All the values of the multi-valued complex funciton $1^{i}$ where $i = \sqrt{- 1}$ , are

purely umaginary

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real and non-negative

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on the unit circle

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equal in real and imaginary parts

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Solution

The correct option is B real and non-negative
Let $x = 1^{i} = [c o s 2 k π + i s i n 2 k π]^{i}$
Where $k$ is an integer.
$= [e^{i 2 k π}]^{i} = e^{- 2 k π}$
$∴$ All values are real and non-negative.

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Let $x = 1^{i} = [c o s 2 k π + i s i n 2 k π]^{i}$
Where $k$ is an integer.
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$∴$ All values are real and non-negative.