If is a complex number in polar coordinates, with say , then one of its square-roots is . But what if one wants to avoid using the exponential function?
This trick was related to me by my colleague Bob Burckel.
Draw the parallelogram generated by and draw its diagonal through . In complex notation that’s simply . By simple geometry, the angle that forms with the positive -axis is half the angle that forms. So it will be enough to renormalize by dividing by and multiplying by and get
One can then check computationally that . Try it, it’s actually not entirely straight-forward.