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 $2\pi i$.

• Proof: From the theorem presented on the Polar Representation with Multiplication of Complex Numbers page we know that $\mid zw \mid = \mid z \mid \mid w \mid$ and $\arg(zw) = \arg(z) + \arg(w) \pmod {2\pi}$, and so:

(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 $2\pi i$. $\blacksquare$