Definition:Natural Logarithm/Complex/Definition 2
Definition
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:
- $\map \Ln z = \map \ln r + i \theta$ 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
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.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.6$. The Logarithm: $(4.23)$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $6$
- 1983: Ian Stewart and David Tall: Complex Analysis (The Hitchhiker's Guide to the Plane) ... (previous) ... (next): $0$ The origins of complex analysis, and a modern viewpoint: $1$. The origins of complex numbers