27.2 The Complex Logarithm
Definition 27.2.1 (Branch of Logarithm).label Let $U \subset \complex$ be a connected open set with $0 \not\in U$ and $f \in C(U; \complex)$, then $f$ is a branch of the logarithm if for every $z \in U$, $z = \exp(f(z))$.
Lemma 27.2.2.label Let $U \subset \complex$ be a connected open set with $0 \not\in U$, and $f, g \in C(U; \complex)$ be two branches of the logarithm, then there exists $k \in \integer$ such that $f - g = 2\pi k i$.
Proof, [Proposition 2.19, Con78]. For each $x \in U$, there exists $k \in \integer$ such that $f(x) - g(x) = 2\pi k i$. Thus $f - g \in C(U; 2\pi i\integer)$. Since $U$ is connected, $(f - g)(U)$ must be a singleton. Therefore there exists $k \in \integer$ such that $f - g = 2\pi k i$.$\square$
Proposition 27.2.3.label Let $U \subset \complex$ be a connected open set with $0 \not\in U$, and $f \in C(U; \complex)$ be a branch of the logartihm, then $f$ is analytic.
Proof. By the Theorem 26.8.1.$\square$
Definition 27.2.4 (Principal Logarithm).label Let $U = \complex \setminus \bracs{z \in \real|z \le 0}$, then there exists a unique mapping $\ell: U \to \complex$ such that:
- (1)
$\ell$ is a branch of the complex logarithm.
- (2)
For each $re^{i\theta}\in U$, $\ell(r^{i\theta}) = \ln r + i\theta$.
The function $\ell$ is the principal logarithm on $U$.