Question

Find the square root of the following complex number:

−i

Answer 1

\[\sqrt{z} = \pm \left[ \sqrt{\frac{\left| z \right| + Re\left( z \right)}{2}} + i\sqrt{\frac{\left| z \right| - Re\left( z \right)}{2}} \right] , \text { if Im }(z) > 0\]

\[\sqrt{z} = \pm \left[ \sqrt{\frac{\left| z \right| + Re\left( z \right)}{2}} - i\sqrt{\frac{\left| z \right| - Re\left( z \right)}{2}} \right] , \text { if Im }(z) < 0\]

\[z = - i, Re\left( z \right) = 0, \left| z \right| = 1\]

\[\text { Here, Im }(z) < 0 \]

\[ \therefore \sqrt{z} = \pm \left[ \sqrt{\frac{\left| z \right| + Re\left( z \right)}{2}} - i\sqrt{\frac{\left| z \right| - Re\left( z \right)}{2}} \right]\]

\[ = \pm \left[ \sqrt{\frac{1}{2}} - i\sqrt{\frac{1}{2}} \right]\]

\[ = \pm \frac{1}{\sqrt{2}}\left( 1 - i \right)\]