Definition:Power (Algebra)/Complex Number

Definition

Let $z, k \in \C$ be complex numbers.

$z$ to the power of $k$ is defined as the multifunction:

$z^k := e^{k \ln \paren z}$

where $e^z$ is the exponential function and $\ln$ is the natural logarithm multifunction.

Principal Branch

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.

Examples

Example: $2^i$

$2^i = \map \cos {\ln 2} + i \map \sin {\ln 2}$

Example: $\paren {2 + i}^4$

$\paren {2 + i}^4 = -7 + 24 i$

Example: $\paren {1 + i \tan \paren {\dfrac {4 m + 1} {4 n} \pi} }^n$

For $m, n \in \Z$ such that $n \ne 0$:

$\paren {1 + i \map \tan {\dfrac {4 m + 1} {4 n} \pi} }^n = \paren {-1}^m \paren {\sec \dfrac {4 m + 1} {4 n} \pi}^n \paren {\dfrac {1 + i} {\sqrt 2} }$

Example: $\paren {1 + \sin \dfrac \pi 5 + i \cos \dfrac \pi 5}^5 + i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5$

$\paren {1 + \sin \dfrac \pi 5 + i \cos \dfrac \pi 5}^5 + i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5 = 0$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $9$