What is the value of iota raised to the power iota?

Question

P.S. I know it is exponential raised to the power -pi/2 , but I wanna know in simple terms (if there is some value) like ,pi, cube root of unity (w),or -w^2.

Answer 1

First of all, it's i, not iota.

i^i = e^(-π/2) as you say. That's the simplest form you're going to get for it, and simpler than the complex cube roots of unity, so I'm not sure why you'd expect something simpler still.

Answer 2

Do you means x^x when x is very small, close to zero?
If so, then let y = x^x
lny = x*lnx = lnx / (1/x)
We wish to find the limit of this as x goes to zero.

L'Hopital's rule,
limit = (1/x) / (-1/x^2) = -x
as x goes to zero, the limit of lny goes to zero.
So y goes to 1.

ie x^x goes to 1 as x tends to zero.
---
If you meant i^i where i = sqrt(-1)
then let y = i^i
Log(y) = i*Log(i)
i = e^(iπ/2)
so Log(i) = iπ/2
Log(y) = i^2 *π/2 = -π/2
So y = i^i = e^(-π/2)

Answer 3

this cannot be simplified further than e^-pi/2