Notes - Complex Analysis MT23, Logarithms

Let $D = \mathbb C \backslash \{x \in \mathbb R : x \le 0\}$. Then $\text{Log} : D \to \mathbb C$ is defined as follows:

\[\text{Log}(z) = \log |z| + i\arg(z)\]

where $\arg(z) \in (-\pi, \pi]$.

The argument in the range $(-\pi, \pi]$.

Just check the definition: for small $h \ne 0$, $\text{Log}(a + h) \ne \text{Log}(a)$ and

\[\frac{\text{Log}(a + h) - \text{Log}(a)}{h} = \frac{\text{Log}(a + h) - \text{Log}(a)}{\exp(\text{Log}(a + h)) - \exp(\text{Log}(a))}\]

Then

\[\lim_{h \to 0} \frac{\exp(\text{Log}(a + h)) - \exp(\text{Log}(a))}{\text{Log}(a + h) - \text{Log}(a)} = \exp'(\text{Log}(a)) = a\]

since when $h \to 0$, $\text{Log}(a + h) - \text{Log}(a) \to 0$ by the continuity of $\text{Log}$. So the limit exists and in fact

\[\text{Log}'(a) = 1/a\]