Complex Power

Well, here's something I asked one of my students, Cassandra, and let's see how many people actually manage to figure it out by, say, the end of the week:

Have a go at it!

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Some of you might be thinking this problem might be adequately solved if one considers Euler’s exponential form of complex numbers:

But here’s another problem; we know that:

In fact, the number i is obtained each time we make a full revolution in the imaginary phase space, such that:

Does that mean now that:

Go figure!

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Well, the answer actually is very simple: you just have to define the allowed values of the argument that define the imaginary number i, such that it can be placed to the power of something. :)

Oh yeah, this answer was contributed by Kenneth Tay Jingyi!