The imaginary tetration unit? ssroot of -1

[JmsNxn]

I was just wondering if anywhere anyone ever looked up a number such that \( \omega^\omega = -1 \), or \( \omega = \text{SuperSquareRoot}(-1) \)?

Is there a representation of \( \omega \) using complex numbers?

I tried to work it out with the lambert W function but I'm not too good with it.

\( \ln(\omega)\cdot \omega = \pi \cdot i \)

I guess technically, there could be a different omega that is defined by:
\( \ln(\omega)\cdot \omega = -\pi \cdot i \) and so on and so forth for all the possible values given by the multivalued nature of the logarithm.

I'm wondering what the principal value is, the one I first asked for.

[bo198214]

I was just wondering if anywhere anyone ever looked up a number such that \( \omega^\omega = -1 \)?

\( (-1)^{-1}=-1 \)?

[JmsNxn]

I was just wondering if anywhere anyone ever looked up a number such that \( \omega^\omega = -1 \)?

\( (-1)^{-1}=-1 \)?

Ohhhhh my god! How did I miss that!?

I guess this kind of makes the square root of negative one more unique in my eyes.


however, there's still

\( ^ff = -1 \)