Describe the Riemann Surface for $z^2 = w + \frac{1}{w}$

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Describe the Riemann Surface for $z^2 = w + \frac{1}{w}$. Not sure if my process is correct. We can see $z = \frac{\sqrt{w^2+1}}{\sqrt{w}}$. So there are branch points at 0, i, -i. We have two layers, with two cuts. One from -i to i and one from 0 to infinity, where we glue the two surfaces together. Is this correct?