My Coding > Mathematics > Complex numbers > Find (1 + i)^(1 + i)

Find (1 + i)^(1 + i)

Let's simplify the power expression of complex numbers (\(1 + i)^{1 + i}\) by using our previously derived equation. This is actually very easy one, becasue \(a = b = c = d = 1\):

\[ a = b = c = d = 1 \\ r = \sqrt{1^2 + 1^2} = \sqrt{2} \\ \Theta = Arg(1, 1) = \frac{\pi}{4} + 2\pi n (a + bi)^{c + id} = r^c\times e^{-d\Theta}\times [\cos(c\Theta + d\ln(r)) + i\sin(c\Theta + d\ln(r))] \\ (1 + i)^{1 + i} = \sqrt{2}\times e^{-(\frac{\pi}{4} + 2\pi n)}\times [\cos(\frac{\pi}{4} + 2\pi n + ln(\sqrt{2})) + i\sin(\frac{\pi}{4} + 2\pi n + ln(\sqrt{2}))]\]

This was a full answer and we can also write principal value:

\[(1 + i)^{1 + i} = \sqrt{2}\times e^{-\frac{\pi}{4}}\times [\cos(\frac{\pi}{4} + ln(\sqrt{2})) + i\sin(\frac{\pi}{4} + ln(\sqrt{2}))]\]

For more detailed explanation, please go to watch our video:


Published: 2023-05-10 00:52:44
Updated: 2023-05-10 01:36:34

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