<s>[INST] What is the expansion? I encounter the following formula in some textbook. However, I can not understand what the expansion in this formula is. Is there anyone giving some tips?
$$
\begin{align}
\alpha &= -\frac{iU}{2 \epsilon} \pm \sqrt{\frac{i\omega}{\epsilon} - \frac{U^{2}}{4 \epsilon^{2}}} \\
&= \frac{iU}{2 \epsilon} \pm \frac{iU}{2 \epsilon} \left [ 1 - \frac{2i\omega \epsilon}{U^{2}} + \frac{2 \omega^{2} \epsilon^{2}}{U^{4}} + O(\epsilon^{3}) \right ]
\end{align}
$$
Thank you!
 [/INST] Hint:
They factored out $\;-\dfrac{U^2}{4\varepsilon^2}=\biggl(\dfrac{iU}{2\varepsilon}\biggr)^2$ under the radical:
$$\sqrt{\frac{i\omega}{\epsilon} - \frac{U^{2}}{4 \epsilon^{2}}}=\sqrt{\frac{-U^2}{4\varepsilon^2}\biggl(1-\frac{4i\omega\varepsilon}{U^2}\biggr)}=\dfrac{iU}{2\varepsilon}\sqrt{1-\frac{4i\omega\varepsilon\mathstrut}{U^2}},$$
and applied the development of $\sqrt{1-x}$ at order $2$:
$$\sqrt{1-x}=1-\frac12 x-\frac18x^2+O(x^3).$$
</s>