Definition:Natural Logarithm/Complex/Definition 2

From ProofWiki
Jump to navigation Jump to search


Let $z \in \C_{\ne 0}$ be a non-zero complex number.

The complex natural logarithm of $z$ is the multifunction defined as:

$\map \ln z := \set {w \in \C: e^w = z}$

Principal Branch

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.


The notation for the natural logarithm function is misleadingly inconsistent throughout the literature. It is written variously as:

$\ln z$
$\log z$
$\Log z$
$\log_e z$

The first of these is commonly encountered, and is the preferred form on $\mathsf{Pr} \infty \mathsf{fWiki}$. However, many who consider themselves serious mathematicians believe this notation to be unsophisticated.

The second and third are ambiguous (it doesn't tell you which base it is the logarithm of).

While the fourth option is more verbose than the others, there is no confusion about exactly what is meant.

Also see

  • Results about logarithms can be found here.

Linguistic Note

The word logarithm comes from the Ancient Greek λόγος (lógos), meaning word or reason, and ἀριθμός (arithmós), meaning number.