# 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:

- $\map \Ln z = \map \ln r + i \theta$ for $\theta \in \hointr 0 {2 \pi}$
- $\map \Ln z = \map \ln r + i \theta$ 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: $6$