Principal square root of a product of complex numbers with positive real part

Question

Given $n$ complex numbers $z_i$ with $\Re z_i>0$, why is it that $$\prod_i\sqrt{z_i}=\sqrt{\prod_i z_i}?$$Numerically, this appears to be the case, however, I don't see an easy way to prove it.

Answer 1

This is not true.

If your branch is defined over all $z$, with $\Re z>0$, but IT IS NOT defined at $$ z_0=r\,\mathrm{e}^{i\vartheta}, \quad \vartheta\in (\pi/2,3\pi/2), $$ then $w=r^{1/3}\,\mathrm{e}^{i\vartheta/3}$ has positive real part, and your branch is not defined on $z_0=w\cdot w\cdot w$.