← Powers

Definitions

The general (multivalued) $a$-th power of $z$ is defined as $$\mathrm{Pow}(z,a)=\exp (a\mathrm{Ln}z),z\ne 0$$ where $\mathrm{Ln}$ is the general Logarithm.

Its principal value is defined as $$z^a=\exp (a\ln z),z\ne 0.$$ where $\ln$ is the principal value of the Logarithm.

Fundamental Properties

$$\mathrm{Pow}(z,n)=z^n,z\in \mathbb{Z}$$ $$z^n=\underset{n\text{ times}}{\underbrace{z\cdot z\cdot \cdots \cdot z}}=1/z^{-n},n\in {\mathbb{Z}}_{\ge \mathrm{𝟘}}$$ $$z^{a+b}=z^a{z}^b,z\ne 0$$ $${(ab)}^z=a^z{b}^z,\text{if }\arg a+\arg b\in (-\pi ,\pi ].$$ $$e^z=\exp z$$

Identities

For complex numbers $a,b,z$, real numbers $x,y$ and integer $n$ the following holds: $$\begin{array}{rl}{(z^a)}^n& =z^{an}\\ {(x^y)}^b& =x^{yb},& & \text{if }x>0\\ {(e^a)}^b& =e^{ab},& & \text{if }\mathrm{Im}a\in (-\pi ,\pi ]\\ \ln x^y& =y\ln x,& & \text{if }x>0.\end{array}$$ More generally, $$\ln z^a=a\ln z+2\pi ik,$$ with $k\in \mathbb{Z}$, such that $\mathrm{Im}(a\ln z+2\pi ik)\in (-\pi ,\pi ].$