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Show that the function $Log(-z) + i\pi$ is a branch of $logz$ analytic in the domain $D_0$ consisting of all points in the plane except those on the nonnegative real axis.

I know that $Log(z)$ is analytic except for a branch cut on the negative real axis, thus that will mean Log(-z) is analytic except for a branch cut on the positive real axis, but then how can I finish the proof more formally?

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Since you've already shown that $Log(-z) +i\pi$ is analytic in the desired region, the only thing left to show is that it is a branch of $\log$, i.e., to show that $\exp(Log(-z) + i\pi) = z$.

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I see now. Thanks a lot Ted. –  Q.matin Jan 28 '13 at 5:18

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