Definition:Natural Logarithm/Complex/Principal Branch

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Definition

The principal branch of the complex natural logarithm is usually defined in one of two ways:

$\Ln \paren z = \ln \paren r + i \theta$ for $\theta \in \hointr 0 {2 \pi}$
$\Ln \paren z = \ln \paren r + i \theta$ for $\theta \in \hointl {-\pi} \pi$

It is important to specify which is in force during a particular exposition.


Notation

Note the capital-letter version of the name of the operator:

$\Ln$

which allows it to be distinguished from its multifunctional counterpart $\ln$.


The forms:

$\Log$
$\Log_e$

can also be found.


Also known as

Some sources refer to the principal branch as the principal value or principal-value, but it is often important to distinguish between the branch of a multifunction and the value of an element under such a mapping.


Sources