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e-z-1-3-i-z-




Question Number 48224 by gunawan last updated on 21/Nov/18
e^z =1−(√3)i  z=..
$${e}^{{z}} =\mathrm{1}−\sqrt{\mathrm{3}}{i} \\ $$$${z}=.. \\ $$
Answered by Smail last updated on 21/Nov/18
z=ln(1−(√3)i)=ln(2((1/2)−i((√3)/2)))  =ln2+ln(e^(−i(π/3)) )=ln2−i(π/3)
$${z}={ln}\left(\mathrm{1}−\sqrt{\mathrm{3}}{i}\right)={ln}\left(\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}}−{i}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right)\right) \\ $$$$={ln}\mathrm{2}+{ln}\left({e}^{−{i}\frac{\pi}{\mathrm{3}}} \right)={ln}\mathrm{2}−{i}\frac{\pi}{\mathrm{3}} \\ $$

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