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i = e^(pi*i*(1/2 + 2*k))
i^(a+b*i)
= e^(pi*i*(1/2 + 2*k))^(a+b*i)
= e^(pi*(1/2 + 2k)*(-b+a*i))
= e^(-pi*b*(1/2+2k) + pi*i*a*(1/2+2k))
= e^(-pi*b*(1/2+2k)) * e^(pi*i*a*(1/2+2k))
If b is not 0, this is a set of points arranged in a spiral. The magnatude of each point is
e^(-pi*b*(1/2+2k))
while the angle is
e^(pi*i*a*(1/2+2k))
If a is rational, the result takes on finitely many angles. If a is irrational, the angles form a dense net. Ie, if b is 0 and a is irrational, then the set of roots is dense on the complex unit circle.
If b is not zero, then the set of roots spirals from 0 out to infinity.
This seems less than useful.
You could identify a particular root as the primary root.
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Last edited by JHVH : 10-29-4004 BC at 09:00 PM. Reason: Time for a rest.
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