# Complex Power Functions

Recall from The Complex Natural Logarithm Function page that if $z \in \mathbb{C} \setminus \{ 0 \}$ then we defined:

(1)Where $\arg(z)$ is specified in some branch of the complex natural logarithm function.

With this definition, we are able to define complex power functions. Let $a, b \in \mathbb{C}$ with $a \neq 0$. We would like a meaningful definition for $a^b$. If $a \neq 0$ then we know that $a = e^{\log a}$ for some branch of the logarithm function. Defining $a^b$ follows naturally.

Definition: If $a, b \in \mathbb{C}$ and $a \neq 0$ then the Complex Power Function $a^b$ is defined as $a^b = (e^{\log a})^b = e^{b \log a}$ with some branch for the complex logarithm function. |

For example, suppose that we want to compute $i^i$. Then from our definition:

(2)In general, since the complex natural logarithm function is a multi-valued function, the power function $a^b$ may take on multiple values. The following theorem tells us when multiple values of $a^b$ occur.

Theorem 1: Let $a, b \in \mathbb{C}$, $a \neq 0$.a) $a^b$ has a unique value if and only if $b \in \mathbb{Z}$.b) If $b \in \mathbb{Q}$, i.e., $\displaystyle{b = \frac{p}{q}}$ where $p, q \in \mathbb{Z}$ ($q \neq 0$), and $p$ and $q$ are in lowest terms, i.e., $\gcd (p, q) = 1$, then $a^b$ has $q$ distinct values which are the $q^{\mathrm{th}}$ roots of $a^p$.c) If $b \in \mathbb{R} \setminus \mathbb{Q}$ ($b$ is irrational) or if $b \in \mathbb{C} \setminus \mathbb{R}$ ($b$ is a nonreal complex number), then $a^b$ takes on infinitely many values. |

*As we will see in the proof below, the various values of $a^b$ differ only by factors of $e^{2kb\pi i}$ where $k \in \mathbb{Z}$ is determined by the choice of branch for the logarithm function.*

**Proof)**Consider the complex natural logarithm function $\log z$ with a chosen branch, say $[0, 2\pi)$. Then:

- For any other branch of the complex natural logarithm function we would have for some $k \in \mathbb{Z}$:

- These values different only by the factor $e^{2kb \pi i}$ where $k \in \mathbb{Z}$ is determined by the choice of branch for the logarithm function. If this value remains the same as $k$ varies then $a^b$ has a single value. If this value differs as $k$ varies then $a^b$ will have multiple (possibly infinite) values. We cover these cases below:

**Proof of a)**Suppose that $b \in \mathbb{Z}$. Then $kb \in \mathbb{Z}$. So $2kb \pi i$ is an integer multiple of $2\pi i$. But from one of the theorems on the Properties of the Complex Exponential Function page we know that $2kb \pi i = 1$ if and only if $kb \in \mathbb{Z}$ (and $kb \in \mathbb{Z}$ if and only if $b \in \mathbb{Z}$. Therefore $a^b$ has a unique value if and only if $b \in \mathbb{Z}$. $\blacksquare$

**Proof of b)**If $b \in \mathbb{Q}$ and $\displaystyle{b = \frac{p}{q}}$ in lowest terms, then we know by the theorem on the nth Roots of Complex Numbers page that $a^p$ has $q$ many $q^{\mathrm{th}}$ roots, i.e., there exists $q$ values for $a^b = a^{p/q}$. $\blacksquare$

**Proof of c)**Suppose that $b$ is irrational and that:

- Then:

- But from the result mentioned in part (a), this happens if and only if $b(k - j) \in \mathbb{Z}$. Since $b$ is irrational, $b(k - j)$ is never an integer - a contradiction. Therefore the equality at $(*)$ never happens. In other words, $a^b$ takes on infinitely many values.

- Now suppose that $b$ is a nonreal complex number. Say $b = x + yi$ with $y \neq 0$. Then:

- $e^{-2ky \pi}$ takes on multiple values as $k$ varies (by choice of the branch for the logarithm function), so $a^b$ takes on infinitely many values.