Complex power

The complex power $z^c$ for $z\ne 0$ and $c\in \mathbb{C}$ is defined via the Complex logarithm:

$z^c=e^{c\log z}.$

Since $\log z$ is multi-valued, $z^c$ is multi-valued too. For practical computation, use the principal branch:

$z^c=e^{c\mathrm{Log}z},z\in \mathbb{C}\setminus (-\infty ,0].$

Worked examples

$i^i$. $\mathrm{Log}(i)=i\pi /2$. So $i^i=e^{i\cdot i\pi /2}=e^{-\pi /2}\approx 0.208$. A real number!

Other branches: $i^i=e^{-\pi /2+2\pi n}$ for $n\in \mathbb{Z}$ — also all real, but at different magnitudes.

$(-1{)}^{1/2}$. $\mathrm{Log}(-1)=i\pi$. So $(-1{)}^{1/2}=e^{(1/2)(i\pi )}=e^{i\pi /2}=i$. The principal square root of $-1$ is $i$ (not $-i$).

The other branch: $(-1{)}^{1/2}=e^{i\pi /2+i\pi }=-i$.

Integer powers

When $c=n$ is a positive integer, the multi-valuedness collapses:

$z^n=e^{n\log z}=e^{n(\ln |z|+i\theta +2\pi ik)}=|z{|}^n{e}^{in\theta }\cdot e^{2\pi ink}=|z{|}^n{e}^{in\theta },$

since $e^{2\pi ink}=1$ for integer $n,k$. So $z^n$ via the formula matches the elementary definition $\underset{n}{\underbrace{z\cdot z\cdots z}}$, with a unique value regardless of branch.

Multi-valuedness only matters for non-integer powers.

Rational powers $z^{1/n}$

For $n$ a positive integer, $z^{1/n}$ has exactly $n$ distinct values:

$z^{1/n}=|z{|}^{1/n}\exp \left(i\frac{\theta +2\pi k}{n}\right),k=0,1,\dots ,n-1.$

Evenly spaced on the circle of radius $|z{|}^{1/n}$. These are the $n$ [[Roots of complex numbers|$n$-th roots]] of $z$.

The principal branch picks the one with $\arg$ in $(-\pi /n,\pi /n]$.

Differentiation

Within a single branch,

$\frac{d}{dz}{z}^c=c{z}^{c-1},$

with the same branch used for $z^c$ and $z^{c-1}$. Mixing branches in the formula gives wrong answers.

In context

• Roots of polynomials and characteristic equations: complex roots of $a^x=b$ via $x={\log }_{a}b=\log b/\log a$.
• Fractional integration / differentiation in signal processing uses $s^c$ for non-integer $c$ — multi-valued, branch-dependent.
• Transfer functions in fractional-order control involve $s^{\alpha }$ for non-integer $\alpha$ — must pick a branch consistently.