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.

The powers z, z squared, z cubed and z to the fourth of a complex number spiralling outwards, each turned by the same angle and stretched by the same factor.

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 $\varphi=\pi/6$ $π /6$ : 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 $n\varphi$ .

$z^{1/n}$ has $n$ values, evenly spaced on a circle of radius $|z|^{1/n}$ . The highlighted one is the principal value $\mathrm{e}^{\frac{1}{n}\operatorname{Log}z}$ .

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 $k\in\mathbb{Z}$ $Z$ . 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 $k\in\mathbb{Z}$ $Z$ ,

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

so $\mathrm{i}^{1/2}$ has the two values $\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\mathrm{i}$ and $-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\mathrm{i}$ $\frac{2}{2} i$ .
$\displaystyle \mathrm{i}^\mathrm{i} = \left(\mathrm{e}^{\mathrm{i}(\pi/2+2\pi k) }\right)^\mathrm{i} = \mathrm{e}^{-\pi/2-2k\pi}\in\mathbb{R},$ where $k\in\mathbb{Z}.$

Prerequisites¶

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