Find the square root of the following complex number:
4i
Solution
\[\sqrt{z} = \pm \left[ \sqrt{\frac{\left| z \right| + Re\left( z \right)}{2}} + i\sqrt{\frac{\left| z \right| - Re\left( z \right)}{2}} \right] , \text { if Im }(z) > 0\]
\[\sqrt{z} = \pm \left[ \sqrt{\frac{\left| z \right| + Re\left( z \right)}{2}} - i\sqrt{\frac{\left| z \right| - Re\left( z \right)}{2}} \right] , \text { if Im }(z) < 0\]
\[ z = 0 + 4i, Re\left( z \right) = 0, \left| z \right| = 4\]
\[ \text { Here, Im }(z) > 0\]
\[ \therefore \sqrt{z} = \pm \left[ \sqrt{\frac{\left| z \right| + Re\left( z \right)}{2}} + i\sqrt{\frac{\left| z \right| - Re\left( z \right)}{2}} \right]\]
\[ = \pm \left[ \sqrt{\frac{4 + 0}{2}} + i\sqrt{\frac{4 - 0}{2}} \right]\]
\[ = \pm \left( \sqrt{2} + i\sqrt{2} \right)\]
\[ = \pm \sqrt{2}\left( 1 + i \right)\]
