Definition:Natural Logarithm/Complex/Principal Branch

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


 * $\operatorname {Ln} \left({z}\right) = \ln \left({r}\right) + i \theta$ for $\theta \in \left[{0 \,.\,.\, \pi}\right)$
 * $\operatorname {Ln} \left({z}\right) = \ln \left({r}\right) + i \theta$ for $\theta \in \left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right]$

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:
 * $\operatorname {Ln}$

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

The forms:
 * $\operatorname {Log}$
 * $\operatorname {Log}_e$

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.