What is i^i?

Answer geometrically by rotating an imaginary number or algebraically by an identity

This is the first homework question in my Signals and Systems class and it was a bit of a mind bender.

One of the rationalizations for this is that raising a number to a complex power is equivalent to a scaling (the real component) and rotation (the imaginary component). Using Euler's Identity, then exponentiation by the base imaginary unit is equivalent to rotation by $90\degree$. The base number can be complex, not just real.

So if the base number in this case is $i$, then it starts on the imaginary axis and rotates onto the real axis.

$$i^i = {\left( e^{\frac{i\pi}{2}}\right)}^i = e^{\frac{i^2\pi}{2}} = e^{-\frac{\pi}{2}} = \boxed{0.2087...}$$

Another method is by DeMoivre's Identity, $e^{ix} = \cos(x) + i\sin(x)$, so when $x = \frac{\pi}{2}$, the identity becomes $e^\frac{i\pi}{2} = i$. Then, the same algebra as previous follows to the same result.