Properties of Complex Power Functions

Recall from the Complex Power Functions page that if $a, b \in \mathbb{C}$, $a \neq 0$ then the complex power function $a^b$ is defined as:

(1)

\begin{align} \quad a^b = e^{b \log a} \end{align}

We will now investigate some of the important properties of complex power functions.

Proposition 1: Let $a, b, c \in \mathbb{C}$. Then: a) If $a \neq 0$ and for a fixed branch of the logarithm function we have that $a^{b + c} = a^b \cdot a^c$. b) If $a, b \neq 0$ and if $\log (ab) = \log (a) + \log (b)$ then $(ab)^c = a^c \cdot b^c$.

For the proof of (b) it is important to note that the condition $\log (ab) = \log (a) + \log (b)$ means that appropriate branches are chosen so that this equality holds. On the Properties of the Complex Natural Logarithm Function page we saw that this equality holds up to an integer multiple of $2\pi i$ and equality can be forced by appropriate branch choices for the logarithm function.

• Proof of a) Fix a branch for the logarithm function. Then:

(2)

\begin{align} \quad a^{b + c} &= e^{(b + c) \log (a)} \\ &= e^{(b + c) \log (a)} \\ &= e^{(b + c) [\log \mid a \mid + i \arg (a)]} \\ &= e^{b[\log \mid a \mid + i \arg(a)] + c[\log \mid a \mid + i \arg(a)]} \\ &= e^{b[\log \mid a \mid + i \arg(a)]} \cdot e^{c[\log \mid a \mid + i \arg (a)]} \\ &= a^b \cdot a^c \quad \blacksquare \end{align}

• Proof of b) Suppose that $\log (ab) = \log(a) + \log(b)$. Then we have that:

(3)

\begin{align} \quad (ab)^c &= e^{c \log (ab)} \\ &= e^{c [\log (a) + \log(b)]} \\ &= e^{c \log (a) + c \log (b)} \\ &= e^{c \log (a)} \cdot e^{c \log (b)} \\ &= a^c \cdot b^c \quad \blacksquare \end{align}