http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/

What I don't get is the place where i

^{i}is mentioned. True, the exponent turns real with a change of base, but what happened to the intuitive understanding now? We normally obtain positive real numbers by accelerating 1 forward and transforming it and we apply that transformation as many times as the exponent says. If the exponent is negative, we transform it backwards, if it is fractional, we transform it fractionally and if it is imaginary, we transform it sideways and rotate it. That's fine. But how about raising i to the exponent? We get i be rotating 1 to the left by pi/2 radians. Thus for real exponents we have to rotate accordingly and use de Moivre's theorem

(cos x + i sin x)

^{n}= cos nx + i sin nx

But what kind of transformation does i

^{i}represent?

And is it true that 1

^{1 - (i ln 2)/2pi}= 2? I got this by taking 1 as e

^{2i.pi}and 2 as e

^{ln 2 + 2i.pi}. Am I allowed to take some non-principal value and transform it like this or will that result in a fallacy like -1 = i

^{2}= i*i = -1

^{1/2}*-1

^{1/2}= (-1*-1)

^{1/2}= 1

^{1/2}= 1?

There is much that I have understood about complex exponentiation, though I feel that there is too much that I have not understood.