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Analytic function proofs

1). Determine the set A such that
For r > 0 let A ={w, w = exp (1/z) where 0<|z|<r}.

2).Prove that there is no branch of the logarithm defined on G= C-{0}.
( C here is the complex plane).
( Hint: suppose such a branch exists and compare this with the principal branch).

I want detailed proofs and please prove every claim you make and justify every step.

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Solution Summary

This shows how to determine a given set and prove a statement regarding the branch of a logarithm.