Definition:Natural Logarithm/Complex/Principal Branch

Definition

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.

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

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.6$. The Logarithm
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $6$