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Define a branch of logarithm on the set $[0,\infty i) $ (the complex plane excluding the positive imaginary axis by $$\log\left(z=re^{i\theta}\right):=log(r)+i \theta,$$ where $\theta \in \left(-\frac{3\pi}{2},\frac{\pi}{2}\right).$

My question is, is there a way to express this branch in terms of the principal branch of the logarithm $\text{Log(z)}$? I believe there will be some kind of angle shift involved, but I am a bit confused on how to proceed.

Any hint/help will be very useful.

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Yes, this neophyte logarithm $\operatorname{nlog}(z)$ can be expressed in terms of the principal logarithm $\operatorname{Log}(z)$.

$$ \operatorname{nlog}(z) = \operatorname{Log}(iz)-i\frac{\pi}{2} $$

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