# Choose branch cut of sqrt so that function is continuous?

I am willing to plot the following function

Plot3D[Im[Sqrt[Conjugate[Sum[w^k*Exp[(1/2)*(Conjugate[w^k]*(x + I*y) - w^k*(x - I*y))]*Sum[w^k*Exp[(-2^(-1))*(Conjugate[w^k]*(x + I*y) -w^k*(x - I*y))], {k, 0, 2}],{k, 0, 2}]]]], {x, -Pi, Pi}, {y, -Pi, Pi}]

The issue then is that Mathematica induces a branch-cut of the square root function so that the above function becomes discontinuous. Is there a way to plot this function so that the graph becomes continuous and smooth?

• Related: How to get smooth square root of complex valued function? However, that question had more to do with a function from $\mathbb{R} \to \mathbb{C}$ while this one appears to be $\mathbb{C} \to \mathbb{R}$. I suspect that will make things more difficult, if not impossible. Nov 8, 2021 at 0:15
• Also, you should provide a definition of w if you want people to be able to reproduce & modify your code. Nov 8, 2021 at 0:19
• Try the option ExclusionStyle-> Nov 8, 2021 at 7:11