Complex Power/Examples

Examples of Complex Powers

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

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

Example: $\paren {2 \cis 50 \degrees}^6$

$\paren {2 \cis 50 \degrees}^6 = 32 - 32 \sqrt 3 i$