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Q: How are imaginary exponents defined?

By Posted On/November 26, 2011

Posted in Best

The original question was: How do you do xi (x to the i power), and how on Earth was it developed? There isn’t really anything to base xi on from previous rules of exponents as it is a completely new idea.

Physicist: Euler, the dude who originally came up with using the imaginary unit, , as a placeholder for (which, despite having no actual solution, is still something we can talk about), also came up with “Euler’s equation“: , where “e” is equal to 2.718281828…

Euler’s (surprising) equation provides a way to talk about complex exponents (“complex” = “involves “). So, Euler not only provided an idea that’s confusing as hell ( ), but also a way to deal with it efficiently.

So, for example (using Euler’s equation and some log properties):

So why is this formalism used, instead of some other set of rules? Like everything else in mathematics, it’s a matter of convenience and self consistency. You’d hope that the usual, old rules would apply in a natural, convenient way. For example, you’d want . And that’s exactly what happens:

(Here I’ve used: Cos(-x) = Cos(x), Sin(-x) = -Sin(x), i2=-1, and the Pythagorean identity.)

Even more important, this technique is used because it recovers the usual rules for real exponents (exponents that don’t involve ), or at the very least doesn’t mess them up. It keeps arithmetic nice and self consistent. After all, when you’re coming up with new math, you want to make sure that you don’t trash what you’ve already got. Euler’s equation is an “analytic continuation” of the exponential function ( ) from the real numbers, to the complex ones. An analytic continuation takes a function defined on a fairly small set, like all the real numbers, and generalizes it to work on a larger set, like all complex numbers (which includes real numbers like “3”, but also includes numbers like “i” and “4-2i”). It’s not obvious, but it turns out that Euler’s equation is the only “nice” way to define complex exponents.

You’ll find (at least, those people who are so inclined will find) that Euler’s equation, and in particular the method for finding imaginary exponents above, is consistent with all the rules of exponentiation. Specifically, , , , and .