Definition:Natural Logarithm/Complex/Principal Branch

From ProofWiki
Jump to navigation Jump to search


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

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

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


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


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

The forms:


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.