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Between imaginary index and imaginary power

With i = \(\sqrt{-1}\),

Let say i = \(e^{\frac{\pi}{2} i}\).

[\(\displaystyle e^{\frac{\pi}{2} i}\)]\(^i\) = [\(\displaystyle e^{\frac{\pi}{2} i}\)]\(^\frac{i^2}{i}\), {Index verses Power}

The whole thing of \(i i\) or \(i \frac{i^2}{i}\) concerning only index and power makes -1 as a real number.

However, | z | = | i | = 1 of Ln | z | = 0 can be said as a case at critical intermediate.

Where [\(\displaystyle e^{\frac{\pi}{2} i}\)]\(^\frac{1}{i}\) = [\(\displaystyle e^{\frac{\pi}{2} i}\)]\(^\frac{i}{i^2}\) can have a genuine cancellation between numerator and denominator,

\(i i\) or \(i \frac{i^2}{i}\) is considered as multiplication between i and i to mix up into real part of the index for a broken value.

To take golden wing of 1 to hold true at critical point, [\(\displaystyle e^{\frac{\pi}{2} i}\)]\(^\frac{1}{i}\) = \(e^ \frac{\pi}{2}\) = 4.8104773809653516554730356667038+ ought to be taken as true rather than a broken range.

Slightly illogical at critical point! This explains why there seem to be contradiction to occur for \(1^1 = 2^1 = 3^1 = \ldots\) with real power of 1 via complex realm. However, golden wing simplified into a real number of 1 while the formation is form of an \(\frac{i}{i}\) rather than an \(i i \).

The simple thing that can be stated is imaginary index and complex indices are valid while imaginary power and complex powers to any given numbers are mostly invalid, unless they are real resultant wings going through \(\frac{i}{i}\) rather than \(i i \).

It is true that \(\frac{a + b \sqrt{-1}}{c + d \sqrt{-1}}\) can always become simply \(e + f \sqrt{-1}\), however, the main thing for a valid case is for no mixing up between Ln | z | part and \(\sqrt{-1}\) Arg z part, when a power is taken by any number consists of complex indices in general.

\(1^1 = 2^1 = 3^1 = \ldots\) is something optional perhaps which is up to you onto how you would think about them. Nevertheless, there is nothing risky which goes against arithmetic of General Number.

We extend critical cases into general cases: Complex indices are valid and complex powers are mostly invalid (or invalid if you like). I personally take both General Number and Golden wing of 1 to be true. This is most beneficial to me.

As a conclusion, we don't find trouble for a given power to general number. Power of complex numbers and above are just invalid#.

Note by Lu Chee Ket
2 years ago

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