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## Homework Statement

Choose a branch that is analytic in the circle |z-2|<1. Then analytically continue this branch along the curve indicated in Fig 5.18. Do the new functional values agree with the old?

[tex]a, 3z^{\frac{2}{3}}[/tex]

[tex]b, (e^z)^\frac{1}{3}[/tex]

Fig 5.18 is basically an ellipse like loop starting at z=2, going around the origin and returning to z=2.

## The Attempt at a Solution

Started with a, well the principle branch satisfies the requirements as the negative real axis isn't included in the circle |z-2| < 1.

And that's it. Not been able to follow this chapter very well and it feels like I'm not understanding more by reading it more. Not very example heavy as well. How do you go about solving this? They did something similar in an example applying the Monodromy theorem but they used a punctured plane as the domain and I cannot see how a loop can be continuously deformed if there is a hole in the middle. Assuming there's something I'm 100% misunderstanding, thus I'm very much at a loss on how to proceed as that's the only example available.