Root of power i from i

Let's calculate the root of power \(i\) from \(i\); \(\sqrt[i]{i} = ?\). For doing this, I will use Euler, or the polar form of complex numbers, and will use the fact that \(i = e^{i(\frac{\pi}{2} + 2\pi n)}\):

\[\sqrt[i]{i} = i^{\frac{1}{i}} = i^{\frac{i}{i\times i}} = i^{-i} = e^{i\times(\frac{\pi}{2} + 2\pi n)\times(-i)} = e^{\frac{\pi}{2} + 2\pi n}\]

Therefore:

\[\sqrt[i]{i} = e^{\frac{\pi}{2} + 2\pi n}\]

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Published: 2023-05-10 01:31:10
Updated: 2023-05-10 01:36:59