Complex Power Functions Examples 1

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 noted that $a^b$ may be single-valued (if $b$ is an integer); has finitely many values (if $b$ is a rational number); or has infinitely many values (if $b$ is irrational or if $b$ is nonreal).

We will now look at some example problems regarding complex power functions.

Example 1

Find all possible values for $2^i$.

Consider the branch of the logarithm function $[0, 2\pi)$. Then for this branch:

(2)

\begin{align} \quad 2^i = e^{i \log 2} =e^{i \left [ \log \mid 2 \mid + i \arg (2) \right ]} = e^{\log(2) i} \end{align}

Since $b = 0 + i$ is a nonreal complex number we know that $2^i$ takes on infinitely many values that differ by a multiple of $e^{2kb\pi i} = e^{2k\pi i^2} = e^{-2k\pi}$, $k \in \mathbb{Z}$. Therefore all possible values for $2^i$ are given for $k \in \mathbb{Z}$ by:

(3)

\begin{align} \quad 2^i = e^{\log(2) i} e^{-2k \pi} \end{align}

Example 2

Prove that if $a, b \in \mathbb{C}$, $a \neq 0$, $b \in \mathbb{R}$ then $\mid a^b \mid = \mid a \mid^b$.

We have that:

(4)

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

The second last equality holds because $b \in \mathbb{R}$.

Example 3

Prove or disprove the following statement: If $a, b \in \mathbb{C}$, $a \neq 0$ then $\mid a^b \mid = \mid a \mid^{\mid b \mid}$.

This statement is false in general. Take $a = 2$ and $b = -3$. Then:

(5)

\begin{align} \quad \mid 2^{-3} \mid = \frac{1}{8} \end{align}

And:

(6)

\begin{align} \quad \mid 2 \mid^{\mid -3 \mid} = 2^3 = 8 \end{align}

Clearly $\displaystyle{\mid 2^{-3} \mid \neq \mid 2 \mid^{\mid -3 \mid}}$.