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Question: What is i^{i}, expressed as a power of e?
Answer:
Since we want to express it as a power of e, we can start by saying that e^{x} = i^{i} for some x. So x = ln(i^{i}) x = i ln(i) Now, i is the square root of 1, so x = i ln(1^{1/2}) x = (i/2) ln(1) But as demonstrated in the sidebar, ln(1) = i So x = (i/2) i x = (i i ) /2 but i i is 1, so x = /2 so i^{i} = e^{/2} (Note that this is a real number! It's about 0.208.) 

Sidebar: What is ln(1)? Euler's Equation states that e^{i} + 1 = 0 Thus, e^{i} = 1 So i = ln(1) 

Thanks to Peter Hartman for showing me this elegant derivation.