Complex function, principal value notation

[Hill]

TL;DR Summary

Notations $Log$, $[]$

When a variable in $[\text { } ]$ means its principal value, $(-\pi,\pi]$, which is correct:
$Log(z^2)=log([z]^2)$ or $Log(z^2)=log([z^2])$ (both, neither)?

 

[FactChecker]

IMO, you are confused. It is the Log that returns the principle value, no matter what the input is. Also, regardless of what the input is, log() does not indicate the principle value. It is a multi-valued function.
$log( z) = ln |z| + i (Arg( z) + 2\pi k)$ for $k \in \mathbb I$.
So the right sides of your two alternative equations are multiple valued.

 

[Hill]

IMO, you are confused. It is the Log that returns the principle value, no matter what the input is. Also, regardless of what the input is, log() does not indicate the principle value. It is a multi-valued function.
$log( z) = ln |z| + i (Arg( z) + 2\pi k)$ for $k \in \mathbb I$.
So the right sides of your two alternative equations are multiple valued.

Thank you. This exercise is the source of my confusion:

What is a role of the square brackets in the first equation? They cannot mean principal values of $z^2$ and of $(-z)^2$ as these functions are single-valued.

 

[WWGD]

@Hill , can you please explain the meaning of '[]'? Is it anything other than a placeholder?

 

[Hill]

@Hill , can you please explain the meaning of '[]'? Is it anything other than a placeholder?

This is how it appears in the text:

 

[FactChecker]

IMO, regardless of whether $[z^2]$ has the principle argument, $Arg(z^2)$, the function $log [z^2]$ is multiple valued.