { n } \sqrt { 1 - \varepsilon ^ { 2 } } }$$

Solve for β (complex solution)

$$\beta =\pi \left(-\omega _{n}\sqrt{1-\epsilon ^{2}}+1\right)$$

$$\omega _{n}\neq 0\text{ and }\epsilon \neq -1\text{ and }\epsilon \neq 1$$

Solve for ε (complex solution)

$$\epsilon =-\frac{\sqrt{-\beta ^{2}+2\pi \beta +\left(\pi \omega _{n}\right)^{2}-\pi ^{2}}}{\pi \omega _{n}}$$

$$\epsilon =\frac{\sqrt{-\beta ^{2}+2\pi \beta +\left(\pi \omega _{n}\right)^{2}-\pi ^{2}}}{\pi \omega _{n}}\text{, }arg(\frac{\pi -\beta }{\omega _{n}})<\pi \text{ and }\beta \neq \pi \text{ and }\omega _{n}\neq 0$$

Solve for β

$$\beta =\pi \left(-\omega _{n}\sqrt{1-\epsilon ^{2}}+1\right)$$

$$\omega _{n}\neq 0\text{ and }|\epsilon |<1$$

Solve for ε

$$\epsilon =\frac{\sqrt{-\beta ^{2}+2\pi \beta +\left(\pi \omega _{n}\right)^{2}-\pi ^{2}}}{\pi |\omega _{n}|}$$

$$\epsilon =-\frac{\sqrt{-\beta ^{2}+2\pi \beta +\left(\pi \omega _{n}\right)^{2}-\pi ^{2}}}{\pi |\omega _{n}|}\text{, }\omega _{n}\neq 0\text{ and }|\omega _{n}|\geq \frac{|\beta -\pi |}{\pi }\text{ and }\beta \leq \pi |\omega _{n}|+\pi \text{ and }\left(\beta \leq \pi -\pi \omega _{n}\text{ or }\beta <\pi \right)\text{ and }\beta \geq -\pi |\omega _{n}|+\pi \text{ and }\left(\beta >\pi \text{ or }\beta \geq \pi -\pi \omega _{n}\right)\text{ and }\beta \neq \pi$$