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Question Number 195035 by mathlove last updated on 22/Jul/23

\sqrt{i} = ?

Answered by alephzero last updated on 22/Jul/23

\sqrt{i} = ?

{(\sqrt{i})}^{2} = i

e^{i π} = - 1

\sqrt{e^{i π}} = e^{i \frac{π}{2}} = i

\sqrt{e^{i \frac{π}{2}}} = e^{i \frac{π}{4}} = \sqrt{i}

e^{i x} = \cos x + i \sin x

e^{i \frac{π}{4}} = \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}

\Rightarrow \sqrt{i} = \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}

Commented by mathlove last updated on 23/Jul/23

t h a n k s s i r

Answered by Frix last updated on 22/Jul/23

i = e^{i \frac{π}{2}}

\sqrt{i} = i^{\frac{1}{2}} = e^{i \frac{π}{4}}

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