system
Given the following post, respond as if you're commenting on Reddit:
human
Does i^i have a fixed, real value?

Given that you can use the identity  e^ix = cos(x) + isin(x) to prove that i^i is real (by letting x=pi/2 and raising both sides to the power of i) that would suggest that i^i = e^-pi/2, however since there are multiple values of x which could work just as well (5pi/2, for instance) and these would give different values, does i^i have a set real value or can it vary or is it just not as simple as I think it is?
gpt
Since we use the natural logarithm to calculate this exponential, and the natural logarithm is a multi-valued function, you should expect multiple solutions. i^i represents all of these values, so in a sense it's a set (e^(2pi k - pi/2) for all k in Z), but more correct is to think of these points as lying on a twisted surface that revolves about the origin infinite times in both directions. Think of an infinitely tall helical parking building - these solutions represent the same parking space on each floor.

Edit: Coming to think of it, the parking space analogy breaks down because the solutions are in different places from level to level, although always on the real line. The point is that each solution is on it's own level.