If-x-C-find-solution-of-3-i-2-e-ix-

Question Number 95897 by john santu last updated on 28/May/20

If x \in C . find solution of

3 + i \sqrt{2} = e^{i x}

Answered by bobhans last updated on 28/May/20

3 + i \sqrt{2} = e^{i x}

z = 3 + i \sqrt{2} \Rightarrow∣ z ∣ = \sqrt{9 + 2} = \sqrt{11}

\arg (z) = \tan^{- 1} (\frac{\sqrt{2}}{3}) = 0.44

\sqrt{11} e^{0.44 i} = e^{i x}

\ln (\sqrt{11}) + 0.44 i = i x

1.199 + 0.44 i = i x

x = \frac{1.199 + 0.44 i}{i} = 0.44 - 1.199 i

Answered by abdomathmax last updated on 28/May/20

we have ∣ 3 + i \sqrt{2} ∣ = \sqrt{11} \Rightarrow 3 + i \sqrt{2} = \sqrt{11} e^{iarctan (\frac{\sqrt{2}}{3})}

(e) \Rightarrow e^{ix} = \sqrt{11} e^{iarctan (\frac{\sqrt{2}}{3})} \Rightarrow ix = \ln (\sqrt{11}) + iarctan (\frac{\sqrt{2}}{3})

\Rightarrow x = - iln (\sqrt{11}) + \arctan (\frac{\sqrt{2}}{3})