I Complex function, principal value notation

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The discussion centers on the interpretation of principal value notation in complex functions, specifically regarding the equations Log(z^2) = log([z]^2) and Log(z^2) = log([z^2]). It is clarified that Log returns the principal value regardless of the input, while log() does not indicate the principal value and is inherently multi-valued. The role of square brackets is questioned, with an emphasis that they cannot represent principal values for single-valued functions like z^2. The confusion arises from the multiple-valued nature of the log function, particularly in relation to the argument of complex numbers. Overall, the conversation highlights the complexities of principal values in logarithmic functions.
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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)?
 
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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.
 
FactChecker said:
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:

1698812854299.png


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.
 
@Hill , can you please explain the meaning of '[]'? Is it anything other than a placeholder?
 
WWGD said:
@Hill , can you please explain the meaning of '[]'? Is it anything other than a placeholder?
This is how it appears in the text:

1698814694007.png
 
IMO, regardless of whether ##[z^2]## has the principle argument, ##Arg(z^2)##, the function ##log [z^2]## is multiple valued.
 
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