Square roots of a complex number for JEE

Formula

\(\sqrt{a+ib}=\pm\left(\sqrt{\dfrac{r+a}{2}}+i\,\operatorname{sgn}(b)\sqrt{\dfrac{r-a}{2}}\right)\),\ \text{where }r=\sqrt{a^2+b^2}\)

Variables: \(a,b\in\mathbb R\)

Conditions: For \(b=0\), choose signs consistently

Explicit evaluation of complex square roots.