Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
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?
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 $\exp(Log(-z) + i\pi) = z$.
