Complex Analysis MT23, Logarithms
Flashcards
@Define $\mathrm{Log}(z)$.
Let $D = \mathbb C \backslash \{x \in \mathbb R : x \le 0\}$. Then $\mathrm{Log} : D \to \mathbb C$ is defined as follows:
\[\mathrm{Log}(z) = \log \vert z \vert + i\arg(z)\]where $\arg(z) \in (-\pi, \pi]$.
@Define the principal argument of $z$.
The argument in the range $(-\pi, \pi]$.
Proofs
@Prove that $\mathrm{Log}(z)$ is holomorphic on $\mathbb C \backslash \{x \in \mathbb R : x \le 0\}$ (assuming the standard branch of $\mathrm{Log}(z)$ defined using an argument in the range $(-\pi, \pi]$).
Just check the definition: for small $h \ne 0$, $\mathrm{Log}(a + h) \ne \mathrm{Log}(a)$ and
\[\frac{\mathrm{Log}(a + h) - \mathrm{Log}(a)}{h} = \frac{\mathrm{Log}(a + h) - \mathrm{Log}(a)}{\exp(\mathrm{Log}(a + h)) - \exp(\mathrm{Log}(a))} \]Then
\[\lim _ {h \to 0} \frac{\exp(\mathrm{Log}(a + h)) - \exp(\mathrm{Log}(a))}{\mathrm{Log}(a + h) - \mathrm{Log}(a)} = \exp'(\mathrm{Log}(a)) = a\]since when $h \to 0$, $\mathrm{Log}(a + h) - \mathrm{Log}(a) \to 0$ by the continuity of $\mathrm{Log}$. So the limit exists and in fact
\[\mathrm{Log}'(a) = 1/a\]