Question

Where in mathematical formulas or studies do we find the transcendental number i^i?

Answer 1

complex numbers maybe..

Answer 2

I'm looking for a theorem or formula where it is seen, and used in the real world.

Answer 3

You're looking at a result of exponential/logarithm functions as further applied to Euler's Identity; that is, \(e^{i\theta}=\cos \theta + i \sin \theta\). How do we prove this (there is a classic maclaurin series expansion of e^(ix) proof, as well as one that is based on formulating a differential equation and using a uniqueness theorem)? For what value of theta does the formula equal i? what happens when we take that expression and take it to the power of i....?