I was just wondering if the rules of exponents still applied to imaginary and complex numbers, like if $(2^4)^i=2^{4i}$ or not and if $(4^i)^i=4^{-1}$, etc
Can I raise a number to the power of i?
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complex-analysis
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1Why is $r$ not raised to the $i$ in these comments? – 2012-09-15
2 Answers
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I think some of the comments are in error.
If $z = r e^{i \theta}$ with $r$ a positive real (if $z = 0$, then $z^i = 0$), $z = r e^{i (\theta + 2 \pi k)}$ for any integer $k$, so $z^i = r^i e^{i^2 (\theta + 2 \pi k)} = e^{i \ln r} e^{- (\theta + 2 \pi k)} = (\cos( \ln r) + i \sin(\ln r)) e^{- (\theta + 2 \pi k)} $.
The principal value is usually the one with $k = 0$, but all the other values are possible.
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The exact same rules apply. A nice example of this is a typical Oxbridge interview question;
$ What\ is\ i^i\ ? $
$ i^i = {(e^{i\frac{\pi}{2}})} ^ i = e^{i^2\frac{\pi}{2}} = e^{-\frac{\pi}{2}} \approx 0.20788 $