Complex Power/Examples/2^i

Example of Complex Power

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


Proof

\(\ds 2^i\) \(=\) \(\ds \map \exp {i \ln 2}\) Definition of Complex Power
\(\ds \) \(=\) \(\ds \map \cos {\ln 2} + i \map \sin {\ln 2}\) De Moivre's Formula

$\blacksquare$


Sources

  • 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $6 \ \text{(iii)}$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.