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What is i to the Power of i?

Asked by Oliver Varban, student, Earl Haig Secondary School on March 6, 1997:
I am interested in knowing what i to the power of i is.
The first question to address is what it means to raise one complex number to the power of another. There is a basic definition of what it means to raise e to a complex power, as described in the answer to an earlier question. Therefore, if z is any complex number for which e^z=i, (e^z)^i = e^(iz) is a possible value for i^i.

What are the possible values for z? Well, if we write z = a + bi, then e^z = e^(a+bi) = e^a e^(bi). By de Moivre's theorem (explained in the answer to an earlier question), e^(i b) = cos(b) + i sin(b), so e^z = e^a ( cos(b) + i sin(b) ). This expression equals i exactly when a=0, cos(b)=0, and sin(b)=1. This occurs when b = pi/2 + 2npi for some integer n, so the possible values of z are 0 + (pi/2 + 2npi)i.

Therefore, the values of i^i are

         zi     [(pi/2 + 2n pi)i]i    - (pi/2 + 2n pi)
        e   =  e                   = e

for any integer n.

Note that there is more than one value for i^i, just as 2 and -2 are both square roots of 4. (However, while the square roots of a number always have the same magnitude even if they differ in sign, the values of i^i have different magnitudes). The principal value of i^i would be e^(-pi/2)--the case where n=0.

It's also interesting to note that all these values of i^i are real numbers.

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