I believe I found a formula for finding the square roots of any imaginary number. The formula is as follows: $\pm \sqrt{ai} = \pm \sqrt{\frac{a}{2}}i \pm \sqrt{\frac{a}{2}}$
and here is my proof:
$\pm \sqrt{ai} = \pm \sqrt{\frac{a}{2}}i \pm \sqrt{\frac{a}{2}} \\ \pm ai = (\pm \sqrt{\frac{a}{2}}i \pm \sqrt{\frac{a}{2}})^2 \\ \pm ai = (\pm \sqrt{\frac{a}{2}}i)^2 \pm 2\sqrt{\frac{a}{2}}\sqrt{\frac{a}{2}}i + \sqrt{\pm \frac{a}{2}}^2 \\ \pm ai = \mp \frac{a}{2} \pm 2 \times \frac{a}{2}i \pm \frac{a}{2} \\ \pm ai = \pm \frac{2a}{2}i \\ \pm ai = \pm ai$
Is this correct? if so, has it been discovered prior to my finding? I can't find it anywhere on google.