Definition:Power (Algebra)/Complex Number

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Definition

Let $z, k \in \C$ be any 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.


Examples

Example: $2^i$

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


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 $n \ne 0$:

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


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