Definition:Power (Algebra)/Complex Number/Principal Branch

Definition

The principal branch of a complex number raised to a complex power is defined as:

$z^k = e^{k \Ln z}$

where $\Ln z$ is the principal branch of the natural logarithm.

Positive Real Base

Let $t > 0$ be a real number and let $k$ be a complex number.

The principal branch of a positive real number raised to a complex power is defined as:

$t^k = e^{k \ln t}$

where $\ln$ is the natural logarithm of a positive real number.