Complex Exponentiation¶

Definition¶

Let $z\neq 0$ and $w$ be complex numbers. The complex power $z^w$ is defined as $\mathrm{e}^{w \log z}$, where the complex logarithm $\log z$ is a multivalued function. The principal value of the complex exponentiation $z^w$ is $\mathrm{e}^{w \operatorname{Log}z}$, where $\operatorname{Log}z$ denotes the principal value of the complex logarithm.

Interpretation¶

For real numbers, $x^y=\mathrm{e}^{y\ln x}$ when $x>0$$0$. The complex power uses the same recipe: take a logarithm of the base, multiply it by the exponent $w$, and apply the complex exponential function. Because $\log z$ has infinitely many values, $z^w$ can have many values too.

For an integer exponent nothing new happens. The figure shows $z,z^2,z^3,z^4$ for $z=r\mathrm{e}^{\mathrm{i}\varphi}$ with $r=1.25$ and $\varphi=\pi/6$$\pi /6$: each further factor $z$ stretches the length by $1.25$ and turns by $\pi/6$, so the points move outwards along a spiral. This is De Moivre's formula: $z^n$ has modulus $r^n$ and argument $n\varphi$.

The values of $\log z$ differ by multiples of $2\pi\mathrm{i}$, so the values of $z^w$ differ by factors $\mathrm{e}^{2\pi\mathrm{i}kw}$ with $k\in\mathbb{Z}$$\mathbb{Z}$. How many different values arise depends only on $w$. For an integer $w$ all these factors equal $1$, and $z^w$ is single-valued. For $w=1/n$ with $n$ a positive integer there are $n$ factors $\mathrm{e}^{2\pi\mathrm{i}k/n}$, giving the $n$ different $n$-th roots of $z$. For an irrational or non-real $w$, the values are all different, so there are infinitely many, as for $\mathrm{i}^{\mathrm{i}}$ in the Examples.

The usual rules for powers need care. With principal values, $(z^a)^b=z^{ab}$ can fail: for $z=-1$$-1$, $a=2$ and $b=\tfrac{1}{2}$$\frac{1}{2}$, the left side is $1^{1/2}=1$$1$, but $z^{ab}=-1$$-1$. For positive real $z$ and real $w$, the principal value agrees with the real power.

Key Results¶

• Complex exponentiation is generally multivalued. For an integer exponent $m$, changing $\log z$ by $2\pi\mathrm{i} k$ multiplies the result by $\mathrm{e}^{2\pi\mathrm{i} km}=1$$1$, so $z^m$ is single-valued.

Examples¶

• The square roots of $\mathrm{i}$: for $k\in\mathbb{Z}$$\mathbb{Z}$,

  $$\mathrm{i}^{1/2}=\left(\mathrm{e}^{\mathrm{i}(\pi/2+2\pi k)}\right)^{1/2} =\mathrm{e}^{\mathrm{i}(\pi/4+\pi k)},$$

  so $\mathrm{i}^{1/2}$ has the two values $\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\mathrm{i}$ and $-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\mathrm{i}$$\frac{\sqrt{2}}{2}\mathrm{i}$.

• $\displaystyle \mathrm{i}^\mathrm{i} = \left(\mathrm{e}^{\mathrm{i}(\pi/2+2\pi k) }\right)^\mathrm{i} = \mathrm{e}^{-\pi/2-2k\pi}\in\mathbb{R},$ where $k\in\mathbb{Z}.$

Prerequisites¶

• Complex Logarithm
• Complex Number

Related Concepts¶

• De Moivre's Formula
• Complex Exponential Function
• Euler's Formula
• Euler's Identity

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: complex power, principal value of complex exponentiation Symbols: element of, integers, phi, pi, real numbers, square root

Last updated: 22 October 2026 19:04