Mistake in my complex exponentiation: where?

[nomadreid]

I am sure I am overlooking something elementary, but playing around with exponentiation (this is not an assignment), I seem to be making a mistake somewhere. Please don't send me a link for a more compact way of getting the correct result; I wish to know what my particular mistake is.

Suppose r,s∈ℂ, r= exp(a+bi) = e^a+bi, and s=c+di, so that
r^s =exp((a+bi)(c+di)) = exp((ac-bd)+(ad+bc)i).
(I am not using "r" as length.)

So far, very straightforward, but the problem comes in the interpretation that this result has a length exp(ac-bd) and angle (ad+bc) Radians.

What bothers me about this is that since r = (exp(a +(b+n2π)i) for n∈ℤ, the above result would come out with a length of
exp(ac-(b+n2π)d) = exp(ac-bd- 2ndπ)
and an angle of (ad+(b+n2π)c)=(ad+ bc+2ncπ):

this means that (with a few exceptions) the length will not be the same as the first result, and the angle will not be equivalent to the first result. But I would imagine that r^s should have a unique (modulo an angle of 2nπ) result, no?

 

[Jonathan Scott]

Complex numbers do not have unique logarithms, so powers are ambiguous. See the Wikipedia entry on Exponentiation, especially the section on Powers of complex numbers.

 

[nomadreid]

Thank you, Jonathan Scott. More interesting than I thought it would be. This encourages me to investigate (from the brief mention in the Wiki article) the possibility of defining r^s, for r∈ℂ, as a function on a Riemann surface. My mistakes lead me into interesting avenues.