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I have taken some math classes recently, but I feel that they opened some new questions.

I am specifically unclear about complex numbers.

I have understood how they can be added, divided, multiplied and subtracted algebraically, and then represented on the Argand diagram (is this the same as the complex plane?), after being converted into the cos (x)+ i sin (x) form, through Euler's Formula (e^ix= cos(x) + i sin (x)).

We have also learned how to find roots of complex numbers with de Moivre's formula: (cos (x) + i sin (x))^n = cos (nx)+ i sin (nx).

I seem to have all the bits of information, but what I am missing is the understanding what happens when we rise a number to the power of i. How (and mostlywhere,) do we represent something like i^i, which according to wolframalpha has actually a numerical value of 0.2078 (4 d.p.) and which can be represented as e^-pi/2. The fact that the latter does not have i in the exponent, indicates to me that I should treat it as a normal numerical value, but I am not even sure if this can be represented on the real axis, if e does not have i in the exponent.

Can somebody please explain to me what rising a number to the power of i do to it in terms of representation on the Argand diagram, or how does one do it algebraically. (as I understand de Moivre's theorem only works for e as a base, but then I still don't understand what rising e^i actually means, apart from making it possible to represent it in the form of cos(x) +i sin (x))

Thank you

EDIT: ln(i)= i pi/2... it still does not make much sense...where does the pi/2 come from in this?

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# What happens when we rise a number to the power of i ?

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