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I'm trying to understand the technical basis of singular foliations for pseudo-Anosov diffeomorphisms, and I've hit a bit of a strange calculation I'm having a hard time verifying/unpacking. In Fathi, Laudenbach and Poénaru, they state without explicitly calculation: $$ \mathrm{Im}\sqrt{z^k dz^2} = r^{k/2} \left(r\cos\left(\frac{k+2}2 \theta\right) \, d\theta + \sin\left(\frac{k+2}2\theta\right) \, dr\right) $$ and I'm having a hard time showing this explicitly. I'm mostly confused by what they mean by taking a square root of the symmetric 2-form $dz^2$. I thought this would be $\mathrm{Im}\left(\phi^* z^k dz \right)$, with $\phi : \mathbb C \to \mathbb C$ being a square root branch, but I'm having trouble showing this explicitly. Is anyone familiar with this notation/calculation?

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