Properties of the Complex Natural Logarithm Function

Properties of the Complex Natural Logarithm Function

Recall from The Complex Natural Logarithm Function page that the natural logarithm function defined for the branch $A_{y_0} = \{ z = x + yi \in \mathbb{C} : x \in \mathbb{R}, y_0 \leq y < y_0 + 2\pi \}$, $\log : \mathbb{C} \setminus \{ 0 \} \to A_{y_0} \}$ is defined for all $z \in \mathbb{C}$ by:

(1)
\begin{align} \quad \log (z) = \log \mid z \mid + i \arg (z) \end{align}

Where $y_0 \leq \arg (z) < y_0 + 2\pi$.

If we remove the restriction on $\arg (z)$ above, then $\log (z)$ is a well-defined multi-valued function on $\mathbb{C} \setminus \{ 0 \}$. We will now look at some properties of this general complex natural logarithm function.

 Proposition 1: Let $z, w \in \mathbb{C} \setminus \{ 0 \}$. Then $\log (zw) = \log (z) + \log (w)$ up to an integer multiple of $2\pi i$.
(2)
\begin{align} \quad \log(zw) &= \log \mid zw \mid + i \arg (zw) \\ &= \log \mid z \mid \mid w \mid + i \arg(z + w) \\ &= \log \mid z \mid + \log \mid w \mid + i [\arg (z) + \arg(w) \pmod {2\pi}] \\ &= \log \mid z \mid + i \arg(z) + \log \mid w \mid + i \arg (w) + 2k\pi i \\ &= \log (z) + \log (w) + 2k \pi i \end{align}
• For some $k \in \mathbb{Z}$. Therefore $\log (zw) = \log (z) + \log (w)$ up to an integer multiple of $2\pi i$. $\blacksquare$