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# Confusing 'i'

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Why is i to the power of i (i^i) equal to .207879...

Guest Mar 23, 2017
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$$\text{Let }x=i^i\\ \ln x = i \ln i$$

Now we need to find the value of ln(i).

$$e^{i\pi} = -1\\ \ln(-1)=i\pi\\ \ln i = \dfrac{1}{2}\ln(-1)= \dfrac{i\pi}{2}$$

Substitute the result into the equation:

$$\ln x = i\left(\dfrac{i\pi}{2}\right)\\ \quad \;\;=-\dfrac{\pi}{2}\\ e^{\ln x}=e^{-\pi /2}\\ x = e^{-\pi/2}\\ \therefore i^i = e^{-\pi /2}$$

e^(-pi /2) approximately equals 0.207879.

MaxWong  Mar 23, 2017