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例題 次の値を$a + bi$の形で表せ．

$\displaystyle 2^{i}$

解答 1. $a^{z} = e^{z\log{a}}$を用いる．


$\displaystyle 2^{i}$ $\displaystyle =$ $\displaystyle e^{i\log{(2)}} = e^{i [\log_{e}{\vert 2\vert} + i(0 + 2n\pi)]}$  
  $\displaystyle =$ $\displaystyle e^{i [\log_{e}{\vert 2\vert}}e^{-2n\pi}$  
  $\displaystyle =$ $\displaystyle e^{-2n\pi}[\cos{(\log_{e}{2})}+i \sin{(\log_{e}{2})}]$