comAttitRize8

2022-07-20

True or false logarithmic branches
Say whether the following are true or false. Give a short proof.
1) $log\left(-z\right)+i\pi$ is a branch of the logarithmic function whose branch cut is the non-negative real axis
2)If $g\left(z\right)$ is a branch of the logarithmic function with domain D and $h\left(z\right)$ is a branch analytic in D, then there is an integer $m$ with $h\left(z\right)=g\left(z\right)+2m\pi i$
Thanks

autarhie6i

Expert

Hint for part $a$: if $z=1$, what happens?
Hint for part $b$: if $g$ and $h$ share the same domain, what do you know about the branch cuts in relation to each other? How does this relate to the integer multiples of $2\pi i$?

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