i in power i

How to calculate \(i\) in power \(i\), or \(i^i\)?

To calculate the expression \(i^i\) it is necessary to use the Euler or exponential form of \(i\). We can express this in polar form as \((1e^{i(\frac{\pi}{2} + 2\pi n)})^i\).

So, let's do it step by step:

\[i = 1e^{i(\frac{\pi}{2} + 2\pi n)}\]

Then we can apply power of \(i\):

\[i^i = (1e^{i(\frac{\pi}{2} + 2\pi n)})^i = i^ie^{i^2(\frac{\pi}{2} + 2\pi n)} = \\= e^{2\pi m}e^{(-\frac{\pi}{2} - 2\pi n)} = e^{(-\frac{\pi}{2} + 2\pi k)}; k = 0, \pm1, \pm2, ...\]

Therefore, the final answer is:

\[i^i = e^{(-\frac{\pi}{2} + 2\pi k)}; k = 0, \pm1, \pm2, ...\]

How to check that this is correct, please watch our video with explanation:

Published: 2023-05-09 01:29:19
Updated: 2023-05-09 01:30:26