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If I have a complex number say, $z = 2 + i$ and I wanted to raise it to a complex power, how can I make sense of it. That is, $z^w = (2+i)^w $ where $w$ is any complex number.

Actually I attempted expanding the $(2+i)^n$ where I thought of $n$ as an integer. As I expand in wolframAlpha online, in addition it also tells that it is periodic with $n = \frac{2 \pi i}{log(2+i)}$ where $log(x)$ is natural log. What does the periodicity mean here. Is it possible to figure out the integer values in the neighborhood of the periodicity as above?

Any help or suggested reading is greatly appreciated.

madeel
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  • Yes, it answer the first part. Represent $2+i$ in the exponential form and then take the complex power, in the end we can take the natural log. For the second part, I would like to understand what does this periodicity imply for the expansion $(2+i)^n$ where periodicity known to be $ n = \frac{2 \pi i}{log(2+i)}$. Can we figure out integer values in the neighborhood of this n? – madeel Aug 25 '22 at 09:27
  • $$(2+i)^{z+n}:=\exp(z\log(2+i)+n\log(2+i))=\exp(z\log(2+i)+2\pi i)=(2+i)^z$. Is that what you asked for? – FShrike Aug 25 '22 at 09:56
  • @Toby Mak: That other answer does not handle complex exponentiation in general, in particular it does not handle complex bases. – emacs drives me nuts Aug 25 '22 at 12:42

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