Problem 1593 (difficulty: 6/10)

Let \(\displaystyle D\) be a simply connected domain that does not contain the origin.

(a) Show that \(\displaystyle 1/z\) has an antiderivative on \(\displaystyle D\).

(b) Show that if \(\displaystyle g'(z)=1/z\) on \(\displaystyle D\), then \(\displaystyle ze^{-g(z)}\) is constant.

(c) Show that \(\displaystyle \log z\) has a continuous branch on \(\displaystyle D\).

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