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I was asked to find $(1+i)^i$, I don't know what to do when there is an imaginary component in the exponent.

since $1+i=\sqrt{2}e^{-\frac{1}{4}i \pi}$ then $(1+i)^i = \sqrt{2}^i e^{\frac{1}{4} \pi}$ but now we run into the same problem again, what is $\sqrt{2}^i$?

Oria Gruber
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1 Answers1

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If $x$ is any positive number then $x^i = e^{i\ln x} = \cos(\ln x) + i\sin(\ln x)$.

Simon S
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  • what about $x^{2i}$? do exponential rules work? can i say its $(x^2)^i$? – Oria Gruber Dec 22 '14 at 15:05
  • IT's kind a weird result. Because that means that $x^i$ is always on the unit circle. $|2^i| = |4^i|$ is counter intuitive. – Oria Gruber Dec 22 '14 at 15:07
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    Here's another way to think about it. $e^{i\theta}$ is on the unit circle and can also be written as $(e^\theta)^i$; $e^\theta$ can be made equal to any positive number. – Simon S Dec 22 '14 at 15:14