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  1. Find an open connected set $G$ and two continuous functions $f$ and $g$ defined on $G$ such that $f(z)^2 g(z)^ 2 =1- z^2$ for all $z\in G$. Can you make $G$ maximal? Are $f$ and $g$ analytic?
  2. Give the principal branch of $\sqrt{1-z}$.
  3. Let $f: G > C$ and $g: G > C$ be branches of $z^a$ and $z^b$ respectively. Show that $fg$ is a branch of $z^{a+b}$ and $f/g$ is a branch of $z^{a-b}$. Suppose that $f(G) ⊆ G$ and $g(G) ⊆ G$ and prove that both $f\circ g$ and $g \circ f$ are branches of $ z^{ab}$.

These are the problem from Conway and i am completely stuck on these. Can anyone help me to solve these problem. Thank you.

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For #2 and #3, what is the definition of "a branch of $z^a$"? What is the definition of "principal branch"? – Greg Martin Nov 29 '12 at 9:31
For #1, why can't you just write $f(z) = \sqrt{1+z}$ and $g(z) = \sqrt{1-z}$? Then squaring both gives the difference of the two squares on the right. – Alfred Yerger Feb 19 at 5:30

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