# convexity and complex square root

by Jerry   Last Updated August 14, 2019 06:20 AM

I have a convex hull enumerated by vertices $${a_1,\cdots, a_n}$$ in the complex plane. I am performing the (complex)square root operation over these vertices. How would the image look like after this operation? whether the convexity is still preserved? Does it going to make a difference if we consider only the principal square root?

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A counterexample: The image of the segment $$[-i, i]$$ under the (principle value of the) square root is not convex.

It contains the points $$z_{1, 2} = \sqrt{\pm i} = \frac{1 \pm i}{2}$$, but not the point $$\frac 12 (z_1 + z_2) = \frac 12$$.

Martin R
August 14, 2019 05:43 AM

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