How do I find value of $i^{1/i}$ and $i^{\sqrt{i}}$ ?
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1First you should expand (simplify) your exponents, then define what you mean by complex number raised to the complex power: it's not a single-valued function in general. – Ruslan Apr 09 '17 at 10:26
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1Hint: Euler's Identity: $$e^{i\theta}=\cos{\theta}+i\sin{\theta}$$ Try to make the RHS equal to $i$ by changing $\theta$. – projectilemotion Apr 09 '17 at 10:30
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Just as an alternative: $i^{\frac{1}{i}}=i^{-1}=\frac{1}{i^i}=\frac{1}{(e^{i\pi /2})^i}=\frac{1}{e^{i^2\pi /2}}=\frac{1}{e^{-\pi /2}}=\frac{1}{\frac{1}{e^{\pi /2}}}=e^{\pi /2}=\sqrt{e^\pi}$ – Thorgott Apr 09 '17 at 10:49
2 Answers
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Replace $i=e^{i\pi/2}$.
The first one becomes $$\left(e^{i\pi/2}\right)^{1/i}=e^{\pi/2}$$
The second one: $$\begin{align} {e^{i\pi/2}}^{(e^{i\pi/4})}&={e^{i\pi/2}}^{(\cos(\pi/4)+i\sin(\pi/4))}\\ &=e^{i\pi/2(\cos(\pi/4)+i\sin(\pi/4))}\\ &=e^{i(\pi/2)\cos(\pi/4)-(\pi/2)\sin(\pi/4)}\\ &=\frac{e^{i(\pi/2)\cos(\pi/4)}}{e^{(\pi/2)\sin(\pi/4)}}\\ &=\frac{\cos((\pi/2)\cos(\pi/4))+i\sin((\pi/2)\cos(\pi/4))}{e^{(\pi/2)\sin(\pi/4)}} \end{align}$$
msm
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@Azra See my hint, that answers your question about why $i=e^{i\pi/2}$. – projectilemotion Apr 09 '17 at 10:34
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1$sin(\pi/4)=cos(\pi/4)=\frac{1}{\sqrt{2}}$ if further simplification is needed. – Thorgott Apr 09 '17 at 10:43
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I suggest that the second one is
$$i^{\sqrt{i}}=i^{(1+i) \sqrt{2}/2}=e^{i\pi/2 \cdot (1+i) \sqrt{2}/2}=e^{\sqrt{2}/4 \cdot \pi(-1+i) }$$
I have checked numerically and find this is equivalent to $i^{\sqrt{i}}$ as well as the result of @msm.
Cye Waldman
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