+9628 \(\text{For those who don't know what is imaginary number: }\\i=\sqrt{-1}\)
We know that \(i^i = e^{-\pi/2}\)
and
\(e^{i\pi} + 1 = 0\), the famous Euler's formula.
We can derive from the Euler's formula that \(\ln(-1) = -i\pi\)
Therefore \(\ln(i) = \ln((-1)^{1/2})=\dfrac{1}{2} \ln(-1)=-\dfrac{i\pi}{2}\)
We take the exponential of both sides:
\(i = e^{-(i\pi)/2}\)
Raise to the power i both sides:
\(i^i = e^{-(-\pi/2)}=\dfrac{1}{i^i}\)
So that\((i^i)^2 = 1\)
Therefore we get \(i^i = \pm 1\)(What is this?????????)
I am so confused, am I wrong?
