Equivalence of Definitions of Complex Natural Logarithm
Theorem
The following definitions of the concept of Complex Natural Logarithm are equivalent:
Definition 1
Let $z = r e^{i \theta}$ be a complex number expressed in exponential form such that $z \ne 0$.
The complex natural logarithm of $z \in \C_{\ne 0}$ is the multifunction defined as:
- $\map \ln z := \set {\map \ln r + i \paren {\theta + 2 k \pi}: k \in \Z}$
where $\map \ln r$ is the natural logarithm of the (strictly) positive real number $r$.
Definition 2
Let $z \in \C_{\ne 0}$ be a non-zero complex number.
The complex natural logarithm of $z$ is the multifunction defined as:
- $\map \ln z := \set {w \in \C: e^w = z}$
Proof
Let $z = r e^{i \theta}$ such that $z \ne 0$.
Let $F = \set {\ln r + i \theta + 2 k \pi i: k \in \Z}$.
Let $G = \set {w \in \C: e^w = z}$.
We will demonstrate that $F = G$.
Definition 1 implies Definition 2
Let $w = x + i y$ such that $w \in F$.
Then:
- $x + i y = \ln r + i \theta + 2 k \pi i$
for some $k \in \Z$.
Equating real and imaginary parts:
- $x = \ln r$
- $y = \theta + 2 k \pi$
Then:
\(\ds e^w\) | \(=\) | \(\ds e^{x + i y}\) | Definition of $w$ | |||||||||||
\(\ds \) | \(=\) | \(\ds e^x e^{i y}\) | Exponential of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{\ln r} e^{i \paren {\theta + 2 k \pi} }\) | Definition of $w$ | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{\ln r} e^{i \theta}\) | Periodicity of Complex Exponential Function | |||||||||||
\(\ds \) | \(=\) | \(\ds r e^{i \theta}\) | Exponential of Natural Logarithm | |||||||||||
\(\ds \) | \(=\) | \(\ds z\) | Definition of $z$ |
Thus:
- $w \in G$
and so:
- $f \subseteq G$
$\Box$
Definition 2 implies Definition 1
Let $w \in G$.
By definition:
- $\exists z \in \C_{\ne 0}: z = e^w = r e^{i \theta}$
Thus:
\(\ds r e^{i \theta}\) | \(=\) | \(\ds e^{\ln r} e^{i \theta}\) | Exponential of Natural Logarithm | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{\ln r + i \theta}\) | Exponential of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds e^w\) | Definition of $w$ |
Thus $w$ is of the form:
- $\ln r + i \theta + k \pi i$
where $k = 0$.
Therefore:
- $w \in F$
and so:
- $G \subseteq F$
$\Box$
So by definition of set equality:
- $F = G$
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.6$. The Logarithm: $(4.24)$
- 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